Part I: Fixed Geometry
Research preprint, version 0.3.3
Disclosure: Primarily AI-generated text. The author reports no understanding of the proofs; see the disclosure at the end of this manuscript.
Introduction and precise scope
Regularized immersed boundary (IB) methods transfer information between a background fluid discretization and an independently represented interface. Adaptive mesh refinement introduces a second transfer: the fluid state must be represented in a different discrete space after refinement or coarsening. Stability of either transfer in isolation does not settle the accuracy of their composition. In particular, a small bulk L 2 L^2 L 2
The relevant ingredients have substantial prior theory. Peskin’s formulation supplies the regularized coupling structure [1 ] . Rigorous IB convergence theory includes Mori’s velocity estimates for periodic Stokes flow and a small-amplitude dynamic problem [2 ] , the analysis of discrete-delta properties and local accuracy by Liu and Mori [3 ] , and their stationary Stokes velocity and pressure estimates in L p L^p L p [4 ] . More recently, Li, Pan, and Ruiz-Álvarez analyzed stationary Stokes discretizations with prescribed outer boundary velocity [5 ] . In that setting the singular interface force is given; here the interface reaction is determined through regularized velocity constraints. These works establish essential IB accuracy results, but do not supply the changing-space argument used below.
Prescribed rigid motion in a background-domain multiplier formulation also has a long history [6 ] . Constraint formulations for rigid bodies and their mobility operators were studied by Kallemov et al. [7 ] ; their reported implementation used uniform grids. Dynamic-mesh Stokes error analysis, including errors introduced by Helmholtz and Stokes projections, predates the present work [8 ] . General stable transfer operators for dynamic AMR were analyzed by Lundquist et al. [9 ] , and projection jumps also appear in analyses of remeshed moving-domain problems [10 ] . Neither projection contractivity nor the observation that remeshing errors can accumulate is a new claim here.
The neighboring adaptive fictitious-domain literature is equally important. Berrone et al. [11 ] proved optimal adaptive convergence for an elliptic fictitious-domain formulation with an unknown boundary multiplier. Heltai and Lei developed regularization error estimates [12 ] and an adaptive analysis for regularized forcing data [13 ] . Their setting emphasizes that the regularization width and the bulk mesh size must be tracked separately. On the algorithmic side, adaptive IB methods go back at least to Roma, Peskin, and Berger [14 ] . Zeng, Bhalla, and Shen developed a DLM–IB framework on dynamically adaptive collocated grids, including subcycling and non-subcycling time advancement [15 ] ; IB, multiple resolution levels, and lattice Green functions have also been combined [16 ] . The present analysis is for a different conforming realization and is not a convergence proof for those implementations. It does not claim that IB with AMR is new.
Computational IB studies also motivate distinguishing boundary enforcement from bulk flow accuracy. He, Yang, Sotiropoulos, and Shen [17 ] introduced an explicit immersed pressure boundary condition to mitigate spurious penetration through thin flexible structures in two-phase flow. For fixed geometries, Yang et al. [18 ] coupled large-eddy simulation with IB for a backward-facing step, while Chen et al. [19 ] used a level-set-based IB method to study airfoil flow transition and vortex evolution. He, Liu, and Shen [20 ] resolved flexible-stem canopy flows with IB, and Zheng et al. [21 ] studied a prescribed flapping plate near a free surface using the two-phase IB framework. These studies provide methodological and application context for accurate interface treatment; their turbulent, multiphase, or deforming-body configurations are not covered by the present Stokes analysis. In particular, their computational results are not used as analytical premises for our convergence proof.
Our result concerns a deliberately specified conforming method. The main construction is a family of coarse, divergence-free comparison functions that vanishes on a collar containing every regularization support used during the computation. These functions satisfy all discrete boundary moments exactly, even when the regularization width and surface partition change. Orthogonal transfer preserves their inner products. We use this fact to obtain estimates with no factor counting refinement or coarsening events. The approximation is constructed, rather than postulated through a mesh-uniform constraint inf – sup \inf\text{--}\sup inf – sup
There are three restrictions with practical consequences. First, the Cartesian velocity space is generated by C 1 C^1 C 1
The prescribed-rotation extension is a fixed-geometry result, not a general moving-body theorem. We construct a common discrete lifting from a cutoff quadratic streamfunction, so that the nonzero regularized moments are feasible for every admissible mesh configuration. The transformed forcing retains both the angular-acceleration term and the endpoint viscous lifting term. The resulting convergence proof uses the homogeneous theory with a vanishing forcing perturbation. The background-domain extension includes prescribed interior forcing to represent rigid rotation; no conclusion about a physical torque is inferred from an uncorrected interface multiplier.
The analysis does not prescribe an error-based marking rule. It permits arbitrary choices of the admissible local enrichments, and therefore concerns stability and convergence under dynamic adaptation, not adaptive optimality. The spatial rate is expressed in the size of the retained coarse grid. Refining a small patch while holding the coarse grid fixed is not a convergence sequence covered by our limit theorem.
Continuous problem and regularized constraints
Let Ω = ( 0 , 1 ) 2 \Omega=(0,1)^2 Ω = ( 0 , 1 ) 2 B ⋐ Ω B\Subset\Omega B ⋐ Ω C 5 C^5 C 5 Γ \Gamma Γ F = Ω ∖ B ‾ F=\Omega\setminus\overline{B} F = Ω ∖ B
U = { v ∈ H 0 1 ( Ω ) 2 : div v = 0 } , ∥ v ∥ U = ∥ ∇ v ∥ L 2 ( Ω ) , (1) U=\{v\in H_0^1(\Omega)^2:\operatorname{div}v=0\},\qquad\|v\|_U=\|\nabla v\|_{L^2(\Omega)}, \tag*{(1)} U = { v ∈ H 0 1 ( Ω ) 2 : div v = 0 } , ∥ v ∥ U = ∥∇ v ∥ L 2 ( Ω ) , (1) V = { v ∈ U : γ Γ v = 0 } , H = V L 2 ( Ω ) 2 , (2) V=\{v\in U:\gamma_\Gamma v=0\},\qquad H=V^{L^2(\Omega)^2}, \tag*{(2)} V = { v ∈ U : γ Γ v = 0 } , H = V L 2 ( Ω ) 2 , (2) a ( v , w ) = ν ( ∇ v , ∇ w ) L 2 ( Ω ) , ν > 0. (3) a(v,w)=\nu(\nabla v,\nabla w)_{L^2(\Omega)},\qquad\nu>0. \tag*{(3)} a ( v , w ) = ν ( ∇ v , ∇ w ) L 2 ( Ω ) , ν > 0. (3) We write ∥ ⋅ ∥ 0 \|\cdot\|_0 ∥ ⋅ ∥ 0 L 2 L^2 L 2 ∥ ⋅ ∥ U \|\cdot\|_U ∥ ⋅ ∥ U H 1 H^1 H 1 U U U
Given u 0 ∈ H u_0\in H u 0 ∈ H f ∈ L 2 ( 0 , T ; U ′ ) f\in L^2(0,T;U') f ∈ L 2 ( 0 , T ; U ′ )
u ∈ L 2 ( 0 , T ; V ) ∩ C ( [ 0 , T ] ; H ) , ⟨ ∂ t u , v ⟩ + a ( u , v ) = ⟨ f , v ⟩ ( v ∈ V ) , u ( 0 ) = u 0 . (4) u\in L^2(0,T;V)\cap C([0,T];H),\qquad\langle\partial_tu,v\rangle+a(u,v)=\langle f,v\rangle\quad(v\in V),\qquad u(0)=u_0. \tag*{(4)} u ∈ L 2 ( 0 , T ; V ) ∩ C ([ 0 , T ] ; H ) , ⟨ ∂ t u , v ⟩ + a ( u , v ) = ⟨ f , v ⟩ ( v ∈ V ) , u ( 0 ) = u 0 . (4) The derivative belongs to L 2 ( 0 , T ; V ′ ) L^2(0,T;V') L 2 ( 0 , T ; V ′ ) V ⊂ H ⊂ V ′ V\subset H\subset V' V ⊂ H ⊂ V ′ [22 ] , Theorem 10.9. The Hilbert-triple chain rule for variational evolution equations [23 Chapter 3] then gives the energy identity
1 2 ∥ u ( t ) ∥ 0 2 + ∫ 0 t a ( u , u ) d s = 1 2 ∥ u 0 ∥ 0 2 + ∫ 0 t ⟨ f , u ⟩ d s . (5) \frac{1}{2}\|u(t)\|_0^2+\int_0^t a(u,u)\,\mathrm{d}s=\frac{1}{2}\|u_0\|_0^2+\int_0^t\langle f,u\rangle\,\mathrm{d}s. \tag*{(5)} 2 1 ∥ u ( t ) ∥ 0 2 + ∫ 0 t a ( u , u ) d s = 2 1 ∥ u 0 ∥ 0 2 + ∫ 0 t ⟨ f , u ⟩ d s . (5) This is a background-domain description. To represent a rigid solid rather than a second interior fluid, take u 0 = 0 u_0=0 u 0 = 0 B B B f f f B B B F F F B B B U U U
Fix a nonnegative, even ρ ∈ C ∞ ( R 2 ) \rho\in C^\infty(\mathbb{R}^2) ρ ∈ C ∞ ( R 2 ) ∫ ρ = 1 \int\rho= 1 ∫ ρ = 1 ρ ε ( x ) = ε − 2 ρ ( x / ε ) \rho_\varepsilon(x) = \varepsilon^{-2}\rho(x/\varepsilon) ρ ε ( x ) = ε − 2 ρ ( x / ε ) E v Ev E v v ∈ H 0 1 ( Ω ) 2 v \in H^1_0(\Omega)^2 v ∈ H 0 1 ( Ω ) 2
J ε v = γ Γ ( ρ ε ∗ E v ) , b ε ( v , μ ) = ∫ Γ μ ⋅ J ε v d s . (6) J_\varepsilon v = \gamma_\Gamma(\rho_\varepsilon* Ev), \qquad b_\varepsilon(v,\mu) = \int_\Gamma\mu\cdot J_\varepsilon v\,ds. \tag*{(6)} J ε v = γ Γ ( ρ ε ∗ E v ) , b ε ( v , μ ) = ∫ Γ μ ⋅ J ε v d s . (6) The force operator is the adjoint of this variational interpolation, retaining the power-compatible coupling of the IB formulation [1 ] . Kernel moment conditions and their consequences for discrete IB accuracy are studied in Liu and Mori [3 ] ; our operator is instead an exact continuum convolution. No sampling or quadrature operator is included in (6) .
Let P \mathcal{P} P Γ \Gamma Γ ℓ \ell ℓ Λ P \Lambda_{\mathcal{P}} Λ P Π P \Pi_{\mathcal{P}} Π P L 2 ( Γ ) 2 L^2(\Gamma)^2 L 2 ( Γ ) 2
b ε ( v h , μ ) = 0 for all μ ∈ Λ P , equivalently Π P J ε v h = 0. (7) b_\varepsilon(v_h,\mu)=0 \quad\text{for all } \mu\in\Lambda_{\mathcal{P}}, \qquad\text{equivalently } \Pi_{\mathcal{P}}J_\varepsilon v_h=0. \tag*{(7)} b ε ( v h , μ ) = 0 for all μ ∈ Λ P , equivalently Π P J ε v h = 0. (7) The surface panels need not coincide with bulk cell intersections. Throughout a run the widths ε \varepsilon ε ε ˉ \bar{\varepsilon} ε ˉ ℓ ˉ \bar{\ell} ℓ ˉ
A physical interface-defect estimate
Lemma 3.1 (Uniform regularized trace and moment defect). For sufficiently small ε > 0 \varepsilon> 0 ε > 0 and v ∈ U v \in U v ∈ U
∥ J ε v ∥ H 1 / 2 ( Γ ) ≤ C ∥ v ∥ U , (8) \|J_\varepsilon v\|_{H^{1/2}(\Gamma)} \le C\|v\|_U, \tag*{(8)} ∥ J ε v ∥ H 1/2 ( Γ ) ≤ C ∥ v ∥ U , (8) ∥ γ Γ v − J ε v ∥ L 2 ( Γ ) ≤ C ε ∥ v ∥ U . (9) \|\gamma_\Gamma v-J_\varepsilon v\|_{L^2(\Gamma)} \le C\sqrt{\varepsilon}\,\|v\|_U. \tag*{(9)} ∥ γ Γ v − J ε v ∥ L 2 ( Γ ) ≤ C ε ∥ v ∥ U . (9) If (7) holds, then
∥ γ Γ v ∥ L 2 ( Γ ) ≤ C ( ε + ℓ ) ∥ v ∥ U . (10) \|\gamma_\Gamma v\|_{L^2(\Gamma)} \le C\left(\sqrt{\varepsilon}+\sqrt{\ell}\right)\|v\|_U. \tag*{(10)} ∥ γ Γ v ∥ L 2 ( Γ ) ≤ C ( ε + ℓ ) ∥ v ∥ U . (10) The constants depend on the fixed geometry and kernel, not on the bulk mesh, panel positions, or the number of panels.
Proof. Zero extension is bounded from H 0 1 ( Ω ) H^1_0(\Omega) H 0 1 ( Ω ) H 1 ( R 2 ) H^1(\mathbb{R}^2) H 1 ( R 2 ) [24 Theorems 3.18–3.19] . Young’s convolution inequality, also used in Heltai and Lei [12 Lemma 3] , gives uniform H 1 H^1 H 1 [24 Theorem 3.10(iii)] , applied on a fixed neighborhood of Γ \Gamma Γ (8) . For w = E v − ρ ε ∗ E v w=Ev-\rho_\varepsilon* Ev w = E v − ρ ε ∗ E v [12 Lemma 4, equation (8), with $s=1] and convolution boundedness give
∥ w ∥ L 2 ( R 2 ) ≤ C ε ∥ v ∥ U , ∥ w ∥ H 1 ( R 2 ) ≤ C ∥ v ∥ U . \|w\|_{L^2(\mathbb{R}^2)} \le C\varepsilon\|v\|_U, \qquad\|w\|_{H^1(\mathbb{R}^2)} \le C\|v\|_U. ∥ w ∥ L 2 ( R 2 ) ≤ C ε ∥ v ∥ U , ∥ w ∥ H 1 ( R 2 ) ≤ C ∥ v ∥ U . The multiplicative trace inequality yields the following bound on a fixed neighborhood of Γ \Gamma Γ [24 Lemma 12.15] , with p = 2 p=2 p = 2 H 1 H^1 H 1
∥ γ Γ w ∥ L 2 ( Γ ) 2 ≤ C ∥ w ∥ L 2 ( R 2 ) ∥ w ∥ H 1 ( R 2 ) ≤ C ε ∥ v ∥ U 2 . \|\gamma_\Gamma w\|_{L^2(\Gamma)}^2 \le C\|w\|_{L^2(\mathbb{R}^2)}\|w\|_{H^1(\mathbb{R}^2)} \le C\varepsilon\|v\|_U^2. ∥ γ Γ w ∥ L 2 ( Γ ) 2 ≤ C ∥ w ∥ L 2 ( R 2 ) ∥ w ∥ H 1 ( R 2 ) ≤ C ε ∥ v ∥ U 2 . This proves (9) .
The panel estimate is the fractional Poincare inequality [24 Lemma 3.26] . For completeness, its direct proof here is short: on an arclength interval I I I ℓ I \ell_I ℓ I z I z_I z I
∫ I ∣ z − z I ∣ 2 d s = 1 2 ℓ I ∫ I ∫ I ∣ z ( s ) − z ( t ) ∣ 2 d s d t ≤ ℓ I 2 ∫ I ∫ I ∣ z ( s ) − z ( t ) ∣ 2 ∣ s − t ∣ 2 d s d t . \int_I |z-z_I|^2\,ds=\frac{1}{2\ell_I}\int_I\int_I |z(s)-z(t)|^2\,ds\,dt \le\frac{\ell_I}{2}\int_I\int_I \frac{|z(s)-z(t)|^2}{|s-t|^2}\,ds\,dt. ∫ I ∣ z − z I ∣ 2 d s = 2 ℓ I 1 ∫ I ∫ I ∣ z ( s ) − z ( t ) ∣ 2 d s d t ≤ 2 ℓ I ∫ I ∫ I ∣ s − t ∣ 2 ∣ z ( s ) − z ( t ) ∣ 2 d s d t . Summing over panels, with the equivalent arclength definition of the surface fractional norm, proves
∥ ( I − Π P ) z ∥ L 2 ( Γ ) ≤ C ℓ ∥ z ∥ H 1 / 2 ( Γ ) . \lVert(I-\Pi_{\mathcal{P}})z\rVert_{L^2(\Gamma)} \le C\sqrt{\ell}\,\lVert z\rVert_{H^{1/2}(\Gamma)}. ∥( I − Π P ) z ∥ L 2 ( Γ ) ≤ C ℓ ∥ z ∥ H 1/2 ( Γ ) . Under (7) , J ε v = ( I − Π P ) J ε v J_{\varepsilon}v=(I-\Pi_{\mathcal{P}})J_{\varepsilon}v J ε v = ( I − Π P ) J ε v (8) , (9) , and the triangle inequality.
Estimate (10) concerns the physical trace of the reconstructed conforming velocity, not a discrete residual that vanishes by definition. No multiplier inf–sup estimate is used. The estimate is useful in a time-integrated norm: viscous dissipation bounds ∫ ∥ u h ∥ U 2 \int\lVert u_h\rVert_{\mathcal{U}}^2 ∫ ∥ u h ∥ U 2
The distinction between a satisfied discrete boundary equation and physical impermeability is also relevant to the spurious-penetration issue addressed computationally by He et al. [17 ] . Their remedy acts through an immersed pressure boundary condition in a two-phase finite-difference solver. Here (10) controls the full velocity trace by regularized moments and viscous energy; it is a different mechanism and does not analyze that pressure-boundary algorithm.
Cartesian spaces and a common constrained approximation family
A concrete two-level conforming construction
Let h = 1 / M h=1/M h = 1/ M M M M T h \mathcal{T}_h T h Ω \Omega Ω h h h Q 3 Q_3 Q 3 S h ⊂ H 0 2 ( Ω ) S_h\subset H^2_0(\Omega) S h ⊂ H 0 2 ( Ω ) ∂ x y \partial_{xy} ∂ x y C 1 C^1 C 1 ∂ Ω \partial\Omega ∂ Ω [25 ] for its tensor-product description and interpolation estimates.
For a coarse cell K ∈ T h K\in\mathcal{T}_h K ∈ T h K K K h / 2 h/2 h /2 B h ( K ) B_h(K) B h ( K ) C 1 C^1 C 1 ∂ K \partial K ∂ K H 0 2 ( Ω ) H^2_0(\Omega) H 0 2 ( Ω ) dim B h ( K ) = 4 \dim B_h(K)=4 dim B h ( K ) = 4 A ⊂ T h A\subset\mathcal{T}_h A ⊂ T h
S h , A = S h + ∑ K ∈ A B h ( K ) , X h , A = curl S h , A , curl ψ = ( ∂ y ψ , − ∂ x ψ ) . (11) S_{h,A}=S_h+\sum_{K\in A}B_h(K),\qquad X_{h,A}=\operatorname{curl}S_{h,A},\qquad\operatorname{curl}\psi=(\partial_y\psi,-\partial_x\psi). \tag*{(11)} S h , A = S h + K ∈ A ∑ B h ( K ) , X h , A = curl S h , A , curl ψ = ( ∂ y ψ , − ∂ x ψ ) . (11) Every X h , A X_{h,A} X h , A U U U div curl ψ = 0 \operatorname{div}\operatorname{curl}\psi=0 div curl ψ = 0 ψ ∈ H 2 \psi\in H^2 ψ ∈ H 2
This construction retains the full coarse space and adds local fine-grid degrees of freedom. All the spaces are contained in the uniform h / 2 h/2 h /2 A A A A A A
For a regularization width ε \varepsilon ε Λ P \Lambda_{\mathcal{P}} Λ P
Z h , A , ε , P = { v h ∈ X h , A : b ε ( v h , μ ) = 0 for every μ ∈ Λ P } . (12) Z_{h,A,\varepsilon,\mathcal{P}}=\{v_h\in X_{h,A}: b_{\varepsilon}(v_h,\mu)=0\ \text{for every }\mu\in\Lambda_{\mathcal{P}}\}. \tag*{(12)} Z h , A , ε , P = { v h ∈ X h , A : b ε ( v h , μ ) = 0 for every μ ∈ Λ P } . (12) These are exact surface-moment constraints. Linear independence of their rows is neither assumed nor needed for the velocity approximation arguments below.
A family present in every configuration
Let ε ˉ \bar{\varepsilon} ε ˉ
D h , ε ˉ = ⋃ { K ∈ T h : K ∩ { x : dist ( x , Γ ) ≤ ε ˉ } ≠ ∅ } , K h , ε ˉ = { v h ∈ curl S h : v h = 0 on D h , ε ˉ } . (13) D_{h,\bar{\varepsilon}}=\bigcup\left\{K\in\mathcal{T}_h:K\cap\{x:\operatorname{dist}(x,\Gamma)\le\bar{\varepsilon}\}\ne\varnothing\right\},\qquad
K_{h,\bar{\varepsilon}}=\{v_h\in\operatorname{curl}S_h:v_h=0\text{ on }D_{h,\bar{\varepsilon}}\}. \tag*{(13)} D h , ε ˉ = ⋃ { K ∈ T h : K ∩ { x : dist ( x , Γ ) ≤ ε ˉ } = ∅ } , K h , ε ˉ = { v h ∈ curl S h : v h = 0 on D h , ε ˉ } . (13) The equalities defining K h , ε ˉ K_{h,\bar{\varepsilon}} K h , ε ˉ velocity ; they do not impose zero streamfunction on the immersed collar. The latter would unnecessarily exclude circulation modes.
If 0 < ε ≤ ε ˉ 0<\varepsilon\le\bar{\varepsilon} 0 < ε ≤ ε ˉ Γ \Gamma Γ D h , ε ˉ D_{h,\bar{\varepsilon}} D h , ε ˉ
K h , ε ˉ ⊂ V , K h , ε ˉ ⊂ Z h , A , ε , P for every A , P , 0 < ε ≤ ε ˉ . (14) K_{h,\bar{\varepsilon}}\subset V,\qquad K_{h,\bar{\varepsilon}}\subset Z_{h,\mathcal{A},\varepsilon,\mathcal{P}}\quad\text{for every }\mathcal{A},\mathcal{P},\quad0<\varepsilon\le\bar{\varepsilon}. \tag*{(14)} K h , ε ˉ ⊂ V , K h , ε ˉ ⊂ Z h , A , ε , P for every A , P , 0 < ε ≤ ε ˉ . (14) In particular, an orthogonal transfer onto any of these constraint spaces fixes every element of K h , ε ˉ K_{h,\bar{\varepsilon}} K h , ε ˉ
Lemma 4.1 (Common recovery family). Let h m → 0 h_m\to0 h m → 0 and ε ˉ m → 0 \bar{\varepsilon}_m\to0 ε ˉ m → 0 For each w ∈ V w\in V w ∈ V there exist w m ∈ K h m , ε ˉ m w_m\in K_{h_m,\bar{\varepsilon}_m} w m ∈ K h m , ε ˉ m such that
∥ w m − w ∥ U ⟶ 0. (15) \lVert w_m-w\rVert_U\longrightarrow0. \tag*{(15)} ∥ w m − w ∥ U ⟶ 0. (15) The sequence w m w_m w m is independent of the refinement sets, the panel partitions, and the individual regularization widths used during run m m m provided those widths do not exceed ε ˉ m \bar{\varepsilon}_m ε ˉ m No relation between h m h_m h m and ε ˉ m \bar{\varepsilon}_m ε ˉ m is required.
Proof. The restrictions of w w w B B B F F F H 1 H^1 H 1 B B B F F F H 1 H^1 H 1 [26 Section III.4] . Their zero extensions and sums form a sequence w ( j ) ∈ C ∞ ( Ω ) 2 w^{(j)}\in C^\infty(\Omega)^2 w ( j ) ∈ C ∞ ( Ω ) 2 w w w U U U j j j w ( j ) w^{(j)} w ( j ) Γ \Gamma Γ
For a fixed such field w ( j ) w^{(j)} w ( j )
ψ ( j ) ( x , y ) = ∫ 0 y w 1 ( j ) ( x , s ) d s . \psi^{(j)}(x,y)=\int_0^y w_1^{(j)}(x,s)\,\mathrm{d}s. ψ ( j ) ( x , y ) = ∫ 0 y w 1 ( j ) ( x , s ) d s . Since div w ( j ) = 0 \operatorname{div}w^{(j)}=0 div w ( j ) = 0 w 1 ( j ) ( x , 0 ) = 0 w_1^{(j)}(x,0)=0 w 1 ( j ) ( x , 0 ) = 0 ∂ y ψ ( j ) = w 1 ( j ) \partial_y\psi^{(j)}=w_1^{(j)} ∂ y ψ ( j ) = w 1 ( j ) ∂ x ψ ( j ) = − w 2 ( j ) \partial_x\psi^{(j)}=-w_2^{(j)} ∂ x ψ ( j ) = − w 2 ( j ) ψ ( j ) = 0 \psi^{(j)}=0 ψ ( j ) = 0 ∇ ψ ( j ) = 0 \nabla\psi^{(j)}=0 ∇ ψ ( j ) = 0 Γ \Gamma Γ ψ ( j ) \psi^{(j)} ψ ( j ) ψ ( j ) \psi^{(j)} ψ ( j )
Let I h ψ ( j ) ∈ S h I_h\psi^{(j)}\in S_h I h ψ ( j ) ∈ S h ], with m = 2 m=2 m = 2
∥ I h ψ ( j ) − ψ ( j ) ∥ H 2 ( Ω ) ≤ C h 2 ∥ ψ ( j ) ∥ H 4 ( Ω ) . (16) \left\lVert I_h\psi^{(j)}-\psi^{(j)}\right\rVert_{H^2(\Omega)}
\le Ch^2\left\lVert\psi^{(j)}\right\rVert_{H^4(\Omega)}. \tag*{(16)} I h ψ ( j ) − ψ ( j ) H 2 ( Ω ) ≤ C h 2 ψ ( j ) H 4 ( Ω ) . (16) If w ( j ) w^{(j)} w ( j ) dist ( x , Γ ) < r j \operatorname{dist}(x,\Gamma)<r_j dist ( x , Γ ) < r j ε ˉ m + 2 h m < r j \bar{\varepsilon}_m+\sqrt{2}h_m<r_j ε ˉ m + 2 h m < r j D h m , ε ˉ m D_{h_m,\bar{\varepsilon}_m} D h m , ε ˉ m curl I h m ψ ( j ) ∈ K h m , ε ˉ m \operatorname{curl} I_{h_m}\psi^{(j)} \in K_{h_m,\bar{\varepsilon}_m} curl I h m ψ ( j ) ∈ K h m , ε ˉ m m m m w ( j ) w^{(j)} w ( j ) U U U
Choose j = j ( m ) j=j(m) j = j ( m ) ∥ w ( j ( m ) ) − w ∥ U → 0 \lVert w^{(j(m))}-w\rVert_U\to0 ∥ w ( j ( m )) − w ∥ U → 0 (15) .
The collar is used to construct comparison functions. The actual discrete solution is constrained only by (12) ; it is not required to vanish on D h , ε ˉ D_{h,\bar{\varepsilon}} D h , ε ˉ
A quantitative collar construction
For a velocity w w w
M ( w ) = ∥ w ∥ W 3 , ∞ ( B ) + ∥ w ∥ W 3 , ∞ ( F ) . \mathcal{M}(w)=\lVert w\rVert_{W^{3,\infty}(B)}+\lVert w\rVert_{W^{3,\infty}(F)}. M ( w ) = ∥ w ∥ W 3 , ∞ ( B ) + ∥ w ∥ W 3 , ∞ ( F ) . All restrictions and traces in the next lemma are understood in the Sobolev sense. A globally H 1 H^1 H 1 Γ \Gamma Γ
Lemma 4.2 (Quantitative common approximation). There are constants c 0 , C > 0 c_0,C>0 c 0 , C > 0 depending only on the fixed geometry, such that the following holds. If h + ε ˉ < c 0 h+\bar{\varepsilon}<c_0 h + ε ˉ < c 0 there is a linear operator Q h , ε ˉ Q_{h,\bar{\varepsilon}} Q h , ε ˉ from V ∩ [ W 3 , ∞ ( B ) 2 × W 3 , ∞ ( F ) 2 ] V\cap[W^{3,\infty}(B)^2\times W^{3,\infty}(F)^2] V ∩ [ W 3 , ∞ ( B ) 2 × W 3 , ∞ ( F ) 2 ] into K h , ε ˉ K_{h,\bar{\varepsilon}} K h , ε ˉ satisfying
∥ Q h , ε ˉ w − w ∥ H 1 ( Ω ) ≤ C h + ε ˉ M ( w ) . (17) \lVert Q_{h,\bar{\varepsilon}}w-w\rVert_{H^1(\Omega)}\le C\sqrt{h+\bar{\varepsilon}}\,\mathcal{M}(w). \tag*{(17)} ∥ Q h , ε ˉ w − w ∥ H 1 ( Ω ) ≤ C h + ε ˉ M ( w ) . (17) For fixed h , ε ˉ h,\bar{\varepsilon} h , ε ˉ this operator commutes with the weak time derivative whenever w w w belongs to the Bochner H 1 H^1 H 1 space in the stated piecewise regularity class, with w ( t ) , ∂ t w ( t ) ∈ V w(t),\partial_t w(t)\in V w ( t ) , ∂ t w ( t ) ∈ V almost everywhere. Hence the same estimate applies almost everywhere to the time derivative of the approximation error, with M ( ∂ t w ) \mathcal{M}(\partial_t w) M ( ∂ t w ) on the right.
Proof. The signed-distance tubular coordinates used below are standard for a smooth fixed interface; see also the geometric setup in Burman et al. [25 ] . The flattening and width-dependent estimates are derived here. As in the preceding proof, set ψ ( x , y ) = ∫ 0 y w 1 ( x , s ) d s \psi(x,y)=\int_0^y w_1(x,s)\,\mathrm{d}s ψ ( x , y ) = ∫ 0 y w 1 ( x , s ) d s curl ψ = w \operatorname{curl}\psi=w curl ψ = w ψ ∈ H 0 2 ( Ω ) \psi\in H^2_0(\Omega) ψ ∈ H 0 2 ( Ω ) ψ \psi ψ W 4 , ∞ W^{4,\infty} W 4 , ∞ B B B F F F W 4 , ∞ W^{4,\infty} W 4 , ∞ C M ( w ) C\mathcal{M}(w) C M ( w ) ∇ ψ = 0 \nabla\psi=0 ∇ ψ = 0 ψ ∣ Γ = c \psi|_\Gamma=c ψ ∣ Γ = c c c c
∣ ψ − c ∣ ≤ C d 2 M ( w ) , ∣ ∇ ψ ∣ ≤ C ∣ d ∣ M ( w ) , (18) |\psi-c|\le Cd^2\mathcal{M}(w),\qquad|\nabla\psi|\le C|d|\mathcal{M}(w), \tag*{(18)} ∣ ψ − c ∣ ≤ C d 2 M ( w ) , ∣∇ ψ ∣ ≤ C ∣ d ∣ M ( w ) , (18) where d d d Γ \Gamma Γ
Choose a fixed smooth cutoff χ \chi χ χ ( s ) = 1 \chi(s)=1 χ ( s ) = 1 ∣ s ∣ ≤ 1 |s|\le1 ∣ s ∣ ≤ 1 χ ( s ) = 0 \chi(s)=0 χ ( s ) = 0 ∣ s ∣ ≥ 2 |s|\ge2 ∣ s ∣ ≥ 2
δ = 4 ( ε ˉ + 2 h ) , ψ δ = ψ − χ ( d / δ ) ( ψ − c ) (19) \delta=4(\bar{\varepsilon}+\sqrt{2}h),\qquad\psi_\delta=\psi-\chi(d/\delta)(\psi-c) \tag*{(19)} δ = 4 ( ε ˉ + 2 h ) , ψ δ = ψ − χ ( d / δ ) ( ψ − c ) (19) inside the tubular neighborhood, and set ψ δ = ψ \psi_\delta=\psi ψ δ = ψ c 0 c_0 c 0 2 δ 2\delta 2 δ ∂ Ω \partial\Omega ∂ Ω c c c ∣ d ∣ ≤ δ |d|\le\delta ∣ d ∣ ≤ δ ψ δ ∈ H 4 ( Ω ) ∩ H 0 2 ( Ω ) \psi_\delta\in H^4(\Omega)\cap H^2_0(\Omega) ψ δ ∈ H 4 ( Ω ) ∩ H 0 2 ( Ω )
The difference ψ δ − ψ \psi_\delta-\psi ψ δ − ψ O ( δ ) O(\delta) O ( δ ) (18) , the zeroth, first, and second derivatives of this difference are bounded pointwise by C δ 2 M ( w ) C\delta^2\mathcal{M}(w) C δ 2 M ( w ) C δ M ( w ) C\delta\mathcal{M}(w) C δ M ( w ) C M ( w ) C\mathcal{M}(w) C M ( w )
∥ ψ δ − ψ ∥ H 2 ( Ω ) ≤ C δ M ( w ) . (20) \lVert\psi_\delta-\psi\rVert_{H^2(\Omega)}\le C\sqrt{\delta}\,\mathcal{M}(w). \tag*{(20)} ∥ ψ δ − ψ ∥ H 2 ( Ω ) ≤ C δ M ( w ) . (20) On the transition region δ < ∣ d ∣ < 2 δ \delta< |d| < 2\delta δ < ∣ d ∣ < 2 δ C δ − 2 M ( w ) C\delta^{-2}\mathcal{M}(w) C δ − 2 M ( w ) ψ − c = O ( δ 2 ) \psi-c=O(\delta^2) ψ − c = O ( δ 2 ) ∇ ψ = O ( δ ) \nabla\psi=O(\delta) ∇ ψ = O ( δ ) ψ \psi ψ δ \delta δ O ( δ ) O(\delta) O ( δ )
∥ ψ δ ∥ H 4 ( Ω ) ≤ C ( 1 + δ − 3 / 2 ) M ( w ) . (21) \|\psi_\delta\|_{H^4(\Omega)} \le C(1+\delta^{-3/2})\mathcal{M}(w). \tag*{(21)} ∥ ψ δ ∥ H 4 ( Ω ) ≤ C ( 1 + δ − 3/2 ) M ( w ) . (21) Define Q h , ε ˉ w = curl I h ψ δ Q_{h,\bar{\varepsilon}}w=\operatorname{curl} I_h\psi_\delta Q h , ε ˉ w = curl I h ψ δ ε ˉ \bar{\varepsilon} ε ˉ ε ˉ + 2 h < δ \bar{\varepsilon}+\sqrt{2}h<\delta ε ˉ + 2 h < δ ψ δ = c \psi_\delta=c ψ δ = c Q h , ε ˉ w ∈ K h , ε ˉ Q_{h,\bar{\varepsilon}}w\in K_{h,\bar{\varepsilon}} Q h , ε ˉ w ∈ K h , ε ˉ (16) , (20) , and (21) gives
∥ Q h , ε ˉ w − w ∥ H 1 ( Ω ) ≤ C ( δ + h 2 ( 1 + δ − 3 / 2 ) ) M ( w ) . \|Q_{h,\bar{\varepsilon}}w-w\|_{H^1(\Omega)}\le C\left(\sqrt{\delta}+h^2(1+\delta^{-3/2})\right)\mathcal{M}(w). ∥ Q h , ε ˉ w − w ∥ H 1 ( Ω ) ≤ C ( δ + h 2 ( 1 + δ − 3/2 ) ) M ( w ) . Because δ ≥ 4 2 h \delta\ge4\sqrt{2}h δ ≥ 4 2 h δ ≤ C ( h + ε ˉ ) \delta\le C(h+\bar{\varepsilon}) δ ≤ C ( h + ε ˉ ) (17) .
Finally, the construction w ↦ ψ w\mapsto\psi w ↦ ψ c c c h , ε ˉ h,\bar{\varepsilon} h , ε ˉ M \mathcal{M} M H 1 ( Ω ) 2 H^1(\Omega)^2 H 1 ( Ω ) 2 (17) and the continuous inclusion of that class in H 1 ( Ω ) 2 H^1(\Omega)^2 H 1 ( Ω ) 2 δ \delta δ
The error in (17) is expressed in the retained coarse mesh size. It proves a robust approximation property, rather than optimality of an adaptive marking strategy. It remains valid when the regularization support intersects coarse–fine boundaries because the comparison function is already zero on the entire support. No uniform estimate for the boundary multiplier is inferred from this construction.
Time stepping and exact energy balance
Choose a time partition 0 = t 0 < ⋯ < t N = T 0=t_0<\cdots<t_N=T 0 = t 0 < ⋯ < t N = T τ n = t n − t n − 1 \tau_n=t_n-t_{n-1} τ n = t n − t n − 1 n n n A n \mathcal{A}^n A n ε n > 0 \varepsilon_n>0 ε n > 0 P n \mathcal{P}^n P n
Z n = { v h ∈ X h , A n : Π P n J ε n v h = 0 } , P n : L 2 ( Ω ) 2 ⟶ Z n (22) Z^n=\{v_h\in X_{h,\mathcal{A}^n}:\Pi_{\mathcal{P}^n}J_{\varepsilon_n}v_h=0\},\qquad P^n:L^2(\Omega)^2\longrightarrow Z^n \tag*{(22)} Z n = { v h ∈ X h , A n : Π P n J ε n v h = 0 } , P n : L 2 ( Ω ) 2 ⟶ Z n (22) be the exact bulk L 2 L^2 L 2 a a a
Set u 0 = P 0 u 0 u^0=P^0u_0 u 0 = P 0 u 0 f n = τ n − 1 ∫ t n − 1 t n f ( t ) d t f^n=\tau_n^{-1}\int_{t_{n-1}}^{t_n}f(t)\,\mathrm{d}t f n = τ n − 1 ∫ t n − 1 t n f ( t ) d t
u n ∈ Z n , ( u n − P n u n − 1 τ n , v h ) + a ( u n , v h ) = ⟨ f n , v h ⟩ ( v h ∈ Z n ) . (23) u^n\in Z^n,\qquad\left(\frac{u^n-P^nu^{n-1}}{\tau_n},v_h\right)+a(u^n,v_h)=\langle f^n,v_h\rangle\qquad(v_h\in Z^n). \tag*{(23)} u n ∈ Z n , ( τ n u n − P n u n − 1 , v h ) + a ( u n , v h ) = ⟨ f n , v h ⟩ ( v h ∈ Z n ) . (23) All bulk inner products without subscripts are in L 2 ( Ω ) 2 L^2(\Omega)^2 L 2 ( Ω ) 2 [8 ] , Algorithm 2.2. Here the projector also enforces the regularized interface moments. Because the mass form plus τ n \tau_n τ n Z n Z^n Z n
Equivalently, one may solve on X h , A n X_{h,\mathcal{A}^n} X h , A n Λ P n \Lambda_{\mathcal{P}^n} Λ P n
Testing by the current velocity and using dual coupling to cancel constraint terms are established energy arguments; see, for example, Boffi et al. [27 Proposition 3] in a distributed-multiplier elastic FSI setting. We use the same elementary energy principle, not their elastic-body theorem. The identity below additionally keeps the loss from transfer between our changing constraint spaces; its contractive-transfer mechanism is consistent with Lundquist et al. [9 Section 2] .
Proposition 5.1 (Energy identity). For every step of (23) ,
∥ u n ∥ 0 2 − ∥ u n − 1 ∥ 0 2 + ∥ u n − u n − 1 ∥ 0 2 + 2 τ n a ( u n , u n ) = 2 τ n ⟨ f n , u n ⟩ . (24) \lVert u^n\rVert_0^2-\lVert u^{n-1}\rVert_0^2+\lVert u^n-u^{n-1}\rVert_0^2+2\tau_n a(u^n,u^n)=2\tau_n\langle f^n,u^n\rangle. \tag*{(24)} ∥ u n ∥ 0 2 − ∥ u n − 1 ∥ 0 2 + ∥ u n − u n − 1 ∥ 0 2 + 2 τ n a ( u n , u n ) = 2 τ n ⟨ f n , u n ⟩ . (24) Moreover,
∥ u n − u n − 1 ∥ 0 2 = ∥ u n − P n u n − 1 ∥ 0 2 + ∥ ( I − P n ) u n − 1 ∥ 0 2 . (25) \lVert u^n-u^{n-1}\rVert_0^2=\lVert u^n-P^n u^{n-1}\rVert_0^2+\lVert(I-P^n)u^{n-1}\rVert_0^2. \tag*{(25)} ∥ u n − u n − 1 ∥ 0 2 = ∥ u n − P n u n − 1 ∥ 0 2 + ∥( I − P n ) u n − 1 ∥ 0 2 . (25) Consequently,
max 0 ≤ n ≤ N ∥ u n ∥ 0 2 + ∑ n = 1 N τ n ∥ u n ∥ U 2 + ∑ n = 1 N ∥ u n − u n − 1 ∥ 0 2 ≤ C ν ( ∥ u 0 ∥ 0 2 + ∥ f ∥ L 2 ( 0 , T ; U ′ ) 2 ) . (26) \max_{0\le n\le N}\lVert u^n\rVert_0^2+\sum_{n=1}^{N}\tau_n\lVert u^n\rVert_U^2+\sum_{n=1}^{N}\lVert u^n-u^{n-1}\rVert_0^2\le C_\nu\left(\lVert u_0\rVert_0^2+\lVert f\rVert_{L^2(0,T;U')}^2\right). \tag*{(26)} 0 ≤ n ≤ N max ∥ u n ∥ 0 2 + n = 1 ∑ N τ n ∥ u n ∥ U 2 + n = 1 ∑ N ∥ u n − u n − 1 ∥ 0 2 ≤ C ν ( ∥ u 0 ∥ 0 2 + ∥ f ∥ L 2 ( 0 , T ; U ′ ) 2 ) . (26) Proof. Orthogonality gives ( P n u n − 1 , u n ) = ( u n − 1 , u n ) (P^n u^{n-1},u^n)=(u^{n-1},u^n) ( P n u n − 1 , u n ) = ( u n − 1 , u n ) (23) with u n u^n u n
2 ( x − y , x ) = ∥ x ∥ 0 2 − ∥ y ∥ 0 2 + ∥ x − y ∥ 0 2 . 2(x-y,x)=\lVert x\rVert_0^2-\lVert y\rVert_0^2+\lVert x-y\rVert_0^2. 2 ( x − y , x ) = ∥ x ∥ 0 2 − ∥ y ∥ 0 2 + ∥ x − y ∥ 0 2 . This is (24) . The two terms u n − P n u n − 1 ∈ Z n u^n-P^n u^{n-1}\in Z^n u n − P n u n − 1 ∈ Z n P n u n − 1 − u n − 1 ⊥ Z n P^n u^{n-1}-u^{n-1}\perp Z^n P n u n − 1 − u n − 1 ⊥ Z n (25) . Sum (24) , use Young’s inequality on the right-hand side, and use Jensen’s inequality for the time averages of f f f
Corollary 5.2 (Time-integrated physical no-slip defect). Let U h ( t ) = u n U_h(t)=u^n U h ( t ) = u n on ( t n − 1 , t n ] (t_{n-1},t_n] ( t n − 1 , t n ] Then
∥ γ Γ U h ∥ L 2 ( 0 , T ; L 2 ( Γ ) ) ≤ C ν , Γ , ρ ( ε ˉ + ℓ ˉ ) ( ∥ u 0 ∥ 0 + ∥ f ∥ L 2 ( 0 , T ; U ′ ) ) . (27) \lVert\gamma_\Gamma U_h\rVert_{L^2(0,T;L^2(\Gamma))}\le C_{\nu,\Gamma,\rho}(\sqrt{\bar\varepsilon}+\sqrt{\bar\ell})\left(\lVert u_0\rVert_0+\lVert f\rVert_{L^2(0,T;U')}\right). \tag*{(27)} ∥ γ Γ U h ∥ L 2 ( 0 , T ; L 2 ( Γ )) ≤ C ν , Γ , ρ ( ε ˉ + ℓ ˉ ) ( ∥ u 0 ∥ 0 + ∥ f ∥ L 2 ( 0 , T ; U ′ ) ) . (27) Proof. Apply Lemma 3.1 on each time interval and then (26) .
Lemma 5.3 (The common comparison cancels transfer). If w h ∈ Z n w_h\in Z^n w h ∈ Z n for every n n n then
( P n u n − 1 , w h ) = ( u n − 1 , w h ) , ( u n − u n − 1 , w h ) = τ n { ⟨ f n , w h ⟩ − a ( u n , w h ) } . (28) (P^n u^{n-1},w_h)=(u^{n-1},w_h),\qquad(u^n-u^{n-1},w_h)=\tau_n\{\langle f^n,w_h\rangle-a(u^n,w_h)\}. \tag*{(28)} ( P n u n − 1 , w h ) = ( u n − 1 , w h ) , ( u n − u n − 1 , w h ) = τ n {⟨ f n , w h ⟩ − a ( u n , w h )} . (28) If z h ( t ) z_h(t) z h ( t ) takes values in a fixed subspace of every Z n Z^n Z n then also P n z h ( t ) = z h ( t ) P^n z_h(t)=z_h(t) P n z h ( t ) = z h ( t ) at all times. Thus no jump of the comparison function is introduced at a mesh change.
Proof. Both statements follow directly from the defining orthogonality of P n P^n P n § 4 supplies the comparison functions required by this lemma.
Convergence with arbitrarily frequent mesh changes
We index complete computations by m m m h m h_m h m m m m
ε ˉ m = max 0 ≤ n ≤ N m ε m n , ℓ ˉ m = max 0 ≤ n ≤ N m ℓ m n , τ max , m = max 1 ≤ n ≤ N m τ m n . \bar\varepsilon_m=\max_{0\le n\le N_m}\varepsilon_m^n,\qquad
\bar\ell_m=\max_{0\le n\le N_m}\ell_m^n,\qquad
\tau_{\max,m}=\max_{1\le n\le N_m}\tau_m^n. ε ˉ m = 0 ≤ n ≤ N m max ε m n , ℓ ˉ m = 0 ≤ n ≤ N m max ℓ m n , τ m a x , m = 1 ≤ n ≤ N m max τ m n . The maxima include the choices used to define the initial projector. Let U m ( t ) = u m n U_m(t)=u_m^n U m ( t ) = u m n ( t n − 1 , t n ] (t_{n-1},t_n] ( t n − 1 , t n ] U m ( 0 ) = u m 0 U_m(0)=u_m^0 U m ( 0 ) = u m 0
Theorem 6.1 (Strong velocity convergence). Let u 0 ∈ H u_0 \in H u 0 ∈ H f ∈ L 2 ( 0 , T ; U ′ ) f \in L^2(0,T;U') f ∈ L 2 ( 0 , T ; U ′ )
h m + ε ˉ m + ℓ ˉ m + τ max , m ⟶ 0. (29) h_m+\bar{\varepsilon}_m+\bar{\ell}_m+\tau_{\max,m}\longrightarrow0. \tag*{(29)} h m + ε ˉ m + ℓ ˉ m + τ m a x , m ⟶ 0. (29) At every time level choose any admissible refinement patch and impose the exact moment constraints in (22) . Let u m 0 = P m 0 u 0 u_m^0=P_m^0u_0 u m 0 = P m 0 u 0 u m n u_m^n u m n (23) , with the time averages of f f f u u u (4) ,
∥ U m − u ∥ L 2 ( 0 , T ; U ) ⟶ 0 , sup 0 ≤ t ≤ T ∥ U m ( t ) − u ( t ) ∥ 0 ⟶ 0. (30) \lVert U_m-u\rVert_{L^2(0,T;U)}\longrightarrow0,\qquad\sup_{0\leq t\leq T}\lVert U_m(t)-u(t)\rVert_0\longrightarrow0. \tag*{(30)} ∥ U m − u ∥ L 2 ( 0 , T ; U ) ⟶ 0 , 0 ≤ t ≤ T sup ∥ U m ( t ) − u ( t ) ∥ 0 ⟶ 0. (30) In addition, the total numerical loss tends to zero:
∑ n = 1 N m { ∥ u m n − P m n u m n − 1 ∥ 0 2 + ∥ ( I − P m n ) u m n − 1 ∥ 0 2 } ⟶ 0. (31) \sum_{n=1}^{N_m}\left\{\lVert u_m^n-P_m^n u_m^{n-1}\rVert_0^2+\lVert(I-P_m^n)u_m^{n-1}\rVert_0^2\right\}\longrightarrow0. \tag*{(31)} n = 1 ∑ N m { ∥ u m n − P m n u m n − 1 ∥ 0 2 + ∥( I − P m n ) u m n − 1 ∥ 0 2 } ⟶ 0. (31) There is no restriction relating N m N_m N m h m h_m h m ε ˉ m \bar{\varepsilon}_m ε ˉ m ℓ ˉ m \bar{\ell}_m ℓ ˉ m (29) .
Proof. Initial approximation and compactness. The common recovery property of Lemma 4.1 implies that, for every w ∈ V w\in V w ∈ V w m w_m w m Z m n Z_m^n Z m n w m → w w_m\to w w m → w U U U V V V H H H P m 0 P_m^0 P m 0
∥ u m 0 − u 0 ∥ 0 ⟶ 0. (32) \lVert u_m^0-u_0\rVert_0\longrightarrow0. \tag*{(32)} ∥ u m 0 − u 0 ∥ 0 ⟶ 0. (32) Indeed, first approximate u 0 u_0 u 0 L 2 L^2 L 2 V V V P m 0 P_m^0 P m 0
The energy bound (26) gives, along a subsequence,
U m ⇀ u ~ in L 2 ( 0 , T ; U ) , U m ⇀ ∗ u ~ in L ∞ ( 0 , T ; L 2 ( Ω ) 2 ) . U_m\rightharpoonup\widetilde{u}\ \text{ in }L^2(0,T;U),\qquad U_m\mathrel{\overset{*}{\rightharpoonup}}\widetilde{u}\ \text{ in }L^\infty(0,T;L^2(\Omega)^2). U m ⇀ u in L 2 ( 0 , T ; U ) , U m ⇀ ∗ u in L ∞ ( 0 , T ; L 2 ( Ω ) 2 ) . By Corollary 5.2, γ Γ U m → 0 \gamma_\Gamma U_m\to0 γ Γ U m → 0 L 2 ( 0 , T ; L 2 ( Γ ) ) L^2(0,T;L^2(\Gamma)) L 2 ( 0 , T ; L 2 ( Γ )) u ~ ∈ L 2 ( 0 , T ; V ) \widetilde{u}\in L^2(0,T;V) u ∈ L 2 ( 0 , T ; V )
The evolution equation from common test functions. Fix w ∈ V w\in V w ∈ V w m → w w_m\to w w m → w U U U g m g_m g m g m ( t n ) = ( u m n , w m ) g_m(t_n)=(u_m^n,w_m) g m ( t n ) = ( u m n , w m )
g m ′ ( t ) = ⟨ f m ( t ) , w m ⟩ − a ( U m ( t ) , w m ) for a.e. t , (33) g_m'(t)=\langle f_m(t),w_m\rangle-a(U_m(t),w_m)\quad\text{for a.e. }t, \tag*{(33)} g m ′ ( t ) = ⟨ f m ( t ) , w m ⟩ − a ( U m ( t ) , w m ) for a.e. t , (33) where f m ( t ) = f m n f_m(t)=f_m^n f m ( t ) = f m n n n n g m ′ g_m' g m ′ L 2 ( 0 , T ) L^2(0,T) L 2 ( 0 , T ) g m ( 0 ) → ( u 0 , w ) g_m(0)\to(u_0,w) g m ( 0 ) → ( u 0 , w ) g m g_m g m C ( [ 0 , T ] ) C([0,T]) C ([ 0 , T ])
sup 0 ≤ t ≤ T ∣ ( U m ( t ) , w ) − g m ( t ) ∣ ≤ C ∥ w − w m ∥ 0 + τ max , m ∥ g m ′ ∥ L 2 ( 0 , T ) ⟶ 0. (34) \sup_{0\leq t\leq T}\left|(U_m(t),w)-g_m(t)\right|\leq C\lVert w-w_m\rVert_0+\sqrt{\tau_{\max,m}}\lVert g_m'\rVert_{L^2(0,T)}\longrightarrow0. \tag*{(34)} 0 ≤ t ≤ T sup ∣ ( U m ( t ) , w ) − g m ( t ) ∣ ≤ C ∥ w − w m ∥ 0 + τ m a x , m ∥ g m ′ ∥ L 2 ( 0 , T ) ⟶ 0. (34) Time averaging on partitions with maximum step tending to zero gives f m → f f_m\to f f m → f L 2 ( 0 , T ; U ′ ) L^2(0,T;U') L 2 ( 0 , T ; U ′ ) (33) , first on a countable dense family in V V V
d d t ( u ~ ( t ) , w ) = ⟨ f ( t ) , w ⟩ − a ( u ~ ( t ) , w ) , ( u ~ ( 0 ) , w ) = ( u 0 , w ) ( w ∈ V ) . \frac{d}{dt}(\widetilde{u}(t),w)=\langle f(t),w\rangle-a(\widetilde{u}(t),w),\qquad(\widetilde{u}(0),w)=(u_0,w)\quad(w\in V). d t d ( u ( t ) , w ) = ⟨ f ( t ) , w ⟩ − a ( u ( t ) , w ) , ( u ( 0 ) , w ) = ( u 0 , w ) ( w ∈ V ) . Consequently ∂ t u ~ ∈ L 2 ( 0 , T ; V ′ ) \partial_t\widetilde{u}\in L^2(0,T;V') ∂ t u ∈ L 2 ( 0 , T ; V ′ ) u ~ \widetilde{u} u (4) . Uniqueness identifies u ~ = u \widetilde{u}=u u = u w ∈ V w\in V w ∈ V
sup 0 ≤ t ≤ T ∣ ( U m ( t ) − u ( t ) , w ) ∣ ⟶ 0. (35) \sup_{0\leq t\leq T}|(U_m(t)-u(t),w)|\longrightarrow0. \tag*{(35)} 0 ≤ t ≤ T sup ∣ ( U m ( t ) − u ( t ) , w ) ∣ ⟶ 0. (35) The same assertion holds for every w ∈ H w \in H w ∈ H
Strong convergence in the dissipative norm. The terminal moment convergence implies
∥ u ( T ) ∥ 0 2 ≤ lim inf m ∥ u m N m ∥ 0 2 . (36) \lVert u(T)\rVert_0^2 \le\liminf_m \lVert u_m^{N_m}\rVert_0^2. \tag*{(36)} ∥ u ( T ) ∥ 0 2 ≤ m lim inf ∥ u m N m ∥ 0 2 . (36) This assertion does not require u m N m u_m^{N_m} u m N m H H H V ⊂ H V \subset H V ⊂ H (24) gives
1 2 ∥ u m N m ∥ 0 2 + ∫ 0 T a ( U m , U m ) d t + 1 2 D m = 1 2 ∥ u m 0 ∥ 0 2 + ∫ 0 T ⟨ f m , U m ⟩ d t , (37) \frac{1}{2}\lVert u_m^{N_m}\rVert_0^2+\int_0^T a(U_m,U_m)\,\mathrm{d}t+\frac{1}{2}D_m=\frac{1}{2}\lVert u_m^0\rVert_0^2+\int_0^T\langle f_m,U_m\rangle\,\mathrm{d}t, \tag*{(37)} 2 1 ∥ u m N m ∥ 0 2 + ∫ 0 T a ( U m , U m ) d t + 2 1 D m = 2 1 ∥ u m 0 ∥ 0 2 + ∫ 0 T ⟨ f m , U m ⟩ d t , (37) where D m = ∑ n = 1 N m ∥ u m n − u m n − 1 ∥ 0 2 D_m=\sum_{n=1}^{N_m}\lVert u_m^n-u_m^{n-1}\rVert_0^2 D m = ∑ n = 1 N m ∥ u m n − u m n − 1 ∥ 0 2 1 2 ∥ u 0 ∥ 0 2 + ∫ 0 T ⟨ f , u ⟩ d t \frac{1}{2}\lVert u_0\rVert_0^2+\int_0^T\langle f,u\rangle\,\mathrm{d}t 2 1 ∥ u 0 ∥ 0 2 + ∫ 0 T ⟨ f , u ⟩ d t (32) , strong convergence of f m f_m f m U m U_m U m L 2 ( 0 , T ; U ) L^2(0,T;U) L 2 ( 0 , T ; U ) (5) , this limit is 1 2 ∥ u ( T ) ∥ 0 2 + ∫ 0 T a ( u , u ) d t \frac{1}{2}\lVert u(T)\rVert_0^2+\int_0^T a(u,u)\,\mathrm{d}t 2 1 ∥ u ( T ) ∥ 0 2 + ∫ 0 T a ( u , u ) d t (36) , and D m ≥ 0 D_m\ge0 D m ≥ 0
∫ 0 T a ( U m , U m ) d t ⟶ ∫ 0 T a ( u , u ) d t , ∥ u m N m ∥ 0 ⟶ ∥ u ( T ) ∥ 0 , D m ⟶ 0. (38) \int_0^T a(U_m,U_m)\,\mathrm{d}t \longrightarrow\int_0^T a(u,u)\,\mathrm{d}t,\qquad\lVert u_m^{N_m}\rVert_0\longrightarrow\lVert u(T)\rVert_0,\qquad D_m\longrightarrow0. \tag*{(38)} ∫ 0 T a ( U m , U m ) d t ⟶ ∫ 0 T a ( u , u ) d t , ∥ u m N m ∥ 0 ⟶ ∥ u ( T ) ∥ 0 , D m ⟶ 0. (38) Since a ( v , v ) = ν ∥ v ∥ U 2 a(v,v)=\nu\lVert v\rVert_U^2 a ( v , v ) = ν ∥ v ∥ U 2 U m → u U_m\to u U m → u L 2 ( 0 , T ; U ) L^2(0,T;U) L 2 ( 0 , T ; U ) [22 Proposition 3.32] . Equation (25) then proves (31) .
There is no hidden terminal component orthogonal to H H H Q H Q_H Q H H H H (35) gives Q H u m N m ⇀ u ( T ) Q_Hu_m^{N_m}\rightharpoonup u(T) Q H u m N m ⇀ u ( T ) H H H
∥ u ( T ) ∥ 0 ≤ lim inf m ∥ Q H u m N m ∥ 0 ≤ lim sup m ∥ Q H u m N m ∥ 0 ≤ lim m ∥ u m N m ∥ 0 = ∥ u ( T ) ∥ 0 . \lVert u(T)\rVert_0\le\liminf_m\lVert Q_Hu_m^{N_m}\rVert_0\le\limsup_m\lVert Q_Hu_m^{N_m}\rVert_0\le\lim_m\lVert u_m^{N_m}\rVert_0=\lVert u(T)\rVert_0. ∥ u ( T ) ∥ 0 ≤ m lim inf ∥ Q H u m N m ∥ 0 ≤ m lim sup ∥ Q H u m N m ∥ 0 ≤ m lim ∥ u m N m ∥ 0 = ∥ u ( T ) ∥ 0 . It follows that Q H u m N m → u ( T ) Q_Hu_m^{N_m}\to u(T) Q H u m N m → u ( T ) ∥ ( I − Q H ) u m N m ∥ 0 → 0 \lVert(I-Q_H)u_m^{N_m}\rVert_0\to0 ∥( I − Q H ) u m N m ∥ 0 → 0
Uniform convergence in the bulk norm. Strong convergence in L 2 ( 0 , T ; U ) L^2(0,T;U) L 2 ( 0 , T ; U )
a ( U m , U m ) ⟶ a ( u , u ) in L 1 ( 0 , T ) , ⟨ f m , U m ⟩ ⟶ ⟨ f , u ⟩ in L 1 ( 0 , T ) . a(U_m,U_m)\longrightarrow a(u,u)\quad\text{in }L^1(0,T),\qquad\langle f_m,U_m\rangle\longrightarrow\langle f,u\rangle\quad\text{in }L^1(0,T). a ( U m , U m ) ⟶ a ( u , u ) in L 1 ( 0 , T ) , ⟨ f m , U m ⟩ ⟶ ⟨ f , u ⟩ in L 1 ( 0 , T ) . The corresponding cumulative integrals converge uniformly in their upper limits. The discrete energy identity at any node, after discarding the nonnegative accumulated loss, and the continuous energy identity therefore give
lim sup m max n ( ∥ u m n ∥ 0 2 − ∥ u ( t n ) ∥ 0 2 ) ≤ 0. \limsup_m\max_n\left(\lVert u_m^n\rVert_0^2-\lVert u(t_n)\rVert_0^2\right)\le0. m lim sup n max ( ∥ u m n ∥ 0 2 − ∥ u ( t n ) ∥ 0 2 ) ≤ 0. By continuity of u : [ 0 , T ] → H u:[0,T]\to H u : [ 0 , T ] → H τ max , m → 0 \tau_{\max,m}\to0 τ m a x , m → 0 t t t U m ( t ) U_m(t) U m ( t ) u ( [ 0 , T ] ) u([0,T]) u ([ 0 , T ]) H H H (35) proves
sup 0 ≤ t ≤ T ∣ ( U m ( t ) − u ( t ) , u ( t ) ) ∣ ⟶ 0. \sup_{0\le t\le T}\left|(U_m(t)-u(t),u(t))\right|\longrightarrow0. 0 ≤ t ≤ T sup ∣ ( U m ( t ) − u ( t ) , u ( t )) ∣ ⟶ 0. Finally, expand
∥ U m ( t ) − u ( t ) ∥ 0 2 = ∥ U m ( t ) ∥ 0 2 − ∥ u ( t ) ∥ 0 2 − 2 ( U m ( t ) − u ( t ) , u ( t ) ) . \lVert U_m(t)-u(t)\rVert_0^2=\lVert U_m(t)\rVert_0^2-\lVert u(t)\rVert_0^2-2(U_m(t)-u(t),u(t)). ∥ U m ( t ) − u ( t ) ∥ 0 2 = ∥ U m ( t ) ∥ 0 2 − ∥ u ( t ) ∥ 0 2 − 2 ( U m ( t ) − u ( t ) , u ( t )) . The preceding two bounds prove uniform bulk convergence.
Remark 6.2 (Why no remeshing count occurs). The cancellation uses a fixed family of comparison functions common to all spaces of one computation, not a bound for each mesh-transfer defect. This differs from the projection-difference terms in Brenner et al. [8 Theorems 3.1 and 3.4] and the remeshing contributions in Gawlik and Lew [10 ] . The dense family is supplied by the retained coarse grid and the collar construction. A collection of unrelated shifted grids need not have such a family. Thus Theorem 6.1 does not assert convergence for arbitrary stable interpolations or for arbitrary changing spaces.
A quantitative estimate for piecewise smooth solutions
We next give an a priori bound for a more regular solution. The estimate is in the retained coarse mesh size; it is not an adaptive optimality statement. The comparison-function and residual-energy strategy is standard; compare the dynamic-mesh Stokes error analysis in Brenner et al. [8 Section 3.1] . The specific additional ingredient here is one constructed comparison operator shared by all admissible spaces, so that its mesh-change residual vanishes exactly. Write
Y = W 3 , ∞ ( B ) 2 × W 3 , ∞ ( F ) 2 , ∥ v ∥ Y = ∥ v ∣ B ∥ W 3 , ∞ ( B ) + ∥ v ∣ F ∥ W 3 , ∞ ( F ) . Y = W^{3,\infty}(B)^2 \times W^{3,\infty}(F)^2,\qquad\|v\|_Y = \|v|_B\|_{W^{3,\infty}(B)} + \|v|_F\|_{W^{3,\infty}(F)}. Y = W 3 , ∞ ( B ) 2 × W 3 , ∞ ( F ) 2 , ∥ v ∥ Y = ∥ v ∣ B ∥ W 3 , ∞ ( B ) + ∥ v ∣ F ∥ W 3 , ∞ ( F ) . The notation u ∈ H 1 ( 0 , T ; Y ) u \in H^1(0,T;Y) u ∈ H 1 ( 0 , T ; Y ) Y Y Y u u u Y Y Y
Theorem 7.1 (Coarse-grid energy error bound). Assume the geometry and exact-integration discretization of the preceding sections. Suppose the target velocity satisfies
u ∈ H 1 ( 0 , T ; Y ) ∩ H 1 ( 0 , T ; V ) . u \in H^1(0,T;Y) \cap H^1(0,T;V). u ∈ H 1 ( 0 , T ; Y ) ∩ H 1 ( 0 , T ; V ) . Suppose also that there is λ ∈ L 2 ( 0 , T ; L 2 ( Γ ) 2 ) \lambda\in L^2(0,T;L^2(\Gamma)^2) λ ∈ L 2 ( 0 , T ; L 2 ( Γ ) 2 ) t t t
( ∂ t u ( t ) , v ) + a ( u ( t ) , v ) + ∫ Γ λ ( t ) ⋅ γ Γ v d s = ⟨ f ( t ) , v ⟩ ( v ∈ U ) . (39) (\partial_t u(t),v) + a(u(t),v) + \int_\Gamma\lambda(t)\cdot\gamma_\Gamma v\,\mathrm{d}s = \langle f(t),v\rangle\qquad(v \in U). \tag*{(39)} ( ∂ t u ( t ) , v ) + a ( u ( t ) , v ) + ∫ Γ λ ( t ) ⋅ γ Γ v d s = ⟨ f ( t ) , v ⟩ ( v ∈ U ) . (39) This assumption specifies the regularity of the continuous reaction; it does not impose an inf-sup condition on the discrete multiplier. For sufficiently small h m + ε ˉ m h_m+\bar{\varepsilon}_m h m + ε ˉ m
max 0 ≤ n ≤ N m ∥ u m n − u ( t n ) ∥ 0 + ( ∑ n = 1 N m τ m n ∥ u m n − u ( t n ) ∥ U 2 ) 1 / 2 ≤ C ν , T , Γ , ρ [ h m + ε ˉ m ( ∥ u ∥ L ∞ ( 0 , T ; Y ) + ∥ ∂ t u ∥ L 2 ( 0 , T ; Y ) ) + ( ε ˉ m + ℓ ˉ m ) ∥ λ ∥ L 2 ( 0 , T ; L 2 ( Γ ) ) + τ max , m ∥ ∂ t u ∥ L 2 ( 0 , T ; U ) ] . (40) \max_{0\le n\le N_m}\|u_m^n-u(t_n)\|_0+\left(\sum_{n=1}^{N_m}\tau_m^n\|u_m^n-u(t_n)\|_U^2\right)^{1/2}
\le C_{\nu,T,\Gamma,\rho}\left[\sqrt{h_m}+\sqrt{\bar{\varepsilon}_m}\left(\|u\|_{L^\infty(0,T;Y)}+\|\partial_tu\|_{L^2(0,T;Y)}\right)
+\left(\sqrt{\bar{\varepsilon}_m}+\sqrt{\bar{\ell}_m}\right)\|\lambda\|_{L^2(0,T;L^2(\Gamma))}
+\tau_{\max,m}\|\partial_tu\|_{L^2(0,T;U)}\right]. \tag*{(40)} 0 ≤ n ≤ N m max ∥ u m n − u ( t n ) ∥ 0 + ( n = 1 ∑ N m τ m n ∥ u m n − u ( t n ) ∥ U 2 ) 1/2 ≤ C ν , T , Γ , ρ [ h m + ε ˉ m ( ∥ u ∥ L ∞ ( 0 , T ; Y ) + ∥ ∂ t u ∥ L 2 ( 0 , T ; Y ) ) + ( ε ˉ m + ℓ ˉ m ) ∥ λ ∥ L 2 ( 0 , T ; L 2 ( Γ )) + τ m a x , m ∥ ∂ t u ∥ L 2 ( 0 , T ; U ) ] . (40) The constant is independent of the selected refined cells, their relative positions to Γ \Gamma Γ O ( h m + ε ˉ m + ℓ ˉ m + τ max , m ) O(\sqrt{h_m}+\sqrt{\bar{\varepsilon}_m}+\sqrt{\bar{\ell}_m}+\tau_{\max,m}) O ( h m + ε ˉ m + ℓ ˉ m + τ m a x , m )
Proof. We suppress the computation index m m m L h = Q h , ε ˉ : V ∩ Y → K h , ε ˉ L_h=Q_{h,\bar{\varepsilon}}:V\cap Y\to K_{h,\bar{\varepsilon}} L h = Q h , ε ˉ : V ∩ Y → K h , ε ˉ Lemma 4.2 , with K h , ε ˉ ⊂ Z n K_{h,\bar{\varepsilon}}\subset Z^n K h , ε ˉ ⊂ Z n n n n
z ( t ) = L h u ( t ) , η ( t ) = u ( t ) − z ( t ) , e n = u n − z ( t n ) , d = h + ε ˉ . z(t)=L_hu(t),\qquad\eta(t)=u(t)-z(t),\qquad e^n=u^n-z(t_n),\qquad d=h+\bar{\varepsilon}. z ( t ) = L h u ( t ) , η ( t ) = u ( t ) − z ( t ) , e n = u n − z ( t n ) , d = h + ε ˉ . The same, time-independent operator is used throughout this computation. The quantitative collar bound and its application to ∂ t u \partial_tu ∂ t u
∥ η ∥ L ∞ ( 0 , T ; U ) ≤ C d ∥ u ∥ L ∞ ( 0 , T ; Y ) , ∥ ∂ t η ∥ L 2 ( 0 , T ; U ) ≤ C d ∥ ∂ t u ∥ L 2 ( 0 , T ; Y ) . (41) \|\eta\|_{L^\infty(0,T;U)}\le C\sqrt{d}\,\|u\|_{L^\infty(0,T;Y)},\qquad\|\partial_t\eta\|_{L^2(0,T;U)}\le C\sqrt{d}\,\|\partial_tu\|_{L^2(0,T;Y)}. \tag*{(41)} ∥ η ∥ L ∞ ( 0 , T ; U ) ≤ C d ∥ u ∥ L ∞ ( 0 , T ; Y ) , ∥ ∂ t η ∥ L 2 ( 0 , T ; U ) ≤ C d ∥ ∂ t u ∥ L 2 ( 0 , T ; Y ) . (41) Here differentiation commutes with L h L_h L h u ‾ n \overline{u}^{n} u n λ ‾ n \overline{\lambda}^{n} λ n ( t n − 1 , t n ) (t_{n-1},t_n) ( t n − 1 , t n ) (39) and subtracting the resulting identity from (23) , using P n z ( t n − 1 ) = z ( t n − 1 ) P^{n}z(t_{n-1})=z(t_{n-1}) P n z ( t n − 1 ) = z ( t n − 1 )
( ( e n − P n e n − 1 ) / τ n , v h ) + a ( e n , v h ) = R n ( v h ) ( v h ∈ Z n ) (42) \left((e^{n}-P^{n}e^{n-1})/\tau_{n},v_{h}\right)+a(e^{n},v_{h})=\mathcal{R}_{n}(v_{h})\qquad(v_{h}\in Z^{n}) \tag*{(42)} ( ( e n − P n e n − 1 ) / τ n , v h ) + a ( e n , v h ) = R n ( v h ) ( v h ∈ Z n ) (42) where
R n ( v h ) = ( τ n − 1 ∫ t n − 1 t n ∂ t η ( t ) d t , v h ) + a ( η ( t n ) , v h ) \mathcal{R}_{n}(v_{h})=\left(\tau_{n}^{-1}\int_{t_{n-1}}^{t_{n}}\partial_{t}\eta(t)\,\mathrm{d}t,v_{h}\right)+a(\eta(t_{n}),v_{h}) R n ( v h ) = ( τ n − 1 ∫ t n − 1 t n ∂ t η ( t ) d t , v h ) + a ( η ( t n ) , v h ) + a ( u ‾ n − u ( t n ) , v h ) + ∫ Γ λ ‾ n ⋅ γ Γ v h d s . (43) \qquad{}+a(\overline{u}^{n}-u(t_{n}),v_{h})+\int_{\Gamma}\overline{\lambda}^{n}\cdot\gamma\Gamma v_{h}\,\mathrm{d}s. \tag*{(43)} + a ( u n − u ( t n ) , v h ) + ∫ Γ λ n ⋅ γ Γ v h d s . (43) All four residuals have a specified origin: comparison differentiation, spatial comparison, time averaging, and the physical boundary reaction. There is no remeshing residual.
Use ∥ v ∥ a = a ( v , v ) = ν ∥ v ∥ U \|v\|_{a}=\sqrt{a(v,v)}=\sqrt{\nu}\|v\|_{U} ∥ v ∥ a = a ( v , v ) = ν ∥ v ∥ U C P C_{P} C P Ω \Omega Ω Lemma 3.1 gives ∣ R n ( v h ) ∣ ≤ r n ∥ v h ∥ a |\mathcal{R}_{n}(v_{h})|\le r_{n}\|v_{h}\|_{a} ∣ R n ( v h ) ∣ ≤ r n ∥ v h ∥ a
r n = C P ν ∥ τ n − 1 ∫ t n − 1 t n ∂ t η d t ∥ 0 + ν ∥ η ( t n ) ∥ U r_{n}=\frac{C_{P}}{\sqrt{\nu}}\left\|\tau_{n}^{-1}\int_{t_{n-1}}^{t_{n}}\partial_{t}\eta\,\mathrm{d}t\right\|_{0}+\sqrt{\nu}\|\eta(t_{n})\|_{U} r n = ν C P τ n − 1 ∫ t n − 1 t n ∂ t η d t 0 + ν ∥ η ( t n ) ∥ U + ν ∥ u ‾ n − u ( t n ) ∥ U + C Γ ν ( ϵ n + ℓ n ) ∥ λ ‾ n ∥ L 2 ( Γ ) . (44) \qquad{}+\sqrt{\nu}\|\overline{u}^{n}-u(t_{n})\|_{U}+\frac{C_{\Gamma}}{\sqrt{\nu}}(\sqrt{\epsilon_{n}}+\sqrt{\ell_{n}})\|\overline{\lambda}^{n}\|_{L^{2}(\Gamma)}. \tag*{(44)} + ν ∥ u n − u ( t n ) ∥ U + ν C Γ ( ϵ n + ℓ n ) ∥ λ n ∥ L 2 ( Γ ) . (44) Test (42) with e n e^{n} e n
∥ e n ∥ 0 2 − ∥ e n − 1 ∥ 0 2 + τ n ∥ e n ∥ a 2 ≤ τ n r n 2 . \|e^{n}\|_{0}^{2}-\|e^{n-1}\|_{0}^{2}+\tau_{n}\|e^{n}\|_{a}^{2}\le\tau_{n}r_{n}^{2}. ∥ e n ∥ 0 2 − ∥ e n − 1 ∥ 0 2 + τ n ∥ e n ∥ a 2 ≤ τ n r n 2 . Therefore
max n ∥ e n ∥ 0 2 + ∑ n τ n ∥ e n ∥ a 2 ≤ 2 ( ∥ e 0 ∥ 0 2 + ∑ n τ n r n 2 ) . (45) \max_{n}\|e^{n}\|_{0}^{2}+\sum_{n}\tau_{n}\|e^{n}\|_{a}^{2}\le2\left(\|e^{0}\|_{0}^{2}+\sum_{n}\tau_{n}r_{n}^{2}\right). \tag*{(45)} n max ∥ e n ∥ 0 2 + n ∑ τ n ∥ e n ∥ a 2 ≤ 2 ( ∥ e 0 ∥ 0 2 + n ∑ τ n r n 2 ) . (45) The initialization is compatible with the common comparison: e 0 = P 0 u 0 − z ( 0 ) = P 0 η ( 0 ) e^{0}=P^{0}u_{0}-z(0)=P^{0}\eta(0) e 0 = P 0 u 0 − z ( 0 ) = P 0 η ( 0 ) ∥ e 0 ∥ 0 ≤ ∥ η ( 0 ) ∥ 0 \|e^{0}\|_{0}\le\|\eta(0)\|_{0} ∥ e 0 ∥ 0 ≤ ∥ η ( 0 ) ∥ 0
Jensen’s inequality controls the averaged ∂ t η \partial_{t}\eta ∂ t η λ ‾ n \overline{\lambda}^{n} λ n
∑ n τ n ∥ u ‾ n − u ( t n ) ∥ U 2 ≤ τ max 2 ∥ ∂ t u ∥ L 2 ( 0 , T ; U ) 2 . (46) \sum_{n}\tau_{n}\|\overline{u}^{n}-u(t_{n})\|_{U}^{2}\le\tau_{\max}^{2}\|\partial_{t}u\|_{L^{2}(0,T;U)}^{2}. \tag*{(46)} n ∑ τ n ∥ u n − u ( t n ) ∥ U 2 ≤ τ m a x 2 ∥ ∂ t u ∥ L 2 ( 0 , T ; U ) 2 . (46) Indeed, on a single interval the norm of u ( t n ) − u ‾ n u(t_{n})-\overline{u}^{n} u ( t n ) − u n τ n ∥ ∂ t u ∥ L 2 ( t n − 1 , t n ; U ) \sqrt{\tau_{n}}\|\partial_{t}u\|_{L^{2}(t_{n-1},t_{n};U)} τ n ∥ ∂ t u ∥ L 2 ( t n − 1 , t n ; U ) (44) , using ( x 1 + ⋯ + x 4 ) 2 ≤ 4 ∑ i x i 2 (x_{1}+\cdots+x_{4})^{2}\le4\sum_{i}x_{i}^{2} ( x 1 + ⋯ + x 4 ) 2 ≤ 4 ∑ i x i 2 (41) proves
∑ n τ n r n 2 ≤ C ν , Γ { δ [ T ∥ u ∥ L ∞ ( 0 , T ; Y ) 2 + ∥ ∂ t u ∥ L 2 ( 0 , T ; Y ) 2 ] + τ max 2 ∥ ∂ t u ∥ L 2 ( 0 , T ; U ) 2 \sum_{n}\tau_{n}r_{n}^{2}\le C_{\nu,\Gamma}\left\{\delta\left[T\|u\|_{L^{\infty}(0,T;Y)}^{2}+\|\partial_{t}u\|_{L^{2}(0,T;Y)}^{2}\right]+\tau_{\max}^{2}\|\partial_{t}u\|_{L^{2}(0,T;U)}^{2}
\right. n ∑ τ n r n 2 ≤ C ν , Γ { δ [ T ∥ u ∥ L ∞ ( 0 , T ; Y ) 2 + ∥ ∂ t u ∥ L 2 ( 0 , T ; Y ) 2 ] + τ m a x 2 ∥ ∂ t u ∥ L 2 ( 0 , T ; U ) 2 + ( ϵ + ℓ ) 2 ∥ λ ∥ L 2 ( 0 , T ; L 2 ( Γ ) ) 2 } . \left.
\qquad{}+(\sqrt{\epsilon}+\sqrt{\ell})^{2}\|\lambda\|_{L^{2}(0,T;L^{2}(\Gamma))}^{2}\right\}. + ( ϵ + ℓ ) 2 ∥ λ ∥ L 2 ( 0 , T ; L 2 ( Γ )) 2 } . Insert this into (45) , take square roots, and use u ‾ n − u ( t n ) = e n − η ( t n ) \overline{u}^{n}-u(t_{n})=e^{n}-\eta(t_{n}) u n − u ( t n ) = e n − η ( t n ) (40) .
Remark 7.2 (Nodal and continuous-time error norms). The same right-hand side as in (40) , with a changed constant, controls ∥ U m − u ∥ L 2 ( 0 , T ; U ) \|U_{m}-u\|_{L^{2}(0,T;U)} ∥ U m − u ∥ L 2 ( 0 , T ; U )
∑ n ∫ t n − 1 t n ∥ u ( t n ) − u ( t ) ∥ U 2 d t ≤ τ max , m 2 ∥ ∂ t u ∥ L 2 ( 0 , T ; U ) 2 . \sum_{n}\int_{t_{n-1}}^{t_{n}}\|u(t_{n})-u(t)\|_{U}^{2}\,\mathrm{d}t\le\tau_{\max,m}^{2}\|\partial_{t}u\|_{L^{2}(0,T;U)}^{2}. n ∑ ∫ t n − 1 t n ∥ u ( t n ) − u ( t ) ∥ U 2 d t ≤ τ m a x , m 2 ∥ ∂ t u ∥ L 2 ( 0 , T ; U ) 2 . The first term in (40) is a maximum over time nodes. For the piecewise constant U m U_{m} U m L ∞ ( 0 , T ; L 2 ) L^{\infty}(0,T;L^{2}) L ∞ ( 0 , T ; L 2 ) τ max , m ∥ ∂ t u ∥ L 2 ( 0 , T ; L 2 ) \sqrt{\tau_{\max,m}}\|\partial_{t}u\|_{L^{2}(0,T;L^{2})} τ m a x , m ∥ ∂ t u ∥ L 2 ( 0 , T ; L 2 ) ∂ t u ∈ L ∞ ( 0 , T ; L 2 ) \partial_{t}u\in L^{\infty}(0,T;L^{2}) ∂ t u ∈ L ∞ ( 0 , T ; L 2 ) Theorem 6.1 does not require this stronger regularity.
Prescribed rotation with fixed immersed geometry
The position of an immersed interface need not change when the material on it moves. A disk rotating about its fixed center is a useful example: its Eulerian boundary remains fixed, but its no-slip velocity is nonzero and may vary in time. We prove an extension for this problem. The common homogeneous comparison family from § 4 is retained, and a second, explicitly constructed coarse-grid function provides an exact lifting of every regularized boundary constraint. Prescribed rigid motion in fictitious-domain formulations predates this construction [6 , 7 ] . The point here is the simultaneous feasibility of one coarse lift for every admitted refinement, regularization width and surface partition.
An exact common lifting
Let B = { x : ∣ x − c ∣ < R } ⋐ Ω B = \{x : \lvert x-c\rvert< R\} \Subset\Omega B = { x : ∣ x − c ∣ < R } ⋐ Ω
J = ( 0 − 1 1 0 ) , r ( x ) = J ( x − c ) , ω ∈ H 1 ( 0 , T ) . J = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}, \qquad r(x) = J(x-c), \qquad\omega\in H^1(0,T). J = ( 0 1 − 1 0 ) , r ( x ) = J ( x − c ) , ω ∈ H 1 ( 0 , T ) . Thus the prescribed boundary velocity is ω ( t ) r \omega(t)r ω ( t ) r ω \omega ω d ∗ > 0 d_* > 0 d ∗ > 0 χ ∈ C 5 ( Ω ) \chi\in C^5(\Omega) χ ∈ C 5 ( Ω ) χ = 1 \chi= 1 χ = 1 d ∗ d_* d ∗ B ‾ \overline{B} B
ψ ( x ) = − 1 2 χ ( x ) ∣ x − c ∣ 2 , g = curl ψ , g h = curl I h ψ . (47) \psi(x) = -\frac{1}{2}\chi(x)\lvert x-c\rvert^2, \qquad g = \operatorname{curl}\psi, \qquad g_h = \operatorname{curl} I_h\psi. \tag*{(47)} ψ ( x ) = − 2 1 χ ( x ) ∣ x − c ∣ 2 , g = curl ψ , g h = curl I h ψ . (47) The sign is consistent with curl ψ = ( ∂ y ψ , − ∂ x ψ ) \operatorname{curl}\psi= (\partial_y\psi,-\partial_x\psi) curl ψ = ( ∂ y ψ , − ∂ x ψ ) g = r g = r g = r χ = 1 \chi= 1 χ = 1 d ∗ d_* d ∗ C 5 C^5 C 5 ψ ∈ H 4 \psi\in H^4 ψ ∈ H 4 g ∈ W 3 , ∞ g \in W^{3,\infty} g ∈ W 3 , ∞ C ∞ C^\infty C ∞ C 5 C^5 C 5
Lemma 8.1 (Common exact rotation lift). Suppose
ε ‾ + 2 h < d ∗ . (48) \overline{\varepsilon}+\sqrt{2}h<d_*. \tag*{(48)} ε + 2 h < d ∗ . (48) Then g h ∈ curl S h ⊂ X h , A g_h \in\operatorname{curl} S_h \subset X_{h,A} g h ∈ curl S h ⊂ X h , A for every admissible refinement patch, and
g h = r on B and on every regularization support centered on Γ , g_h=r \quad\text{on } B \text{ and on every regularization support centered on } \Gamma, g h = r on B and on every regularization support centered on Γ , γ Γ g h = r , J ε g h = r ( 0 < ε ≤ ε ‾ ) , (49) \gamma_\Gamma g_h=r, \qquad J_\varepsilon g_h=r \quad(0<\varepsilon\leq\overline{\varepsilon}), \tag*{(49)} γ Γ g h = r , J ε g h = r ( 0 < ε ≤ ε ) , (49) ∥ g h − g ∥ H 1 ( Ω ) ≤ C g h 2 . (50) \lVert g_h-g\rVert_{H^1(\Omega)}\leq Cg h^2. \tag*{(50)} ∥ g h − g ∥ H 1 ( Ω ) ≤ C g h 2 . (50) Consequently the nonempty affine constraint space is exactly
ω n g h + Z n = { v h ∈ X h , A n : Π P n ( J ε n v h − ω n r ) = 0 } , ω n = ω ( t n ) . (51) \omega_n g_h+Z^n=\{v_h\in X_{h,A_n}:\Pi_{P_n}(J_{\varepsilon_n}v_h-\omega_n r)=0\}, \qquad\omega_n=\omega(t_n). \tag*{(51)} ω n g h + Z n = { v h ∈ X h , A n : Π P n ( J ε n v h − ω n r ) = 0 } , ω n = ω ( t n ) . (51) These assertions require neither independence nor an inf--sup bound for the surface constraints.
Proof. Every coarse cell meeting the ε ‾ \overline{\varepsilon} ε Γ \Gamma Γ ( ε ‾ + 2 h ) (\overline{\varepsilon}+\sqrt{2}h) ( ε + 2 h ) B ‾ \overline{B} B B B B ψ = − ∣ x − c ∣ 2 / 2 \psi=-\lvert x-c\rvert^2/2 ψ = − ∣ x − c ∣ 2 /2 Ω \Omega Ω ρ \rho ρ
∫ ρ ε ( z ) d z = 1 and ∫ z ρ ε ( z ) d z = 0. \int\rho_\varepsilon(z)\,dz=1 \quad\text{and} \quad\int z\rho_\varepsilon(z)\,dz=0. ∫ ρ ε ( z ) d z = 1 and ∫ z ρ ε ( z ) d z = 0. Hence, for x ∈ Γ x\in\Gamma x ∈ Γ
J ε g h ( x ) = ∫ ρ ε ( z ) r ( x − z ) d z = r ( x ) − J ∫ z ρ ε ( z ) d z = r ( x ) . J_\varepsilon g_h(x)=\int\rho_\varepsilon(z)r(x-z)\,dz=r(x)-J\int z\rho_\varepsilon(z)\,dz=r(x). J ε g h ( x ) = ∫ ρ ε ( z ) r ( x − z ) d z = r ( x ) − J ∫ z ρ ε ( z ) d z = r ( x ) . This is affine reproduction by zeroth and first kernel moments, the continuum counterpart of the familiar IB moment conditions [1 , 3 ] ; the displayed equality proves the exact property needed for our lift without a discrete-delta convergence theorem. The interpolation estimate (16) , applied to the fixed smooth ψ \psi ψ (50) . Finally subtracting ω n g h \omega_n g_h ω n g h Z n Z^n Z n
Continuous and discrete centered equations
Subtracting a lift is the standard reduction of nonhomogeneous essential boundary data to a homogeneous variational problem; see, for example, Ern and Guermond [28 Chapter 33] for its elliptic form. The time-dependent load and the common discrete lift needed here are specified explicitly below, rather than inferred from that elliptic result. Let f ∈ L 2 ( 0 , T ; U ′ ) f \in L^2(0,T;U') f ∈ L 2 ( 0 , T ; U ′ ) u 0 = ω ( 0 ) g + w 0 u_0=\omega(0)g+w_0 u 0 = ω ( 0 ) g + w 0 w 0 ∈ H w_0\in H w 0 ∈ H
⟨ F ( t ) , v ⟩ = ⟨ f ( t ) , v ⟩ − ω ′ ( t ) ( g , v ) − ω ( t ) a ( g , v ) , v ∈ U . (52) \langle\mathcal{F}(t),v\rangle=\langle f(t),v\rangle-\omega'(t)(g,v)-\omega(t)a(g,v),\qquad v\in U. \tag*{(52)} ⟨ F ( t ) , v ⟩ = ⟨ f ( t ) , v ⟩ − ω ′ ( t ) ( g , v ) − ω ( t ) a ( g , v ) , v ∈ U . (52) This functional belongs to L 2 ( 0 , T ; U ′ ) L^2(0,T;U') L 2 ( 0 , T ; U ′ ) w w w (4) with initial value w 0 w_0 w 0 F \mathcal{F} F
G ( t ) = ω ( t ) g , u ( t ) = w ( t ) + G ( t ) . (53) G(t)=\omega(t)g,\qquad u(t)=w(t)+G(t). \tag*{(53)} G ( t ) = ω ( t ) g , u ( t ) = w ( t ) + G ( t ) . (53) Then u ∈ G + L 2 ( 0 , T ; V ) u\in G+L^2(0,T;V) u ∈ G + L 2 ( 0 , T ; V ) u ∈ C ( [ 0 , T ] ; L 2 ) u\in C([0,T];L^2) u ∈ C ([ 0 , T ] ; L 2 ) γ Γ u = ω r \gamma_\Gamma u=\omega r γ Γ u = ω r
⟨ ∂ t u , v ⟩ + a ( u , v ) = ⟨ f , v ⟩ ( v ∈ V ) . \langle\partial_tu,v\rangle+a(u,v)=\langle f,v\rangle\qquad(v\in V). ⟨ ∂ t u , v ⟩ + a ( u , v ) = ⟨ f , v ⟩ ( v ∈ V ) . Conversely any solution in this affine class has the stated form by subtracting G G G
Initialize by the true affine projection of the physical initial data:
u h 0 = ω 0 g h + P 0 ( u 0 − ω 0 g h ) , u_h^0=\omega_0g_h+P^0(u_0-\omega_0g_h), u h 0 = ω 0 g h + P 0 ( u 0 − ω 0 g h ) , w h 0 = P 0 ( w 0 + ω 0 ( g − g h ) ) . (54) w_h^0=P^0(w_0+\omega_0(g-g_h)). \tag*{(54)} w h 0 = P 0 ( w 0 + ω 0 ( g − g h )) . (54) The difference from the centered initialization P 0 w 0 P^0w_0 P 0 w 0 C g h 2 ∣ ω 0 ∣ C_gh^2|\omega_0| C g h 2 ∣ ω 0 ∣
δ ω n = ω n − ω n − 1 τ n = 1 τ n ∫ t n − 1 t n ω ′ ( t ) d t , ω ˉ n = 1 τ n ∫ t n − 1 t n ω ( t ) d t . \delta\omega_n=\frac{\omega_n-\omega_{n-1}}{\tau_n}=\frac{1}{\tau_n}\int_{t_{n-1}}^{t_n}\omega'(t)\,\mathrm{d}t,\qquad\bar{\omega}_n=\frac{1}{\tau_n}\int_{t_{n-1}}^{t_n}\omega(t)\,\mathrm{d}t. δ ω n = τ n ω n − ω n − 1 = τ n 1 ∫ t n − 1 t n ω ′ ( t ) d t , ω ˉ n = τ n 1 ∫ t n − 1 t n ω ( t ) d t . The centered backward Euler scheme is
w h n ∈ Z n , u h n = w h n + ω n g h , w_h^n\in Z^n,\qquad u_h^n=w_h^n+\omega_ng_h, w h n ∈ Z n , u h n = w h n + ω n g h , ( w h n − P n w h n − 1 τ n , v h ) + a ( w h n , v h ) = ⟨ F h n , v h ⟩ ( v h ∈ Z n ) , (55) \left(\frac{w_h^n-P^nw_h^{n-1}}{\tau_n},v_h\right)+a(w_h^n,v_h)=\langle\mathcal{F}_h^n,v_h\rangle\qquad(v_h\in Z^n), \tag*{(55)} ( τ n w h n − P n w h n − 1 , v h ) + a ( w h n , v h ) = ⟨ F h n , v h ⟩ ( v h ∈ Z n ) , (55) ⟨ F h n , v ⟩ = ⟨ f n , v ⟩ − δ ω n ( g h , v ) − ω n a ( g h , v ) . (56) \langle\mathcal{F}_h^n,v\rangle=\langle f^n,v\rangle-\delta\omega_n(g_h,v)-\omega_na(g_h,v). \tag*{(56)} ⟨ F h n , v ⟩ = ⟨ f n , v ⟩ − δ ω n ( g h , v ) − ω n a ( g h , v ) . (56) The velocity exists uniquely by positive definiteness on Z n Z^n Z n (51) at every time level. Although g h g_h g h not belong to Z n Z^n Z n P n g h = g h P^ng_h=g_h P n g h = g h
Remark 8.2 (Affine projection and a potentially inconsistent reset). For a fixed step the genuine L 2 L^2 L 2
P n a f f q = ω n g h + P n ( q − ω n g h ) . P_n^{\mathrm{aff}}q=\omega_ng_h+P^n(q-\omega_ng_h). P n aff q = ω n g h + P n ( q − ω n g h ) . Orthogonality gives ( P n a f f q − q , v h ) = 0 (P_n^{\mathrm{aff}}q-q,v_h)=0 ( P n aff q − q , v h ) = 0 v h ∈ Z n v_h\in Z^n v h ∈ Z n (55) is equivalent to the physical velocity equation
( u h n − P n a f f u h n − 1 τ n , v h ) + a ( u h n , v h ) = ⟨ f n , v h ⟩ , v h ∈ Z n . \left(\frac{u_h^n-P_n^{\mathrm{aff}}u_h^{n-1}}{\tau_n},v_h\right)+a(u_h^n,v_h)=\langle f^n,v_h\rangle,\qquad v_h\in Z^n. ( τ n u h n − P n aff u h n − 1 , v h ) + a ( u h n , v h ) = ⟨ f n , v h ⟩ , v h ∈ Z n . Indeed, writing u h n = w h n + ω n g h u_h^n=w_h^n+\omega_ng_h u h n = w h n + ω n g h δ ω n ( g h , v h ) \delta\omega_n(g_h,v_h) δ ω n ( g h , v h ) (56) . A true affine projection therefore retains the prescribed acceleration.
It is different from the lift-reset map
R n q = ω n g h + P n ( q − ω n − 1 g h ) , P n a f f q = R n q − τ n δ ω n P n g h . \mathcal{R}_n q=\omega_n g_h+P^n(q-\omega_{n-1}g_h),\qquad\mathcal{P}^{\mathrm{aff}}_n q=\mathcal{R}_n q-\tau_n\delta\omega_nP^n g_h. R n q = ω n g h + P n ( q − ω n − 1 g h ) , P n aff q = R n q − τ n δ ω n P n g h . Replacing the affine projection by R n \mathcal{R}_n R n − δ ω n ( g h , v h ) -\delta\omega_n(g_h,v_h) − δ ω n ( g h , v h )
Centered energy and the physical boundary defect
Let
D ω = ∥ w 0 ∥ 0 + ∥ f ∥ L 2 ( 0 , T ; U ′ ) + ∥ ω ′ ∥ L 2 ( 0 , T ) + ∥ ω ∥ L ∞ ( 0 , T ) . D_\omega=\|w_0\|_0+\|f\|_{L^2(0,T;U')}+\|\omega'\|_{L^2(0,T)}+\|\omega\|_{L^\infty(0,T)}. D ω = ∥ w 0 ∥ 0 + ∥ f ∥ L 2 ( 0 , T ; U ′ ) + ∥ ω ′ ∥ L 2 ( 0 , T ) + ∥ ω ∥ L ∞ ( 0 , T ) . Constants below may depend on the fixed lifting, ν \nu ν T T T
Proposition 8.3 (Centered energy and unchanged interface order). At every step of (55) ,
∥ w h n ∥ 0 2 − ∥ w h n − 1 ∥ 0 2 + ∥ w h n − w h n − 1 ∥ 0 2 + 2 τ n a ( w h n , w h n ) = 2 τ n { ⟨ f n , w h n ⟩ − δ ω n ( g h , w h n ) − ω n a ( g h , w h n ) } . (57) \begin{aligned}
\|w_h^n\|_0^2-\|w_h^{n-1}\|_0^2+\|w_h^n-w_h^{n-1}\|_0^2+2\tau_n a(w_h^n,w_h^n)\\
=2\tau_n\{\langle f^n,w_h^n\rangle-\delta\omega_n(g_h,w_h^n)-\omega_n a(g_h,w_h^n)\}. \tag*{(57)}
\end{aligned} ∥ w h n ∥ 0 2 − ∥ w h n − 1 ∥ 0 2 + ∥ w h n − w h n − 1 ∥ 0 2 + 2 τ n a ( w h n , w h n ) = 2 τ n {⟨ f n , w h n ⟩ − δ ω n ( g h , w h n ) − ω n a ( g h , w h n )} . (57) The increment has the orthogonal decomposition
∥ w h n − w h n − 1 ∥ 0 2 = ∥ w h n − P n w h n − 1 ∥ 0 2 + ∥ ( I − P n ) w h n − 1 ∥ 0 2 . (58) \|w_h^n-w_h^{n-1}\|_0^2=\|w_h^n-P^nw_h^{n-1}\|_0^2+\|(I-P^n)w_h^{n-1}\|_0^2. \tag*{(58)} ∥ w h n − w h n − 1 ∥ 0 2 = ∥ w h n − P n w h n − 1 ∥ 0 2 + ∥ ( I − P n ) w h n − 1 ∥ 0 2 . (58) Moreover,
max n ∥ w h n ∥ 0 2 + ∑ n τ n ∥ w h n ∥ U 2 + ∑ n ∥ w h n − w h n − 1 ∥ 0 2 ≤ C D ω 2 . (59) \max_n\|w_h^n\|_0^2+\sum_n\tau_n\|w_h^n\|_U^2+\sum_n\|w_h^n-w_h^{n-1}\|_0^2\le CD_\omega^2. \tag*{(59)} n max ∥ w h n ∥ 0 2 + n ∑ τ n ∥ w h n ∥ U 2 + n ∑ ∥ w h n − w h n − 1 ∥ 0 2 ≤ C D ω 2 . (59) Let U h ( t ) = u h n U_h(t)=u_h^n U h ( t ) = u h n W h ( t ) = w h n W_h(t)=w_h^n W h ( t ) = w h n and ω τ ( t ) = ω n \omega_\tau(t)=\omega_n ω τ ( t ) = ω n on ( t n − 1 , t n ] (t_{n-1},t_n] ( t n − 1 , t n ] Then
∥ γ Γ U h − ω τ r ∥ L 2 ( 0 , T ; L 2 ( Γ ) ) ≤ C ( ε ˉ + ℓ ˉ ) D ω , (60) \|\gamma_\Gamma U_h-\omega_\tau r\|_{L^2(0,T;L^2(\Gamma))}\le C(\sqrt{\bar\varepsilon}+\sqrt{\bar\ell})D_\omega, \tag*{(60)} ∥ γ Γ U h − ω τ r ∥ L 2 ( 0 , T ; L 2 ( Γ )) ≤ C ( ε ˉ + ℓ ˉ ) D ω , (60) ∥ γ Γ U h − ω r ∥ L 2 ( 0 , T ; L 2 ( Γ ) ) ≤ C ( ε ˉ + ℓ ˉ ) D ω + τ max ∥ r ∥ L 2 ( Γ ) ∥ ω ′ ∥ L 2 ( 0 , T ) . (61) \|\gamma_\Gamma U_h-\omega r\|_{L^2(0,T;L^2(\Gamma))}\le C(\sqrt{\bar\varepsilon}+\sqrt{\bar\ell})D_\omega+\tau_{\max}\|r\|_{L^2(\Gamma)}\|\omega'\|_{L^2(0,T)}. \tag*{(61)} ∥ γ Γ U h − ω r ∥ L 2 ( 0 , T ; L 2 ( Γ )) ≤ C ( ε ˉ + ℓ ˉ ) D ω + τ m a x ∥ r ∥ L 2 ( Γ ) ∥ ω ′ ∥ L 2 ( 0 , T ) . (61) In particular there is no spatial lifting-error term in these trace bounds.
Proof. Test (55) with w h n w_h^n w h n (57) , and the two components in (58) are orthogonal. The lift estimate gives a uniform bound on ∥ g h ∥ U \|g_h\|_U ∥ g h ∥ U ∥ g h ∥ 0 \|g_h\|_0 ∥ g h ∥ 0
( ∑ n τ n ∥ F h n ∥ U ′ 2 ) 1 / 2 ≤ ∥ f ∥ L 2 ( 0 , T ; U ′ ) + C P ∥ g h ∥ 0 ∥ ω ′ ∥ L 2 ( 0 , T ) + ν T ∥ g h ∥ U ∥ ω ∥ L ∞ ( 0 , T ) . \left(\sum_n\tau_n\|F_h^n\|_{U'}^2\right)^{1/2}\le\|f\|_{L^2(0,T;U')}+C_P\|g_h\|_0\|\omega'\|_{L^2(0,T)}+\nu\sqrt{T}\|g_h\|_U\|\omega\|_{L^\infty(0,T)}. ( n ∑ τ n ∥ F h n ∥ U ′ 2 ) 1/2 ≤ ∥ f ∥ L 2 ( 0 , T ; U ′ ) + C P ∥ g h ∥ 0 ∥ ω ′ ∥ L 2 ( 0 , T ) + ν T ∥ g h ∥ U ∥ ω ∥ L ∞ ( 0 , T ) . The initial norm is at most ∥ w 0 ∥ 0 + C g h 2 ∣ ω 0 ∣ \|w_0\|_0+C_gh^2|\omega_0| ∥ w 0 ∥ 0 + C g h 2 ∣ ω 0 ∣ (59) . Since γ Γ U h − ω τ r = γ Γ W h \gamma_\Gamma U_h-\omega_\tau r=\gamma_\Gamma W_h γ Γ U h − ω τ r = γ Γ W h (60) . Finally the fundamental theorem of calculus and Cauchy–Schwarz give
∥ ω τ − ω ∥ L 2 ( 0 , T ) ≤ τ max ∥ ω ′ ∥ L 2 ( 0 , T ) , ∥ ω τ − ω ∥ L ∞ ( 0 , T ) ≤ τ max ∥ ω ′ ∥ L 2 ( 0 , T ) . (62) \|\omega_\tau-\omega\|_{L^2(0,T)}\le\tau_{\max}\|\omega'\|_{L^2(0,T)},\qquad\|\omega_\tau-\omega\|_{L^\infty(0,T)}\le\sqrt{\tau_{\max}}\|\omega'\|_{L^2(0,T)}. \tag*{(62)} ∥ ω τ − ω ∥ L 2 ( 0 , T ) ≤ τ m a x ∥ ω ′ ∥ L 2 ( 0 , T ) , ∥ ω τ − ω ∥ L ∞ ( 0 , T ) ≤ τ m a x ∥ ω ′ ∥ L 2 ( 0 , T ) . (62) The triangle inequality proves (61) .
Proposition 8.4 (Background kinetic energy and constraint work). Set G h n = ω n g h G_h^n=\omega_n g_h G h n = ω n g h
W Γ n = ( ( u h n − u h n − 1 ) / τ n , G h n ) + a ( u h n , G h n ) − ⟨ f n , G h n ⟩ . (63) \mathcal{W}_{\Gamma}^n=\left((u_h^n-u_h^{n-1})/\tau_n,G_h^n\right)+a(u_h^n,G_h^n)-\langle f^n,G_h^n\rangle. \tag*{(63)} W Γ n = ( ( u h n − u h n − 1 ) / τ n , G h n ) + a ( u h n , G h n ) − ⟨ f n , G h n ⟩ . (63) Then the physical-velocity variable satisfies the exact background-domain identity
∥ u h n ∥ 0 2 − ∥ u h n − 1 ∥ 0 2 + ∥ u h n − u h n − 1 ∥ 0 2 + 2 τ n a ( u h n , u h n ) = 2 τ n ⟨ f n , u h n ⟩ + 2 τ n ν W Γ n . (64) \lVert u_h^n\rVert_0^2-\lVert u_h^{n-1}\rVert_0^2+\lVert u_h^n-u_h^{n-1}\rVert_0^2+2\tau_n a(u_h^n,u_h^n)
=2\tau_n\langle f^n,u_h^n\rangle+2\tau_n\nu\mathcal{W}_{\Gamma}^n. \tag*{(64)} ∥ u h n ∥ 0 2 − ∥ u h n − 1 ∥ 0 2 + ∥ u h n − u h n − 1 ∥ 0 2 + 2 τ n a ( u h n , u h n ) = 2 τ n ⟨ f n , u h n ⟩ + 2 τ n ν W Γ n . (64) The value of W Γ n \mathcal{W}_{\Gamma}^n W Γ n G h n G_h^n G h n
( ( u h n − u h n − 1 ) / τ n , v h ) + a ( u h n , v h ) + b ϵ n ( v h , λ h n ) = ⟨ f n , v h ⟩ ( v h ∈ X h , A n ) , (65) \left((u_h^n-u_h^{n-1})/\tau_n,v_h\right)+a(u_h^n,v_h)+b_{\epsilon_n}(v_h,\lambda_h^n)=\langle f^n,v_h\rangle\quad(v_h\in X_{h,A^n}), \tag*{(65)} ( ( u h n − u h n − 1 ) / τ n , v h ) + a ( u h n , v h ) + b ϵ n ( v h , λ h n ) = ⟨ f n , v h ⟩ ( v h ∈ X h , A n ) , (65) then W Γ n = − b ϵ n ( G h n , λ h n ) \mathcal{W}_{\Gamma}^n=-b_{\epsilon_n}(G_h^n,\lambda_h^n) W Γ n = − b ϵ n ( G h n , λ h n )
Proof. Let
R n ( v ) = ( ( u h n − u h n − 1 ) / τ n , v ) + a ( u h n , v ) − ⟨ f n , v ⟩ . R_n(v)=\left((u_h^n-u_h^{n-1})/\tau_n,v\right)+a(u_h^n,v)-\langle f^n,v\rangle. R n ( v ) = ( ( u h n − u h n − 1 ) / τ n , v ) + a ( u h n , v ) − ⟨ f n , v ⟩ . The centered scheme and projection orthogonality give R n ( v h ) = 0 R_n(v_h)=0 R n ( v h ) = 0 v h ∈ Z n v_h\in Z^n v h ∈ Z n u h n − G h n = w h n ∈ Z n u_h^n-G_h^n=w_h^n\in Z^n u h n − G h n = w h n ∈ Z n R n ( u h n ) = R n ( G h n ) = W Γ n R_n(u_h^n)=R_n(G_h^n)=\mathcal{W}_{\Gamma}^n R n ( u h n ) = R n ( G h n ) = W Γ n 2 τ n 2\tau_n 2 τ n (64) . Any two lifts in the current affine space differ by an element of Z n Z^n Z n R n R_n R n R n ∣ Z n = 0 R_n|_{Z^n}=0 R n ∣ Z n = 0 − R n -R_n − R n (65) exists, without a row-rank assumption. Testing that equation with G h n G_h^n G h n X h , A n X_{h,A^n} X h , A n G h n G_h^n G h n W Γ n \mathcal{W}_{\Gamma}^n W Γ n
Reduction to the homogeneous convergence theorem
The discrete load in (56) is not exactly the time average of (52) . The next estimate controls this difference independently of the mesh history.
Lemma 8.5 (Lifting load perturbation). Let F ‾ n \overline{F}^n F n F F F n n n E h n = F h n − F ‾ n \mathcal{E}_h^n=F_h^n-\overline{F}^n E h n = F h n − F n
⟨ E h n , v ⟩ = − δ ω n ( g h − g , v ) − ω n a ( g h − g , v ) − ( ω n − ω ‾ n ) a ( g , v ) , (66) \langle\mathcal{E}_h^n,v\rangle=-\delta\omega_n(g_h-g,v)-\omega_n a(g_h-g,v)-(\omega_n-\overline{\omega}_n)a(g,v), \tag*{(66)} ⟨ E h n , v ⟩ = − δ ω n ( g h − g , v ) − ω n a ( g h − g , v ) − ( ω n − ω n ) a ( g , v ) , (66) ( ∑ n τ n ∥ E h n ∥ U ′ 2 ) 1 / 2 ≤ C [ h 2 ∥ ω ′ ∥ L 2 ( 0 , T ) + ∥ ω ∥ L ∞ ( 0 , T ) ] + τ max ∥ ω ′ ∥ L 2 ( 0 , T ) . (67) \left(\sum_n\tau_n\lVert\mathcal{E}_h^n\rVert_{U'}^2\right)^{1/2}
\le C\left[h^2\left\lVert\omega'\right\rVert_{L^2(0,T)}+\lVert\omega\rVert_{L^\infty(0,T)}\right]+\tau_{\max}\left\lVert\omega'\right\rVert_{L^2(0,T)}. \tag*{(67)} ( n ∑ τ n ∥ E h n ∥ U ′ 2 ) 1/2 ≤ C [ h 2 ∥ ω ′ ∥ L 2 ( 0 , T ) + ∥ ω ∥ L ∞ ( 0 , T ) ] + τ m a x ∥ ω ′ ∥ L 2 ( 0 , T ) . (67) Proof. Time averaging (52) and subtracting gives (66) . Its first two terms are bounded in U ′ U' U ′ C P h 2 C g ∣ δ ω n ∣ C_P h^2 C_g|\delta\omega_n| C P h 2 C g ∣ δ ω n ∣ ν C g h 2 ∣ ω n ∣ \nu C_g h^2|\omega_n| ν C g h 2 ∣ ω n ∣
∑ n τ n ∣ ω n − ω ‾ n ∣ 2 ≤ τ max 2 ∥ ω ′ ∥ L 2 ( 0 , T ) 2 , \sum_n\tau_n|\omega_n-\overline{\omega}_n|^2\le\tau_{\max}^2\lVert\omega'\rVert_{L^2(0,T)}^2, n ∑ τ n ∣ ω n − ω n ∣ 2 ≤ τ m a x 2 ∥ ω ′ ∥ L 2 ( 0 , T ) 2 , which follows interval by interval from the fundamental theorem of calculus, as in (46) . Jensen’s inequality for δ ω n \delta\omega_n δ ω n ∑ n τ n ∣ ω n ∣ 2 ≤ T ∥ ω ∥ L ∞ 2 \sum_n\tau_n|\omega_n|^2\le T\lVert\omega\rVert_{L^\infty}^2 ∑ n τ n ∣ ω n ∣ 2 ≤ T ∥ ω ∥ L ∞ 2
Theorem 8.6 (Strong convergence for prescribed fixed-disk rotation). Let f ∈ L 2 ( 0 , T ; U ′ ) f \in L^2(0,T;U') f ∈ L 2 ( 0 , T ; U ′ ) ω ∈ H 1 ( 0 , T ) \omega\in H^1(0,T) ω ∈ H 1 ( 0 , T ) w 0 ∈ H w_0 \in H w 0 ∈ H (54) . Suppose (29) holds and, for all sufficiently large m m m (48) holds. The admissible patch, kernel width, and surface panel partition may change at every step. Then the physical velocities satisfy
∥ U m − u ∥ L 2 ( 0 , T ; U ) ⟶ 0 , sup 0 ≤ t ≤ T ∥ U m ( t ) − u ( t ) ∥ 0 ⟶ 0. (68) \lVert U_m-u\rVert_{L^2(0,T;U)}\longrightarrow0,\qquad\sup_{0\le t\le T}\lVert U_m(t)-u(t)\rVert_0\longrightarrow0. \tag*{(68)} ∥ U m − u ∥ L 2 ( 0 , T ; U ) ⟶ 0 , 0 ≤ t ≤ T sup ∥ U m ( t ) − u ( t ) ∥ 0 ⟶ 0. (68) Also the sum of the two centered losses in (58) tends to zero. None of these statements requires a regular interface multiplier or a restriction on the number of mesh changes.
Proof. For each run let y h n ∈ Z n y_h^n\in Z^n y h n ∈ Z n (23) with initial value P 0 w 0 P^0w_0 P 0 w 0 F ‾ n \overline{F}^n F n ( w 0 , F ) (w_0,\mathcal{F}) ( w 0 , F ) Y m Y_m Y m w w w (30) , and its total loss tends to zero.
Set e h n = w h n − y h n e_h^n=w_h^n-y_h^n e h n = w h n − y h n
( e h n − P n e h n − 1 τ n , v h ) + a ( e h n , v h ) = ⟨ E h n , v h ⟩ , e h 0 = ω 0 P 0 ( g − g h ) . \left(\frac{e_h^n-P^ne_h^{n-1}}{\tau_n},v_h\right)+a(e_h^n,v_h)=\langle\mathcal{E}_h^n,v_h\rangle,\qquad e_h^0=\omega_0P^0(g-g_h). ( τ n e h n − P n e h n − 1 , v h ) + a ( e h n , v h ) = ⟨ E h n , v h ⟩ , e h 0 = ω 0 P 0 ( g − g h ) . The same energy argument as in Proposition 5.1 yields
max n ∥ e h n ∥ 0 2 + ∑ n τ n ∥ e h n ∥ U 2 + ∑ n ∥ e h n − e h n − 1 ∥ 0 2 ≤ C ν ( ∥ e h 0 ∥ 0 2 + ∑ n τ n ∥ E h n ∥ U ′ 2 ) . (69) \max_n\lVert e_h^n\rVert_0^2+\sum_n\tau_n\lVert e_h^n\rVert_U^2+\sum_n\lVert e_h^n-e_h^{n-1}\rVert_0^2\le C_\nu\left(\lVert e_h^0\rVert_0^2+\sum_n\tau_n\lVert\mathcal{E}_h^n\rVert_{U'}^2\right). \tag*{(69)} n max ∥ e h n ∥ 0 2 + n ∑ τ n ∥ e h n ∥ U 2 + n ∑ ∥ e h n − e h n − 1 ∥ 0 2 ≤ C ν ( ∥ e h 0 ∥ 0 2 + n ∑ τ n ∥ E h n ∥ U ′ 2 ) . (69) The initial error is at most C g h 2 ∣ ω 0 ∣ C_g h^2|\omega_0| C g h 2 ∣ ω 0 ∣ W m → w W_m\to w W m → w L 2 ( 0 , T ; U ) L^2(0,T;U) L 2 ( 0 , T ; U )
∑ n ∥ w h n − w h n − 1 ∥ 0 2 ≤ 2 ∑ n ∥ y h n − y h n − 1 ∥ 0 2 + 2 ∑ n ∥ e h n − e h n − 1 ∥ 0 2 ⟶ 0 , \sum_n\lVert w_h^n-w_h^{n-1}\rVert_0^2\le2\sum_n\lVert y_h^n-y_h^{n-1}\rVert_0^2+2\sum_n\lVert e_h^n-e_h^{n-1}\rVert_0^2\longrightarrow0, n ∑ ∥ w h n − w h n − 1 ∥ 0 2 ≤ 2 n ∑ ∥ y h n − y h n − 1 ∥ 0 2 + 2 n ∑ ∥ e h n − e h n − 1 ∥ 0 2 ⟶ 0 , which gives the centered loss assertion by (58) .
Finally U m = W m + ω τ m g h m U_m=W_m+\omega_{\tau_m}g_{h_m} U m = W m + ω τ m g h m u = w + ω g u=w+\omega g u = w + ω g
∥ ω τ m g h m − ω g ∥ L 2 ( 0 , T ; U ) ≤ T ∥ ω ∥ L ∞ ∥ g h m − g ∥ U + ∥ ω τ m − ω ∥ L 2 ( 0 , T ) ∥ g ∥ U , \lVert\omega_{\tau_m}g_{h_m}-\omega g\rVert_{L^2(0,T;U)}
\le\sqrt{T}\lVert\omega\rVert_{L^\infty}\lVert g_{h_m}-g\rVert_U+\lVert\omega_{\tau_m}-\omega\rVert_{L^2(0,T)}\lVert g\rVert_U, ∥ ω τ m g h m − ω g ∥ L 2 ( 0 , T ; U ) ≤ T ∥ ω ∥ L ∞ ∥ g h m − g ∥ U + ∥ ω τ m − ω ∥ L 2 ( 0 , T ) ∥ g ∥ U , and its supremum bulk norm is bounded by
∥ ω ∥ L ∞ ∥ g h m − g ∥ 0 + ∥ ω τ m − ω ∥ L ∞ ( 0 , T ) ∥ g ∥ 0 . \lVert\omega\rVert_{L^\infty}\lVert g_{h_m}-g\rVert_0+\lVert\omega_{\tau_m}-\omega\rVert_{L^\infty(0,T)}\lVert g\rVert_0. ∥ ω ∥ L ∞ ∥ g h m − g ∥ 0 + ∥ ω τ m − ω ∥ L ∞ ( 0 , T ) ∥ g ∥ 0 . Equations (50) and (62) show that both expressions tend to zero. This proves (68) , including time zero with the specified initialization.
Quantitative velocity estimate
Suppose in addition that w = u − ω g ∈ H 1 ( 0 , T ; Y ) ∩ H 1 ( 0 , T ; V ) w=u-\omega g\in H^1(0,T;Y)\cap H^1(0,T;V) w = u − ω g ∈ H 1 ( 0 , T ; Y ) ∩ H 1 ( 0 , T ; V ) λ ∈ L 2 ( 0 , T ; L 2 ( Γ ) 2 ) \lambda\in L^2(0,T;L^2(\Gamma)^2) λ ∈ L 2 ( 0 , T ; L 2 ( Γ ) 2 )
( ∂ t u , v ) + a ( u , v ) + ∫ Γ λ ⋅ γ Γ v d s = ⟨ f , v ⟩ ( v ∈ U ) (70) (\partial_tu,v)+a(u,v)+\int_\Gamma\lambda\cdot\gamma_\Gamma v\,\mathrm{d}s=\langle f,v\rangle\qquad(v\in U) \tag*{(70)} ( ∂ t u , v ) + a ( u , v ) + ∫ Γ λ ⋅ γ Γ v d s = ⟨ f , v ⟩ ( v ∈ U ) (70) almost everywhere. Define
B m ( w , λ ) = h m + ε ‾ m ( ∥ w ∥ L ∞ ( 0 , T ; Y ) + ∥ ∂ t w ∥ L 2 ( 0 , T ; Y ) ) + ( ε ‾ m + ℓ ‾ m ) ∥ λ ∥ L 2 ( 0 , T ; L 2 ( Γ ) ) + τ max , m ∥ ∂ t w ∥ L 2 ( 0 , T ; U ) . \mathcal{B}_m(w,\lambda)=\sqrt{h_m+\overline{\varepsilon}_m}\left(\lVert w\rVert_{L^\infty(0,T;Y)}+\lVert\partial_tw\rVert_{L^2(0,T;Y)}\right)
+(\sqrt{\overline{\varepsilon}_m}+\sqrt{\overline{\ell}_m})\lVert\lambda\rVert_{L^2(0,T;L^2(\Gamma))}
+\tau_{\max,m}\lVert\partial_tw\rVert_{L^2(0,T;U)}. B m ( w , λ ) = h m + ε m ( ∥ w ∥ L ∞ ( 0 , T ; Y ) + ∥ ∂ t w ∥ L 2 ( 0 , T ; Y ) ) + ( ε m + ℓ m ) ∥ λ ∥ L 2 ( 0 , T ; L 2 ( Γ )) + τ m a x , m ∥ ∂ t w ∥ L 2 ( 0 , T ; U ) . Theorem 8.7 (Energy error with a prescribed rotation lift). Under the preceding regularity and geometric assumptions,
max n ∥ u m n − u ( t n ) ∥ 0 + ( ∑ n τ m n ∥ u m n − u ( t n ) ∥ U 2 ) 1 / 2 ≤ C [ B m ( w , λ ) + h m 2 { ∥ ω ∥ L ∞ ( 0 , T ) + ∥ ω ′ ∥ L 2 ( 0 , T ) } + τ max , m ∥ ω ′ ∥ L 2 ( 0 , T ) ] . (71) \max_n \lVert u_m^n-u(t_n)\rVert_0+\left(\sum_n\tau_m^n\lVert u_m^n-u(t_n)\rVert_U^2\right)^{1/2}
\le C\left[\mathcal{B}_m(w,\lambda)+h_m^2\{\lVert\omega\rVert_{L^\infty(0,T)}+\lVert\omega'\rVert_{L^2(0,T)}\}+\tau_{\max,m}\lVert\omega'\rVert_{L^2(0,T)}\right]. \tag*{(71)} n max ∥ u m n − u ( t n ) ∥ 0 + ( n ∑ τ m n ∥ u m n − u ( t n ) ∥ U 2 ) 1/2 ≤ C [ B m ( w , λ ) + h m 2 {∥ ω ∥ L ∞ ( 0 , T ) + ∥ ω ′ ∥ L 2 ( 0 , T ) } + τ m a x , m ∥ ω ′ ∥ L 2 ( 0 , T ) ] . (71) The constant may depend on the fixed smooth lift, but is independent of the refinement history and of the number of mesh changes.
Proof. Subtracting the exact lift from (70) gives
( ∂ t w , v ) + a ( w , v ) + ∫ Γ λ ⋅ γ Γ v d s = ⟨ F , v ⟩ ( v ∈ U ) . (\partial_t w,v)+a(w,v)+\int_\Gamma\lambda\cdot\gamma_\Gamma v\,\mathrm{d}s=\langle\mathcal{F},v\rangle\qquad(v\in U). ( ∂ t w , v ) + a ( w , v ) + ∫ Γ λ ⋅ γ Γ v d s = ⟨ F , v ⟩ ( v ∈ U ) . Thus Theorem 7.1 applies to the auxiliary velocities y h n y_h^n y h n w w w F \mathcal{F} F λ \lambda λ C B m ( w , λ ) C\mathcal{B}_m(w,\lambda) C B m ( w , λ ) C g h 2 ∣ ω 0 ∣ C_g h^2|\omega_0| C g h 2 ∣ ω 0 ∣ (69) and (67) bound w h n − y h n w_h^n-y_h^n w h n − y h n (71) . At each node,
u h n − u ( t n ) = ( y h n − w ( t n ) ) + ( w h n − y h n ) + ω n ( g h − g ) . u_h^n-u(t_n)=(y_h^n-w(t_n))+(w_h^n-y_h^n)+\omega_n(g_h-g). u h n − u ( t n ) = ( y h n − w ( t n )) + ( w h n − y h n ) + ω n ( g h − g ) . The last term has maximum bulk norm at most C g h 2 ∥ ω ∥ L ∞ C_g h^2\lVert\omega\rVert_{L^\infty} C g h 2 ∥ ω ∥ L ∞ U U U C g h 2 T ∥ ω ∥ L ∞ C_g h^2\sqrt{T}\lVert\omega\rVert_{L^\infty} C g h 2 T ∥ ω ∥ L ∞
The same right-hand side, with a changed constant, controls ∥ U m − u ∥ L 2 ( 0 , T ; U ) \lVert U_m-u\rVert_{L^2(0,T;U)} ∥ U m − u ∥ L 2 ( 0 , T ; U ) τ max , m ∥ ∂ t u ∥ L 2 ( 0 , T ; U ) \tau_{\max,m}\lVert\partial_tu\rVert_{L^2(0,T;U)} τ m a x , m ∥ ∂ t u ∥ L 2 ( 0 , T ; U ) ∂ t u = ∂ t w + ω ′ g \partial_tu=\partial_tw+\omega'g ∂ t u = ∂ t w + ω ′ g O ( τ max , m ) O(\sqrt{\tau_{\max,m}}) O ( τ m a x , m ) H 1 H^1 H 1 (71) does not assert a first-order bound for that reconstruction norm.
Rigid interior interpretation and the scope of the energy law
The affine background-domain problem describes a rigidly rotating interior at the continuum level if the interior data are chosen consistently. Specifically suppose
u 0 ∣ B = ω ( 0 ) r , ⟨ f ( t ) , v ⟩ = ω ′ ( t ) ( r , v ) L 2 ( B ) u_0|_B=\omega(0)r,\qquad\langle f(t),v\rangle=\omega'(t)(r,v)_{L^2(B)} u 0 ∣ B = ω ( 0 ) r , ⟨ f ( t ) , v ⟩ = ω ′ ( t ) ( r , v ) L 2 ( B ) for every divergence-free v ∈ H 0 1 ( B ) 2 v\in H_0^1(B)^2 v ∈ H 0 1 ( B ) 2 Ω \Omega Ω ω ′ r \omega' r ω ′ r B B B u ∣ B = ω r u|_B=\omega r u ∣ B = ω r w = u − ω g w=u-\omega g w = u − ω g g = r g=r g = r B B B r r r
a B ( r , v ) = ν ∫ B ∇ r : ∇ v = 0 ( v ∈ H 0 1 ( B ) 2 ) . a_B(r,v)=\nu\int_B\nabla r:\nabla v=0\qquad(v\in H_0^1(B)^2). a B ( r , v ) = ν ∫ B ∇ r : ∇ v = 0 ( v ∈ H 0 1 ( B ) 2 ) . Thus w ∣ B w|_B w ∣ B w ∣ B = 0 w|_B=0 w ∣ B = 0 B B B
Equation (57) is an exact energy law for the centered velocity, whereas (64) identifies the work of the prescribed discrete constraint in the background kinetic-energy balance. The latter is not automatically an exterior-fluid torque law. The interior kinetic energy and its matching body-force work must be identified when passing from the background domain to the physical fluid, and a physical torque requires the appropriate exterior stress and its approximation. For example, a rigid rotation has a B ( ω r , ω r ) = 2 ν ∣ B ∣ ω 2 a_B(\omega r,\omega r)=2\nu|B|\omega^2 a B ( ω r , ω r ) = 2 ν ∣ B ∣ ω 2 u ∈ H 0 1 ( Ω ) 2 u\in H^1_0(\Omega)^2 u ∈ H 0 1 ( Ω ) 2
ν ∫ Ω ∣ ∇ u ∣ 2 = 2 ν ∫ Ω ∣ D ( u ) ∣ 2 , D ( u ) = 1 2 ( ∇ u + ∇ u T ) . \nu\int_{\Omega}|\nabla u|^2=2\nu\int_{\Omega}|D(u)|^2,\qquad D(u)=\frac{1}{2}(\nabla u+\nabla u^{\mathsf{T}}). ν ∫ Ω ∣∇ u ∣ 2 = 2 ν ∫ Ω ∣ D ( u ) ∣ 2 , D ( u ) = 2 1 ( ∇ u + ∇ u T ) . If u = ω r u=\omega r u = ω r B B B λ \lambda λ (70) includes the interface balance between the two background subdomains; it is not identified here with exterior physical traction or a convergent torque. No multiplier estimate is needed for the centered energy or qualitative convergence.
The result covers prescribed rotation of a disk whose geometric interface is fixed. It does not cover translation of the disk, rotation of a nonsymmetric body that changes the interface, an angular velocity determined by a coupled rigid-body equation, or unrestricted changes of the lifting function during a run. Those cases would require additional arguments beyond the two common families constructed here.
Manufactured-solution verification
We give a small numerical verification study of the specified conforming method. It checks an implementation against an exact nonstationary solution, following the manufactured-solution principle [29 ] . The first five subsections concern a stationary body; § 9.6 then verifies prescribed rotation of the same fixed geometric interface. The tests address bulk error, the physical interface trace, and mesh transfer; they are not benchmarks for a conventional MAC implementation.
An exact solution with nonzero interface shear
Let B B B R = 0.2 R=0.2 R = 0.2 c = ( 1 / 2 , 1 / 2 ) c=(1/2,1/2) c = ( 1/2 , 1/2 ) Ω = ( 0 , 1 ) 2 \Omega=(0,1)^2 Ω = ( 0 , 1 ) 2
X = x − c 1 , Y = y − c 2 , s = X 2 + Y 2 − R 2 , Q = x 2 ( 1 − x ) 2 y 2 ( 1 − y ) 2 , s + = max ( s , 0 ) . X=x-c_1,\qquad Y=y-c_2,\qquad s=X^2+Y^2-R^2,\qquad Q=x^2(1-x)^2y^2(1-y)^2,\qquad s_+=\max(s,0). X = x − c 1 , Y = y − c 2 , s = X 2 + Y 2 − R 2 , Q = x 2 ( 1 − x ) 2 y 2 ( 1 − y ) 2 , s + = max ( s , 0 ) . With A = 1000 A=1000 A = 1000 ν = 0.1 \nu=0.1 ν = 0.1 T = 0.5 T=0.5 T = 0.5
ψ 0 = A Q s + 2 , U = curl ψ 0 , u ( x , t ) = a ( t ) U ( x ) , a ( t ) = 1 − e − t , p = 0. (72) \psi_0=AQs_+^2,\qquad U=\operatorname{curl}\psi_0,\qquad u(x,t)=a(t)U(x),\qquad a(t)=1-e^{-t},\qquad p=0. \tag*{(72)} ψ 0 = A Q s + 2 , U = curl ψ 0 , u ( x , t ) = a ( t ) U ( x ) , a ( t ) = 1 − e − t , p = 0. (72) The velocity is divergence free, vanishes inside B B B Γ \Gamma Γ Γ \Gamma Γ H 1 H^1 H 1 Theorem 7.1 .
The ordinary bulk force is obtained by differentiating the exterior polynomial, not from a discrete residual:
f ( x , t ) = e − t U − ν ( 1 − e − t ) Δ U ( x ∈ F ) , f = 0 ( x ∈ B ) . (73) f(x,t)=e^{-t}U-\nu(1-e^{-t})\Delta U\quad(x\in F),\qquad f=0\quad(x\in B). \tag*{(73)} f ( x , t ) = e − t U − ν ( 1 − e − t ) Δ U ( x ∈ F ) , f = 0 ( x ∈ B ) . (73) For explicit evaluation, set
L = s 2 Δ Q + 8 s ( X Q x + Y Q y ) + ( 16 s + 8 R 2 ) Q . L=s^2\Delta Q+8s(XQ_x+YQ_y)+(16s+8R^2)Q. L = s 2 Δ Q + 8 s ( X Q x + Y Q y ) + ( 16 s + 8 R 2 ) Q . In F F F Δ U = A ( L y , − L x ) \Delta U=A(L_y,-L_x) Δ U = A ( L y , − L x ) n = ( cos θ , sin θ ) n=(\cos\theta,\sin\theta) n = ( cos θ , sin θ ) (39) is
λ = ν ∂ n u = 8 ν A a ( t ) R 2 Q ( sin θ , − cos θ ) on Γ . (74) \lambda=\nu\partial_n u=8\nu Aa(t)R^2Q(\sin\theta,-\cos\theta)\quad\text{on }\Gamma. \tag*{(74)} λ = ν ∂ n u = 8 ν A a ( t ) R 2 Q ( sin θ , − cos θ ) on Γ. (74) Figure 1. Left: one enrichment pattern for M = 16 M=16 M = 16 T = 0.5 T=0.5 T = 0.5
Thus the interface shear is nonzero. The solution does not vanish in an open collar of the interface, and its first spatial derivatives jump across Γ \Gamma Γ (73) ; the constraint supplies the interface reaction. Independent symbolic differentiation and integration by parts check the forcing and the sign in (74) .
The load is averaged exactly in time. On [ t n − 1 , t n ] [t_{n-1},t_n] [ t n − 1 , t n ] d n = ( e − t n − 1 − e − t n ) / τ n d_n=(e^{-t_{n-1}}-e^{-t_n})/\tau_n d n = ( e − t n − 1 − e − t n ) / τ n f ˉ n = d n U − ν ( 1 − d n ) Δ U \bar{f}^{n}=d_nU-\nu(1-d_n)\Delta U f ˉ n = d n U − ν ( 1 − d n ) Δ U F F F expm1.
Discrete realization and numerical accuracy
The implementation uses exactly the spaces of § 4 . All coarse and local enrichment functions are represented in one fixed hierarchical master basis. If z K z_K z K
R h = { K : ∥ z K − c ∥ − R < 1.5 h } , \mathcal{R}_h=\{K:\lVert z_K-c\rVert-R<1.5h\}, R h = { K : ∥ z K − c ∥ − R < 1.5 h } , and alternate the two patches
A 0 = { K ∈ R h : ( z K ) 1 < c 1 + h / 4 } , A 1 = { K ∈ R h : ( z K ) 1 > c 1 − h / 4 } . \mathcal{A}_0=\{K\in\mathcal{R}_h:(z_K)_1<c_1+h/4\},\qquad\mathcal{A}_1=\{K\in\mathcal{R}_h:(z_K)_1>c_1-h/4\}. A 0 = { K ∈ R h : ( z K ) 1 < c 1 + h /4 } , A 1 = { K ∈ R h : ( z K ) 1 > c 1 − h /4 } . Both retain the complete coarse space. Their boundaries intersect the regularization region; the two patterns are not nested.
For the regularizer, let
k ( z ) = exp [ − 1 / ( 1 − z 2 ) ] 1 ∣ z ∣ < 1 ∫ − 1 1 exp [ − 1 / ( 1 − r 2 ) ] d r , ρ ε ( x , y ) = 2 ε 2 k ( 2 x ε ) k ( 2 y ε ) . k(z)=\frac{\exp[-1/(1-z^2)]\,\mathbb{1}_{|z|<1}}{\int_{-1}^{1}\exp[-1/(1-r^2)]\,\mathrm{d}r},\qquad\rho_\varepsilon(x,y)=\frac{2}{\varepsilon^2}k\left(\frac{\sqrt{2}x}{\varepsilon}\right)k\left(\frac{\sqrt{2}y}{\varepsilon}\right). k ( z ) = ∫ − 1 1 exp [ − 1/ ( 1 − r 2 )] d r exp [ − 1/ ( 1 − z 2 )] 1 ∣ z ∣ < 1 , ρ ε ( x , y ) = ε 2 2 k ( ε 2 x ) k ( ε 2 y ) . This is a smooth even kernel supported in a square contained in the radius-ε \varepsilon ε ε = 0.75 h \varepsilon=0.75h ε = 0.75 h P = ⌈ 2 π R / h ⌉ P=\lceil2\pi R/h\rceil P = ⌈ 2 π R / h ⌉ ℓ = 2 π R / P ≤ h \ell=2\pi R/P\le h ℓ = 2 π R / P ≤ h
Bulk mass and viscosity matrices use four-point Gauss rules per coordinate on the h / 2 h/2 h /2
The force and error integrals respect the exact circular cut: cells are split into exterior intervals using the circle equation, with additional breaks at circle–cell-edge intersections. Uncut cells use order 10, and cut intervals use order 24. The convolution integral is split at every master-grid line before using order 16 in each coordinate; surface moments use order 10 on each panel. Physical trace integration is independent of the constraint residual and splits the circle into arcs at all master-grid crossings.
The constraints are solved through an orthonormal basis for their row space and a Schur complement. The singular-value cutoff is 10 − 12 10^{-12} 1 0 − 12 6.47 × 10 − 4 6.47 \times10^{-4} 6.47 × 1 0 − 4 10 − 10 10^{-10} 1 0 − 10 10 − 14 10^{-14} 1 0 − 14 M = 16 M=16 M = 16
The step right-hand side uses the cross-mesh mass pairing with the old velocity. By orthogonality, this is equivalent to the constrained L 2 L^2 L 2 (23) ; it does not use nodal interpolation. All energy diagnostics use the same physical mass matrix. The numerical surface integrals are approximations to the exact moments assumed in the theory. Consequently, the computations are a controlled approximate realization, and the quadrature checks below do not replace a quadrature-consistency theorem.
Velocity convergence and physical slip
Let U h U_h U h
E 0 = ∥ u h N − u ( T ) ∥ L 2 ( Ω ) ∥ u ( T ) ∥ L 2 ( Ω ) , E 1 = ∥ U h − u ∥ L 2 ( 0 , T ; U ) ∥ u ∥ L 2 ( 0 , T ; U ) , D Γ = ∥ γ Γ U h ∥ L 2 ( 0 , T ; L 2 ( Γ ) ) . \begin{aligned}
E_0 &= \frac{\lVert u_h^N-u(T)\rVert_{L^2(\Omega)}}{\lVert u(T)\rVert_{L^2(\Omega)}}, \qquad
E_1 = \frac{\lVert U_h-u\rVert_{L^2(0,T;U)}}{\lVert u\rVert_{L^2(0,T;U)}},\\
D_\Gamma&= \lVert\gamma_\Gamma U_h\rVert_{L^2(0,T;L^2(\Gamma))}.
\end{aligned} E 0 D Γ = ∥ u ( T ) ∥ L 2 ( Ω ) ∥ u h N − u ( T ) ∥ L 2 ( Ω ) , E 1 = ∥ u ∥ L 2 ( 0 , T ; U ) ∥ U h − u ∥ L 2 ( 0 , T ; U ) , = ∥ γ Γ U h ∥ L 2 ( 0 , T ; L 2 ( Γ )) . The volume norms include the body interior. In E 1 E_1 E 1
Table 1 uses N = 400 N=400 N = 400 h h h ε \varepsilon ε ℓ \ell ℓ Theorem 6.1 , nor do they establish a sharp spatial rate. In particular, a rate faster than the half-order upper bound is not a contradiction of Theorem 7.1 .
Table 1. Velocity errors and physical no-slip defect, N = 400 N=400 N = 400 E 0 , E 1 E_0,E_1 E 0 , E 1 D Γ D_\Gamma D Γ
Mesh switching, position sensitivity, and energy
At M = 32 M=32 M = 32 N = 400 N=400 N = 400 ε \varepsilon ε Table 2 shows that the final relative velocity error stays between 2.31271% and 2.31398%. The accumulated projection losses are positive: the switches genuinely discard resolved velocity components. The result supports robustness for these histories; it does not assert identical errors for arbitrary histories or replace the event-count-independent theorem.
Table 2. Switching frequency at fixed M = 32 M = 32 M = 32 N = 400 N = 400 N = 400 ∑ n ∥ ( I − P n ) u n − 1 ∥ 0 2 \sum_n \lVert(I - P^n)u^{n-1}\rVert_0^2 ∑ n ∥( I − P n ) u n − 1 ∥ 0 2
Figure 2. The data in Table 1 . Physical slip is evaluated on the sharp circle, rather than inferred from the regularized constraint residual.
At M = 24 M = 24 M = 24 c = ( 1 / 2 + α h , 1 / 2 ) c = (1/2 + \alpha h, 1/2) c = ( 1/2 + α h , 1/2 ) α ∈ { 0 , 1 / 4 , 1 / 2 , 3 / 4 } \alpha\in\{0, 1/4, 1/2, 3/4\} α ∈ { 0 , 1/4 , 1/2 , 3/4 } Table 3 . The manufactured force is recomputed for each center. The errors remain of comparable size, with no anomalous growth in these four alignments. Moving the disk also changes the exact solution relative to the outer walls, so this is a position-sensitivity test, not an isolated test of matrix conditioning.
Table 3. Four fixed interface positions, M = 24 M = 24 M = 24 N = 400 N = 400 N = 400
Across all 26 saved experiment runs, the largest absolute residual of (24) is 1.87 × 10 − 17 1.87 \times10^{-17} 1.87 × 1 0 − 17 (25) is 2.17 × 10 − 19 2.17 \times10^{-19} 2.17 × 1 0 − 19 5.33 × 10 − 17 5.33 \times10^{-17} 5.33 × 1 0 − 17
A separate unforced test uses M = 16 M = 16 M = 16 N = 200 N = 200 N = 200
Figure 3. Left: the fixed-time-step switching experiment. Right: kinetic energy in the separate unforced decay test.
projection of U U U (72) , and a patch change at every step. Its kinetic energy decreases at every step, ending at 1.81 × 10 − 7 1.81 \times10^{-7} 1.81 × 1 0 − 7
Checks against quadrature and temporal contamination
For the M = 16 M = 16 M = 16 N = 200 N = 200 N = 200 ( 16 , 10 , 24 ) (16, 10, 24) ( 16 , 10 , 24 ) ( 24 , 16 , 32 ) (24, 16, 32) ( 24 , 16 , 32 ) 5.67 × 10 − 8 5.67 \times10^{-8} 5.67 × 1 0 − 8 L 2 L^2 L 2 2.17 × 10 − 7 2.17 \times10^{-7} 2.17 × 1 0 − 7 H 1 H^1 H 1 (74) also checks the body-force assembly.
To isolate time discretization, keep the M = 32 M = 32 M = 32 N = 100 , 200 , 400 , 800 N = 100, 200, 400, 800 N = 100 , 200 , 400 , 800 L 2 L^2 L 2 1.9766 × 10 − 3 1.9766 \times10^{-3} 1.9766 × 1 0 − 3 9.8702 × 10 − 4 9.8702 \times10^{-4} 9.8702 × 1 0 − 4 4.9319 × 10 − 4 4.9319 \times10^{-4} 4.9319 × 1 0 − 4 2.0026 2.0026 2.0026 2.0013 2.0013 2.0013 N N N 400 400 400 800 800 800 E 0 E_0 E 0 1.13535 % 1.13535\% 1.13535% 1.12074 % 1.12074\% 1.12074%
The companion provides the solver, exact-solution verifier, independent implementation checks, all 26 parameter files, final coefficient vectors, and diagnostic histories, and scripts that regenerate the tables and figures. The results support the implementation and the stated qualitative trends. They do not prove adaptive optimality, multiplier convergence, or general robustness of other IB discretizations.
Prescribed rotation on fixed geometry
We also impose a nonzero prescribed velocity on the same circular interface. The disk rotates with angular velocity ω ( t ) = a ( t ) = 1 − e − t \omega(t) = a(t) = 1 - e^{-t} ω ( t ) = a ( t ) = 1 − e − t r ( x ) = ( − Y , X ) r(x) = (-Y,X) r ( x ) = ( − Y , X ) σ = X 2 + Y 2 \sigma= X^2 + Y^2 σ = X 2 + Y 2
ψ g = − 1 2 σ χ ( σ ) , g = curl ψ g , \psi_g = -\frac{1}{2}\sigma\chi(\sigma), \qquad g = \operatorname{curl}\psi_g, ψ g = − 2 1 σ χ ( σ ) , g = curl ψ g , χ ( σ ) = 1 − 2772 ∫ 0 η ( σ ) q 5 ( 1 − q ) 5 d q , \chi(\sigma) = 1 - 2772\int_0^{\eta(\sigma)}q^5(1-q)^5\,\mathrm{d}q, χ ( σ ) = 1 − 2772 ∫ 0 η ( σ ) q 5 ( 1 − q ) 5 d q , η ( σ ) = min { 1 , max { 0 , σ − 0.35 2 0.47 2 − 0.35 2 } } . \eta(\sigma) = \min\left\{1,\max\left\{0,\frac{\sigma-0.35^2}{0.47^2-0.35^2}\right\}\right\}. η ( σ ) = min { 1 , max { 0 , 0.4 7 2 − 0.3 5 2 σ − 0.3 5 2 } } . The cutoff is C 5 C^5 C 5 g = r g=r g = r ∣ x − c ∣ ≤ 0.35 \lvert x-c\rvert\le0.35 ∣ x − c ∣ ≤ 0.35 g = 0 g=0 g = 0 ∣ x − c ∣ ≥ 0.47 \lvert x-c\rvert\ge0.47 ∣ x − c ∣ ≥ 0.47 U U U (72) , the exact solution and ordinary bulk force are
u ( x , t ) = a ( t ) ( g + U ) , f = e − t ( g + U ) − ν ( 1 − e − t ) ( Δ g + Δ U ) . (75) u(x,t)=a(t)(g+U),\qquad f=e^{-t}(g+U)-\nu(1-e^{-t})(\Delta g+\Delta U). \tag*{(75)} u ( x , t ) = a ( t ) ( g + U ) , f = e − t ( g + U ) − ν ( 1 − e − t ) ( Δ g + Δ U ) . (75) Here Δ U \Delta U Δ U (73) . In particular, u = a ( t ) r u=a(t)r u = a ( t ) r f = e − t r f=e^{-t}r f = e − t r a U aU a U T = 0.5 T=0.5 T = 0.5 L 2 ( Ω ) L^2(\Omega) L 2 ( Ω ) a g ag a g
Let g h = curl I h ψ g g_h=\operatorname{curl}I_h\psi_g g h = curl I h ψ g R + ε + 2 h < 0.35 R+\varepsilon+\sqrt{2}h<0.35 R + ε + 2 h < 0.35 g h = r g_h=r g h = r w h n = u h n − a ( t n ) g h w_h^n=u_h^n-a(t_n)g_h w h n = u h n − a ( t n ) g h M h M_h M h K h K_h K h ( ∇ v , ∇ z ) (\nabla v,\nabla z) ( ∇ v , ∇ z ) b v b_v b v b Δ v b_{\Delta v} b Δ v
d n ( b U + b g − M h g h ) − ν ( 1 − d n ) ( b Δ U + b Δ g ) − ν a ( t n ) K h g h . (76) d_n(b_U+b_g-M_hg_h)-\nu(1-d_n)(b_{\Delta U}+b_{\Delta g})-\nu a(t_n)K_hg_h. \tag*{(76)} d n ( b U + b g − M h g h ) − ν ( 1 − d n ) ( b Δ U + b Δ g ) − ν a ( t n ) K h g h . (76) The last coefficient is the endpoint value a ( t n ) a(t_n) a ( t n ) g h g_h g h (75) , not manufactured from the discrete matrices. The same exact mass pairing implements transfer of the centered unknown between the two enrichment patterns.
For this extension only, kernel subdivision contains each master-grid cut and its reflection about the evaluation point. The resulting symmetric nonnegative Gaussian weights are normalized to mass one in each coordinate. This ensures affine moment reproduction without negative correction weights. Diagnostics retain the mass defect before normalization, the normalized mass and first-moment defects, and the assembled discrepancy between C h g h C_hg_h C h g h
Table 4 uses M = 16 , 32 , 64 M=16,32,64 M = 16 , 32 , 64 N = 400 N=400 N = 400 E 0 E_0 E 0 E 1 E_1 E 1 (75) , including the body interior. To distinguish boundary error from time reconstruction error, we report both
D Γ , c o n t = ∥ γ Γ U h − a ( t ) r ∥ L 2 ( 0 , T ; L 2 ( Γ ) ) , D Γ , n o d e 2 = ∑ n τ n ∥ γ Γ u h n − a ( t n ) r ∥ L 2 ( Γ ) 2 . D_{\Gamma,\mathrm{cont}}=\left\|\gamma_{\Gamma}U_h-a(t)r\right\|_{L^2(0,T;L^2(\Gamma))},\qquad
D_{\Gamma,\mathrm{node}}^2=\sum_n\tau_n\left\|\gamma_{\Gamma}u_h^n-a(t_n)r\right\|_{L^2(\Gamma)}^2. D Γ , cont = ∥ γ Γ U h − a ( t ) r ∥ L 2 ( 0 , T ; L 2 ( Γ )) , D Γ , node 2 = n ∑ τ n ∥ γ Γ u h n − a ( t n ) r ∥ L 2 ( Γ ) 2 . The latter equals the time-integrated sharp trace of w h n w_h^n w h n r r r a ( t ) a(t) a ( t ) a ( t n ) a(t_n) a ( t n ) N N N
The coarse M = 16 M=16 M = 16 M = 64 M = 64 M = 64 L 2 L^2 L 2 a ( T ) U a(T)U a ( T ) U a ( t ) g a(t)g a ( t ) g
Table 4. Fixed-geometry prescribed rotation, N = 400 N = 400 N = 400
Figure 4. Total-velocity errors and sharp-interface slip for prescribed rotation. The two slip curves distinguish the continuous prescribed motion from its endpoint reconstruction on each time interval.
At fixed M = 32 M = 32 M = 32 L 2 L^2 L 2 M = 16 M = 16 M = 16 5.61 × 10 − 9 5.61 \times10^{-9} 5.61 × 1 0 − 9 L 2 L^2 L 2 8.95 × 10 − 9 8.95 \times10^{-9} 8.95 × 1 0 − 9 H 1 H^1 H 1
Across these eight runs, the largest absolute residual of the centered energy identity is 1.94 × 10 − 17 1.94 \times10^{-17} 1.94 × 1 0 − 17 5.14 × 10 − 16 5.14 \times10^{-16} 5.14 × 1 0 − 16 8.10 × 10 − 15 8.10 \times10^{-15} 8.10 × 1 0 − 15 2.06 × 10 − 8 2.06 \times10^{-8} 2.06 × 1 0 − 8 4.45 × 10 − 16 4.45 \times10^{-16} 4.45 × 1 0 − 16 10 − 18 10^{-18} 1 0 − 18
Figure 5. Left: switching frequency at fixed discretization parameters. Right: absolute residuals of the centered energy identity and the full reconstructed-velocity identity, including constraint work. The latter work is an algebraic constraint functional, not a reported hydrodynamic torque.
What the result does and does not imply
The estimates permit a completely different set of enriched coarse cells at every time step. Neither nesting in time nor a bound on the number of coarsening events is required. Surface panels and regularization widths may change at the same instants. Their influence enters through their maximal widths and through the common collar used in the proof, rather than through a sum of eventwise consistency errors.
This conclusion is conditional on a concrete structural choice, not on arbitrary interpolation: the full coarse space is retained, the method is conforming in a common physical Hilbert space, and transfer is the exact constrained L 2 L^2 L 2 [8 , 10 ] is therefore not contradicted.
The half-order spatial estimate does not quantify the gain obtained by an adaptive marking rule. It is a robust upper bound in terms of the coarse grid and a collar construction. It does not establish optimality, a pointwise velocity rate, a uniform-in-time physical trace rate, pressure convergence, or convergence of the multiplier. The results do not imply that a poorly conditioned multiplier system is harmless for finite-precision computation. They show that velocity convergence can be proved without assuming an inf–sup estimate for that system.
The exact-integration qualification is essential. A marker implementation introduces an additional consistency term, for example
J h , ε q u a d v h − J ε v h , J^{\mathrm{quad}}_{h,\varepsilon} v_h - J_{\varepsilon} v_h, J h , ε quad v h − J ε v h , after an appropriate surface reconstruction. That term is absent from the exact-integral method and has not been estimated here. § 9 checks quadrature sensitivity for the reported computations, without adding a quadrature-consistency theorem. The same theoretical limitation applies to approximate projections, changing mass quadrature, solver tolerances, subcycling, and conventional coarse–fine MAC stencils. Proving bounds for these modifications is a separate extension, not a consequence of replacing their matrices by the notation in this paper.
Prescribed rotation of a fixed disk is covered by the common lifting constructed in § 8 . For general prescribed rigid-body motion, the interface itself moves; a collar of the entire swept interface is generally too large to be a dense approximation device. A time-dependent recovery construction and its transport estimates are therefore needed. Fluid-driven motion also requires consistent momentum and total-energy coupling, including the treatment of fictitious interior fluid. These are future research modules, not consequences asserted by the fixed-geometry theorem.
For example, the prescribed heave and pitch of a plate in Zheng et al. [21 ] change its geometric interface, unlike rotation of a circular disk about its center. That application therefore requires a moving-geometry analysis even before its free-surface and nonlinear-flow effects are considered. Likewise, the flexible stems in He et al. [20 ] require structural dynamics beyond the rigid-body scope of this series.
Formal verification and reproducibility
The accompanying Lean project checks the algebraic projection and discrete energy layer of the analysis. It was compiled with Lean 4.19.0 [30 ] and Mathlib [31 ] at commit c44e0c8 (the full hash is pinned in the companion). The project contains 26 checked theorems in three source modules. The Energy.lean module proves 12 Hilbert-space identities, scalar energy estimates, telescoping statements, and a conditional composition of the two interface bounds. The Projection.lean module proves five results using Mathlib’s actual orthogonal projection onto complete subspaces, rather than treating all projection properties as hypotheses. Finite-dimensional spaces such as Z n \mathbb{Z}^{n} Z n Affine.lean module adds nine checked statements for actual affine projections, the change of lifting, the centered prescribed-rotation equation, and its energy balances.
Table 11.
The final build and the explicit #print axioms checks passed. The listed foundational axioms are only propext, Classical.choice, and Quot.sound; there is no sorryAx and no user-declared axiom. The companion contains the pinned project manifests, source hashes, commands, and compiler logs.
This is not a Lean formalization of Theorems 6.1 , 7.1 , or the prescribed-rotation extension in their entirety. Zero extension, convolution and trace estimates, fractional panel approximation, the finite element and collar construction, solenoidal density, the evolution equation, and passage to the strong limit are proved in the mathematical text. In particular, the Lean theorem that composes the interface bounds does not establish its analytic hypotheses. The verified layer checks an important part of the argument but does not certify originality or replace review of the remaining analysis.
The manuscript and companion were developed primarily by AI agents; the detailed disclosure at the end of this manuscript states the author’s current understanding of the proofs. The manufactured-solution computations in § 9 are implementation checks and numerical illustrations, not formal proofs. The reproducibility package contains the complete LaTeX sources, bibliographic database, Lean sources, numerical solver, raw experiment data, and independent implementation checks. Downloaded dependency caches are excluded. The original 17 Lean theorems remain unchanged; nine affine identities were added for version 0.3. None of these theorems certifies the numerical implementation or the analytic construction of the common lifting.
A kernel obstruction to an unrestricted multiplier assumption
The following elementary calculation motivates avoiding an unproved multiplier stability hypothesis. It is a diagnostic example, not a separate claim of novelty; Fourier analysis of IB kernels and planar boundaries has an established literature [32 ] , and rigid-body mobility conditioning is discussed in Kallemov et al. [7 ] .
On an even periodic row with N N N h h h ϕ ( 0 ) = 1 / 2 \phi(0) = 1/2 ϕ ( 0 ) = 1/2 ϕ ( ± 1 ) = 1 / 4 \phi(\pm1) = 1/4 ϕ ( ± 1 ) = 1/4
( A λ ) j = 1 4 λ j − 1 + 1 2 λ j + 1 4 λ j + 1 . (A\lambda)_j = \frac{1}{4}\lambda_{j-1} + \frac{1}{2}\lambda_j + \frac{1}{4}\lambda_{j+1}. ( A λ ) j = 4 1 λ j − 1 + 2 1 λ j + 4 1 λ j + 1 . For λ j = ( − 1 ) j \lambda_j = (-1)^j λ j = ( − 1 ) j A λ = 0 A\lambda= 0 A λ = 0
The Fourier symbol is
m ( θ ) = 1 2 + 1 2 cos θ = cos 2 ( θ / 2 ) . m(\theta) = \frac{1}{2} + \frac{1}{2}\cos\theta= \cos^2(\theta/2). m ( θ ) = 2 1 + 2 1 cos θ = cos 2 ( θ /2 ) . After deleting the exact Nyquist mode, the neighboring mode still has
m ( π − 2 π / N ) = sin 2 ( π / N ) ⟶ 0. m(\pi- 2\pi/N) = \sin^2(\pi/N) \longrightarrow0. m ( π − 2 π / N ) = sin 2 ( π / N ) ⟶ 0. Thus deleting the exact kernel alone does not produce a uniform ℓ 2 \ell^2 ℓ 2
AI use, human understanding, and verification limits
The research direction, scope, and revision requests were provided by Sida He. OpenAI ChatGPT/Codex agents generated most of the mathematical derivations, manuscript text, numerical implementation, and Lean formalization, and performed automated literature searches and internal checks. This work is therefore disclosed as primarily AI-generated text, rather than merely language editing.
At the preparation date of this version, the author reports that he does not understand the mathematical proofs and has not independently verified them. The intended Hexagon disclosure is Human understanding: no parts , with respect to the proofs. No independent human expert verification or external peer review of the proofs is claimed. Internal reviews by additional AI agents do not constitute independent human mathematical review.
The Lean checks described above cover selected algebraic identities and finite-step arguments only. They do not formalize the full PDE convergence theorems or establish their analytic hypotheses. Numerical experiments are implementation checks and illustrations, not proofs. The original proof source, dependency pins, numerical results, and recorded build logs are retained in the companion reproducibility material. This preparation revision does not claim a new Lean compilation or a new numerical experiment. The manuscript is offered for critical scrutiny; its correctness and novelty are not certified by these disclosures or by repository acceptance.
Relationship to companion manuscripts. The other parts of this series are Part II [33 ] , and Part III [34 ] . They are cited as unpublished companion manuscripts; no public identifier or acceptance is claimed. Each part has its own hypotheses and scope. The series is not asserted to be a chain of special-case theorems.
Availability of reproducibility material. A separate versioned package has been prepared with the Lean source, dependency lockfiles, numerical code, and archived results. No public repository URL or archive identifier has been assigned to that package at the preparation date of this manuscript. The mathematical source submitted to Hexagon does not itself contain the Lean project.
