Introduction
In this paper, we consider the Gross–Pitaevskii eigenvalue problem
− Δ u + V u + β ∣ u ∣ 2 u = λ u in Ω , u = 0 on ∂ Ω , ∥ u ∥ L 2 ( Ω ) = 1. (1) -\Delta u + V u + \beta|u|^2 u = \lambda u \quad\text{in } \Omega,\qquad u=0 \quad\text{on } \partial\Omega,\qquad\lVert u\rVert_{L^2(\Omega)}=1. \tag*{(1)} − Δ u + V u + β ∣ u ∣ 2 u = λ u in Ω , u = 0 on ∂ Ω , ∥ u ∥ L 2 ( Ω ) = 1. (1) where Ω ⊂ R d \Omega\subset\mathbb{R}^d Ω ⊂ R d d ∈ { 1 , 2 , 3 } d \in\{1,2,3\} d ∈ { 1 , 2 , 3 } V V V β ∈ R \beta\in\mathbb{R} β ∈ R (1) arises in quantum physics as a model for stationary states of Bose–Einstein condensates. Among these stationary states, the ground state is of particular importance. It is defined as a minimizer of the Gross–Pitaevskii energy
E ( v ) : = ∫ Ω ( 1 2 ∣ ∇ v ∣ 2 + 1 2 V ∣ v ∣ 2 + β 4 ∣ v ∣ 4 ) d x E(v) := \int_{\Omega} \left(\frac{1}{2}|\nabla v|^2+\frac{1}{2}V|v|^2+\frac{\beta}{4}|v|^4\right)\,\mathrm{d}x E ( v ) := ∫ Ω ( 2 1 ∣∇ v ∣ 2 + 2 1 V ∣ v ∣ 2 + 4 β ∣ v ∣ 4 ) d x under the normalization constraint, i.e., as a solution of
min v ∈ H 0 1 ( Ω ) E ( v ) subject to ∥ v ∥ L 2 ( Ω ) = 1. (2) \min_{v\in H_0^1(\Omega)} E(v) \qquad\text{subject to} \qquad\lVert v\rVert_{L^2(\Omega)}=1. \tag*{(2)} v ∈ H 0 1 ( Ω ) min E ( v ) subject to ∥ v ∥ L 2 ( Ω ) = 1. (2) Every minimizer of (2) satisfies the Euler–Lagrange equation (1) , where the eigenvalue λ \lambda λ (1) , and the corresponding eigenvalue is referred to as the ground state eigenvalue. Since the ground state is in general not available in closed form, its efficient and reliable numerical approximation is of central importance.
The numerical computation of the ground state involves two main aspects: the spatial discretization of the problem and the design of iterative solvers for the resulting nonlinear discrete problem. Regarding the latter, one may start from either the eigenvalue problem (1) or the constrained minimization problem (2) . Starting from (1) , the spatial discretization leads to a nonlinear algebraic eigenvalue problem. A widely used approach for such problems is the self-consistent field (SCF) iteration [9 , 11 , 25 ] , which only requires the solution of a linear eigenvalue problem in each iteration. Starting from (1.2), a variety of optimization-based approaches are available, including Riemannian optimization methods [2 , 3 , 22 ] , preconditioned nonlinear conjugate gradient methods [4], regularized Newton methods [44 ] , J J J [1 , 39 ] , and gradient flow methods [7 , 17 , 18 , 21 , 37 , 33 , 40 , 45 ] . For a comprehensive overview of numerical methods for computing the ground state, we refer to the recent review [34 ] .
In this paper, we focus on the first aspect, namely, the spatial discretization and its numerical analysis. Existing results in this direction include both a priori and a posteriori error estimates. General finite-dimensional approximations of the Gross–Pitaevskii eigenvalue problem were studied by Zhou in [46 ] , while related nonlinear eigenvalue problems were studied in [15 , 47 ] . These studies established optimal H 1 H^1 H 1 [10 ] , where sharp estimates for the ground state energy and eigenvalue were established, together with L 2 L^2 L 2 H 1 H^1 H 1 [16 , 38 ] , while a mass-lumped linear FEM that preserves the positivity of the discrete ground state was proposed and analyzed in [30 ] . Generalized finite element approximations of the Gross–Pitaevskii problem were studied in [35 , 36 ] , and finite-dimensional approximations of the Kohn–Sham equation were analyzed in [13 ] . Besides these a priori results, a posteriori error estimates and adaptive FEMs were developed in [14 , 16 , 26 ] , and adaptive FEMs for the Kohn–Sham equation were proposed in [12 ] . More recently, it was shown in [29 ] that Galerkin approximations of the ground state superconverge to the H 1 H^1 H 1
In this work, we analyze linear finite element discretizations of (1) , including both the standard FEM and the mass-lumped FEM. For the standard FEM, the uniqueness of the discrete ground state up to sign for sufficiently small mesh sizes can be deduced from the analysis in [16 , 38 ] . Here we present an alternative proof based on a perturbation argument for the linearized operator, which only requires the H 1 H^1 H 1 [16 , 19 , 20 , 31 ] , a perturbation approach was used to transfer a posteriori error estimates and adaptive finite element techniques from the associated source problem to the eigenvalue problem. Following a similar strategy, we combine this perturbation argument with existing L 2 L^2 L 2 H 1 H^1 H 1 L p L^p L p W 1 , p W^{1,p} W 1 , p W 1 , p W^{1,p} W 1 , p
The remainder of this paper is organized as follows. § 2 collects some preliminaries, including properties of the Gross–Pitaevskii eigenvalue problem, the finite element discretization, and several technical results. In § 3 , we prove the uniqueness of the discrete ground state of the standard FEM up to sign for sufficiently small mesh sizes and establish convergence rates in the L p L^p L p W 1 , p W^{1,p} W 1 , p § 4 is devoted to the corresponding error analysis and superconvergence results for the mass-lumped FEM. Numerical experiments illustrating the theoretical results are presented in § 5 , and concluding remarks are given in § 6 .
Preliminaries
2. Preliminaries. We adopt the standard notation for Sobolev spaces W k , p ( Ω ) W^{k,p}(\Omega) W k , p ( Ω ) ∥ ⋅ ∥ W k , p ( Ω ) \lVert\cdot\rVert_{W^{k,p}(\Omega)} ∥ ⋅ ∥ W k , p ( Ω ) ∣ ⋅ ∣ W k , p ( Ω ) \lvert\cdot\rvert_{W^{k,p}(\Omega)} ∣ ⋅ ∣ W k , p ( Ω ) k = 0 k=0 k = 0 L p ( Ω ) : = W 0 , p ( Ω ) L^p(\Omega):=W^{0,p}(\Omega) L p ( Ω ) := W 0 , p ( Ω ) p = 2 p=2 p = 2 H k ( Ω ) : = W k , 2 ( Ω ) H^k(\Omega):=W^{k,2}(\Omega) H k ( Ω ) := W k , 2 ( Ω ) L 2 ( Ω ) L^2(\Omega) L 2 ( Ω ) ( ⋅ , ⋅ ) L 2 ( Ω ) (\cdot,\cdot)_{L^2(\Omega)} ( ⋅ , ⋅ ) L 2 ( Ω ) H 0 1 ( Ω ) H^1_0(\Omega) H 0 1 ( Ω ) H 1 ( Ω ) H^1(\Omega) H 1 ( Ω ) ∂ Ω \partial\Omega ∂ Ω C C C h h h
The Gross–Pitaevskii eigenvalue problem
Throughout this paper, the computational domain is taken to be Ω = ( a , b ) d \Omega=(a,b)^d Ω = ( a , b ) d V ∈ L ∞ ( Ω ) V\in L^\infty(\Omega) V ∈ L ∞ ( Ω ) V ≥ 0 V\ge0 V ≥ 0 Ω \Omega Ω β ≥ 0 \beta\ge0 β ≥ 0
M : = { u ∈ H 0 1 ( Ω ) : ∥ u ∥ L 2 ( Ω ) = 1 } . \mathcal{M}:=\{u\in H_0^1(\Omega):\|u\|_{L^2(\Omega)}=1\}. M := { u ∈ H 0 1 ( Ω ) : ∥ u ∥ L 2 ( Ω ) = 1 } . Under the above assumptions, the following result holds; see [6 , 10 , 34 ] .
Proposition 2.1. The Gross–Pitaevskii eigenvalue problem (1) admits infinitely many real eigenvalues { λ k } k ≥ 1 \{\lambda_k\}_{k\ge1} { λ k } k ≥ 1 { u k } k ≥ 1 ⊂ M \{u_k\}_{k\ge1}\subset\mathcal{M} { u k } k ≥ 1 ⊂ M
λ 1 < λ 2 ≤ ⋯ ≤ λ k ≤ ⋯ , λ k → ∞ as k → ∞ . \lambda_1<\lambda_2\le\cdots\le\lambda_k\le\cdots,\qquad\lambda_k\to\infty\quad\text{as }k\to\infty. λ 1 < λ 2 ≤ ⋯ ≤ λ k ≤ ⋯ , λ k → ∞ as k → ∞. In particular, the smallest eigenvalue λ 1 \lambda_1 λ 1 u 1 u_1 u 1 u 1 > 0 u_1>0 u 1 > 0 u 1 < 0 u_1<0 u 1 < 0 Ω \Omega Ω
Let u ∗ u^* u ∗ L 2 L^2 L 2 (1) and let λ ∗ : = λ 1 \lambda^*:=\lambda_1 λ ∗ := λ 1 μ ∈ R \mu\in\mathbb{R} μ ∈ R u ≠ 0 u\ne0 u = 0
− Δ u + V u + β ∣ u ∗ ∣ 2 u = μ u . (3) -\Delta u+Vu+\beta|u^*|^2u=\mu u. \tag*{(3)} − Δ u + V u + β ∣ u ∗ ∣ 2 u = μu . (3) It follows directly from (1) that ( λ ∗ , u ∗ ) (\lambda^*,u^*) ( λ ∗ , u ∗ ) (3) . Moreover, the following result holds; see, e.g., [10 , 33 ] .
Proposition 2.2. The smallest eigenvalue μ 1 \mu_1 μ 1 (3) is simple and coincides with the ground state eigenvalue of (1) , i.e., μ 1 = λ ∗ \mu_1=\lambda^* μ 1 = λ ∗
Note that the ground state u ∗ u^* u ∗
− Δ u ∗ = ( λ ∗ − V − β ∣ u ∗ ∣ 2 ) u ∗ in Ω , u ∗ = 0 on ∂ Ω , (4) -\Delta u^*=(\lambda^*-V-\beta|u^*|^2)u^*\quad\text{in }\Omega,\qquad u^*=0\quad\text{on }\partial\Omega, \tag*{(4)} − Δ u ∗ = ( λ ∗ − V − β ∣ u ∗ ∣ 2 ) u ∗ in Ω , u ∗ = 0 on ∂ Ω , (4) which can be viewed as a Poisson equation for u ∗ u^* u ∗ V ∈ L ∞ ( Ω ) V\in L^\infty(\Omega) V ∈ L ∞ ( Ω ) H 0 1 ( Ω ) ↪ L 6 ( Ω ) H_0^1(\Omega)\hookrightarrow L^6(\Omega) H 0 1 ( Ω ) ↪ L 6 ( Ω ) d ≤ 3 d\le3 d ≤ 3 (4) belongs to L 2 ( Ω ) L^2(\Omega) L 2 ( Ω ) H 2 H^2 H 2 [28 ] yields u ∗ ∈ H 2 ( Ω ) u^*\in H^2(\Omega) u ∗ ∈ H 2 ( Ω )
Finite element discretization
Let T h \mathcal{T}_h T h Ω \Omega Ω h : = max K ∈ T h diam ( K ) h:=\max_{K\in\mathcal{T}_h}\operatorname{diam}(K) h := max K ∈ T h diam ( K ) N h \mathcal{N}_h N h T h \mathcal{T}_h T h K ∈ T h K\in\mathcal{T}_h K ∈ T h N K \mathcal{N}_K N K K K K
V h : = { v h ∈ C 0 ( Ω ‾ ) : v h ∣ K ∈ P 1 ( K ) for all K ∈ T h } V_h:=\{v_h\in C^0(\overline{\Omega}):v_h|_K\in\mathbb{P}_1(K)\text{ for all }K\in\mathcal{T}_h\} V h := { v h ∈ C 0 ( Ω ) : v h ∣ K ∈ P 1 ( K ) for all K ∈ T h } be the continuous piecewise linear Lagrange finite element space associated with T h \mathcal{T}_h T h V h 0 : = V h ∩ H 0 1 ( Ω ) V_h^0:=V_h\cap H_0^1(\Omega) V h 0 := V h ∩ H 0 1 ( Ω )
For v ∈ C ( Ω ‾ ) v\in C(\overline{\Omega}) v ∈ C ( Ω ) I h v ∈ V h I_hv\in V_h I h v ∈ V h H 2 ( Ω ) ↪ C ( Ω ‾ ) H^2(\Omega)\hookrightarrow C(\overline{\Omega}) H 2 ( Ω ) ↪ C ( Ω ) d ≤ 3 d\le3 d ≤ 3 I h v I_hv I h v v ∈ H 2 ( Ω ) v\in H^2(\Omega) v ∈ H 2 ( Ω ) [8 ] .
Proposition 2.3. There exists a constant C > 0 C>0 C > 0 h h h v ∈ H 2 ( Ω ) v\in H^2(\Omega) v ∈ H 2 ( Ω )
∥ v − I h v ∥ L 2 ( Ω ) + h ∥ v − I h v ∥ H 1 ( Ω ) ≤ C h 2 ∥ v ∥ H 2 ( Ω ) . (5) \|v-I_hv\|_{L^2(\Omega)}+h\|v-I_hv\|_{H^1(\Omega)}\le Ch^2\|v\|_{H^2(\Omega)}. \tag*{(5)} ∥ v − I h v ∥ L 2 ( Ω ) + h ∥ v − I h v ∥ H 1 ( Ω ) ≤ C h 2 ∥ v ∥ H 2 ( Ω ) . (5) For u ∈ W 0 1 , 1 ( Ω ) u \in W^{1,1}_0(\Omega) u ∈ W 0 1 , 1 ( Ω ) R h u ∈ V h 0 R_hu \in V^0_h R h u ∈ V h 0
∫ Ω ∇ R h u ⋅ ∇ v h d x = ∫ Ω ∇ u ⋅ ∇ v h d x ∀ v h ∈ V h 0 ; (6) \int_{\Omega} \nabla R_hu \cdot\nabla v_h\,\mathrm{d}x = \int_{\Omega} \nabla u \cdot\nabla v_h\,\mathrm{d}x \qquad\forall v_h \in V^0_h; \tag*{(6)} ∫ Ω ∇ R h u ⋅ ∇ v h d x = ∫ Ω ∇ u ⋅ ∇ v h d x ∀ v h ∈ V h 0 ; (6) see, e.g., [24 ] . This is well defined since ∇ v h ∈ L ∞ ( Ω ) \nabla v_h \in L^\infty(\Omega) ∇ v h ∈ L ∞ ( Ω ) v h ∈ V h 0 v_h \in V^0_h v h ∈ V h 0 Ω \Omega Ω [8 ] .
Proposition 2.4. There exists a constant C > 0 C > 0 C > 0 h h h u ∈ H 2 ( Ω ) ∩ H 0 1 ( Ω ) u \in H^2(\Omega) \cap H^1_0(\Omega) u ∈ H 2 ( Ω ) ∩ H 0 1 ( Ω )
∥ u − R h u ∥ L 2 ( Ω ) + h ∥ ∇ ( u − R h u ) ∥ L 2 ( Ω ) ≤ C h 2 ∥ u ∥ H 2 ( Ω ) . (7) \lVert u-R_hu\rVert_{L^2(\Omega)}+h\lVert\nabla(u-R_hu)\rVert_{L^2(\Omega)}\leq Ch^2\lVert u\rVert_{H^2(\Omega)}. \tag*{(7)} ∥ u − R h u ∥ L 2 ( Ω ) + h ∥ ∇ ( u − R h u ) ∥ L 2 ( Ω ) ≤ C h 2 ∥ u ∥ H 2 ( Ω ) . (7) To establish convergence in L p L^p L p W 1 , p W^{1,p} W 1 , p h h h
Proposition 2.5. Let 1 < p ≤ ∞ 1 < p \leq\infty 1 < p ≤ ∞ C p > 0 C_p > 0 C p > 0 h h h u ∈ W 0 1 , p ( Ω ) u \in W^{1,p}_0(\Omega) u ∈ W 0 1 , p ( Ω )
∥ ∇ R h u ∥ L p ( Ω ) ≤ C p ∥ ∇ u ∥ L p ( Ω ) . (8) \lVert\nabla R_hu\rVert_{L^p(\Omega)}\leq C_p\lVert\nabla u\rVert_{L^p(\Omega)}. \tag*{(8)} ∥ ∇ R h u ∥ L p ( Ω ) ≤ C p ∥ ∇ u ∥ L p ( Ω ) . (8) Moreover, for all u ∈ L ∞ ( Ω ) ∩ H 0 1 ( Ω ) u \in L^\infty(\Omega) \cap H^1_0(\Omega) u ∈ L ∞ ( Ω ) ∩ H 0 1 ( Ω )
∥ R h u ∥ L ∞ ( Ω ) ≤ C ∣ ln h ∣ ∥ u ∥ L ∞ ( Ω ) . (9) \lVert R_hu\rVert_{L^\infty(\Omega)}\leq C|\ln h|\lVert u\rVert_{L^\infty(\Omega)}. \tag*{(9)} ∥ R h u ∥ L ∞ ( Ω ) ≤ C ∣ ln h ∣ ∥ u ∥ L ∞ ( Ω ) . (9) The first estimate is proved in [24 Corollary 3.2] for d = 2 , 3 d=2,3 d = 2 , 3 [42 ] for 2 ≤ p ≤ ∞ 2 \leq p \leq\infty 2 ≤ p ≤ ∞ d = 2 d=2 d = 2 u u u [43 Theorem 2] for d = 2 d=2 d = 2 [41 Theorem 12] for d = 3 d=3 d = 3 u ∈ L ∞ ( Ω ) ∩ H 0 1 ( Ω ) u \in L^\infty(\Omega) \cap H^1_0(\Omega) u ∈ L ∞ ( Ω ) ∩ H 0 1 ( Ω )
Proposition 2.6. Let 1 ≤ p < ∞ 1 \leq p < \infty 1 ≤ p < ∞ q : = max { p , 2 } q := \max\{p,2\} q := max { p , 2 } C p > 0 C_p > 0 C p > 0 h h h u ∈ W 2 , q ( Ω ) ∩ H 0 1 ( Ω ) u \in W^{2,q}(\Omega) \cap H^1_0(\Omega) u ∈ W 2 , q ( Ω ) ∩ H 0 1 ( Ω )
∥ u − R h u ∥ L p ( Ω ) + h ∥ ∇ ( u − R h u ) ∥ L p ( Ω ) ≤ C p h 2 ∥ u ∥ W 2 , q ( Ω ) . \lVert u-R_hu\rVert_{L^p(\Omega)}+h\lVert\nabla(u-R_hu)\rVert_{L^p(\Omega)}\leq C_ph^2\lVert u\rVert_{W^{2,q}(\Omega)}. ∥ u − R h u ∥ L p ( Ω ) + h ∥ ∇ ( u − R h u ) ∥ L p ( Ω ) ≤ C p h 2 ∥ u ∥ W 2 , q ( Ω ) . Moreover, there exists a constant C > 0 C > 0 C > 0 h h h u ∈ W 2 , ∞ ( Ω ) ∩ H 0 1 ( Ω ) u \in W^{2,\infty}(\Omega) \cap H^1_0(\Omega) u ∈ W 2 , ∞ ( Ω ) ∩ H 0 1 ( Ω )
∥ u − R h u ∥ L ∞ ( Ω ) ≤ C h 2 ∣ ln h ∣ ∥ u ∥ W 2 , ∞ ( Ω ) , ∥ ∇ ( u − R h u ) ∥ L ∞ ( Ω ) ≤ C h ∥ u ∥ W 2 , ∞ ( Ω ) . \lVert u-R_hu\rVert_{L^\infty(\Omega)}\leq Ch^2|\ln h|\lVert u\rVert_{W^{2,\infty}(\Omega)},\qquad\lVert\nabla(u-R_hu)\rVert_{L^\infty(\Omega)}\leq Ch\lVert u\rVert_{W^{2,\infty}(\Omega)}. ∥ u − R h u ∥ L ∞ ( Ω ) ≤ C h 2 ∣ ln h ∣ ∥ u ∥ W 2 , ∞ ( Ω ) , ∥ ∇ ( u − R h u ) ∥ L ∞ ( Ω ) ≤ C h ∥ u ∥ W 2 , ∞ ( Ω ) . For 1 ≤ p < 2 1 \leq p < 2 1 ≤ p < 2 p = 2 p=2 p = 2 Ω \Omega Ω 2 ≤ p < ∞ 2 \leq p < \infty 2 ≤ p < ∞ [42 ] for d = 2 d=2 d = 2 d = 3 d=3 d = 3 W 2 , p ′ W^{2,p'} W 2 , p ′ [23 , 28 ] . The L ∞ L^\infty L ∞ u − R h u = ( I − R h ) ( u − I h u ) u-R_hu=(I-R_h)(u-I_hu) u − R h u = ( I − R h ) ( u − I h u ) d = 1 d=1 d = 1 R h u = I h u R_hu=I_hu R h u = I h u
Some technical results
In this subsection, we collect some technical results that will be used in the subsequent analysis. The first one is a W 1 , p W^{1,p} W 1 , p [27 Corollary 1] .
Lemma 2.7. Let 2 ≤ p < ∞ 2 \le p < \infty 2 ≤ p < ∞ F ∈ [ L p ( Ω ) ] d \mathbf{F} \in[L^{p}(\Omega)]^{d} F ∈ [ L p ( Ω ) ] d u ∈ H 0 1 ( Ω ) u \in H_{0}^{1}(\Omega) u ∈ H 0 1 ( Ω )
( ∇ u , ∇ v ) L 2 ( Ω ) = ( F , ∇ v ) L 2 ( Ω ) ∀ v ∈ H 0 1 ( Ω ) . (\nabla u,\nabla v)_{L^{2}(\Omega)}=(\mathbf{F},\nabla v)_{L^{2}(\Omega)} \qquad\forall v \in H_{0}^{1}(\Omega). ( ∇ u , ∇ v ) L 2 ( Ω ) = ( F , ∇ v ) L 2 ( Ω ) ∀ v ∈ H 0 1 ( Ω ) . Then u ∈ W 0 1 , p ( Ω ) u \in W_{0}^{1,p}(\Omega) u ∈ W 0 1 , p ( Ω ) C p > 0 C_{p}>0 C p > 0 u u u F \mathbf{F} F
∥ ∇ u ∥ L p ( Ω ) ≤ C p ∥ F ∥ L p ( Ω ) . \|\nabla u\|_{L^{p}(\Omega)} \le C_{p}\|\mathbf{F}\|_{L^{p}(\Omega)}. ∥∇ u ∥ L p ( Ω ) ≤ C p ∥ F ∥ L p ( Ω ) . Combining Lemma 2.7 with the stability of the Ritz projection, we obtain the following discrete regularity result.
Lemma 2.8. Let g ∈ L 2 ( Ω ) g \in L^{2}(\Omega) g ∈ L 2 ( Ω ) F ∈ [ L 6 ( Ω ) ] d \mathbf{F} \in[L^{6}(\Omega)]^{d} F ∈ [ L 6 ( Ω ) ] d z h ∈ V h 0 z_{h} \in V_{h}^{0} z h ∈ V h 0
( ∇ z h , ∇ v h ) L 2 ( Ω ) = ( g , v h ) L 2 ( Ω ) + ( F , ∇ v h ) L 2 ( Ω ) ∀ v h ∈ V h 0 . (\nabla z_{h},\nabla v_{h})_{L^{2}(\Omega)}=(g,v_{h})_{L^{2}(\Omega)}+(\mathbf{F},\nabla v_{h})_{L^{2}(\Omega)} \qquad\forall v_{h} \in V_{h}^{0}. ( ∇ z h , ∇ v h ) L 2 ( Ω ) = ( g , v h ) L 2 ( Ω ) + ( F , ∇ v h ) L 2 ( Ω ) ∀ v h ∈ V h 0 . Then there exists a constant C > 0 C>0 C > 0 h h h g g g F \mathbf{F} F
∥ ∇ z h ∥ L 6 ( Ω ) ≤ C ( ∥ g ∥ L 2 ( Ω ) + ∥ F ∥ L 6 ( Ω ) ) . \|\nabla z_{h}\|_{L^{6}(\Omega)} \le C\bigl(\|g\|_{L^{2}(\Omega)}+\|\mathbf{F}\|_{L^{6}(\Omega)}\bigr). ∥∇ z h ∥ L 6 ( Ω ) ≤ C ( ∥ g ∥ L 2 ( Ω ) + ∥ F ∥ L 6 ( Ω ) ) . Proof. Let w g , w F ∈ H 0 1 ( Ω ) w_{g},w_{\mathbf{F}} \in H_{0}^{1}(\Omega) w g , w F ∈ H 0 1 ( Ω )
( ∇ w g , ∇ v ) L 2 ( Ω ) = ( g , v ) L 2 ( Ω ) , ( ∇ w F , ∇ v ) L 2 ( Ω ) = ( F , ∇ v ) L 2 ( Ω ) , ∀ v ∈ H 0 1 ( Ω ) . (\nabla w_{g},\nabla v)_{L^{2}(\Omega)}=(g,v)_{L^{2}(\Omega)}, \qquad(\nabla w_{\mathbf{F}},\nabla v)_{L^{2}(\Omega)}=(\mathbf{F},\nabla v)_{L^{2}(\Omega)}, \qquad\forall v \in H_{0}^{1}(\Omega). ( ∇ w g , ∇ v ) L 2 ( Ω ) = ( g , v ) L 2 ( Ω ) , ( ∇ w F , ∇ v ) L 2 ( Ω ) = ( F , ∇ v ) L 2 ( Ω ) , ∀ v ∈ H 0 1 ( Ω ) . By the H 2 H^{2} H 2 p = 6 p=6 p = 6 H 1 ( Ω ) ↪ L 6 ( Ω ) H^{1}(\Omega) \hookrightarrow L^{6}(\Omega) H 1 ( Ω ) ↪ L 6 ( Ω ) d ≤ 3 d \le3 d ≤ 3
∥ ∇ w g ∥ L 6 ( Ω ) ≤ C ∥ w g ∥ H 2 ( Ω ) ≤ C ∥ g ∥ L 2 ( Ω ) , ∥ ∇ w F ∥ L 6 ( Ω ) ≤ C ∥ F ∥ L 6 ( Ω ) . \|\nabla w_{g}\|_{L^{6}(\Omega)} \le C\|w_{g}\|_{H^{2}(\Omega)} \le C\|g\|_{L^{2}(\Omega)}, \qquad\|\nabla w_{\mathbf{F}}\|_{L^{6}(\Omega)} \le C\|\mathbf{F}\|_{L^{6}(\Omega)}. ∥∇ w g ∥ L 6 ( Ω ) ≤ C ∥ w g ∥ H 2 ( Ω ) ≤ C ∥ g ∥ L 2 ( Ω ) , ∥∇ w F ∥ L 6 ( Ω ) ≤ C ∥ F ∥ L 6 ( Ω ) . By the definition of R h R_{h} R h v h ∈ V h 0 v_{h} \in V_{h}^{0} v h ∈ V h 0
( ∇ R h ( w g + w F ) , ∇ v h ) L 2 ( Ω ) = ( g , v h ) L 2 ( Ω ) + ( F , ∇ v h ) L 2 ( Ω ) , (\nabla R_{h}(w_{g}+w_{\mathbf{F}}),\nabla v_{h})_{L^{2}(\Omega)}=(g,v_{h})_{L^{2}(\Omega)}+(\mathbf{F},\nabla v_{h})_{L^{2}(\Omega)}, ( ∇ R h ( w g + w F ) , ∇ v h ) L 2 ( Ω ) = ( g , v h ) L 2 ( Ω ) + ( F , ∇ v h ) L 2 ( Ω ) , and hence z h = R h ( w g + w F ) z_{h}=R_{h}(w_{g}+w_{\mathbf{F}}) z h = R h ( w g + w F ) Proposition 2.5 with p = 6 p=6 p = 6
∥ ∇ z h ∥ L 6 ( Ω ) ≤ C ∥ ∇ ( w g + w F ) ∥ L 6 ( Ω ) ≤ C ( ∥ g ∥ L 2 ( Ω ) + ∥ F ∥ L 6 ( Ω ) ) , \|\nabla z_{h}\|_{L^{6}(\Omega)} \le C\|\nabla(w_{g}+w_{\mathbf{F}})\|_{L^{6}(\Omega)} \le C(\|g\|_{L^{2}(\Omega)}+\|\mathbf{F}\|_{L^{6}(\Omega)}), ∥∇ z h ∥ L 6 ( Ω ) ≤ C ∥∇ ( w g + w F ) ∥ L 6 ( Ω ) ≤ C ( ∥ g ∥ L 2 ( Ω ) + ∥ F ∥ L 6 ( Ω ) ) , which completes the proof.
For a fixed u ∈ H 0 1 ( Ω ) u \in H_{0}^{1}(\Omega) u ∈ H 0 1 ( Ω )
a u ( v , w ) : = ∫ Ω ( ∇ v ⋅ ∇ w + V v w + β ∣ u ∣ 2 v w ) d x , v , w ∈ H 0 1 ( Ω ) , a_{u}(v,w):=\int_{\Omega}\left(\nabla v\cdot\nabla w+Vvw+\beta|u|^{2}vw\right)\,\mathrm{d}x,\qquad v,w \in H_{0}^{1}(\Omega), a u ( v , w ) := ∫ Ω ( ∇ v ⋅ ∇ w + V v w + β ∣ u ∣ 2 v w ) d x , v , w ∈ H 0 1 ( Ω ) , which is an inner product on H 0 1 ( Ω ) H_{0}^{1}(\Omega) H 0 1 ( Ω ) ∥ ⋅ ∥ a u \|\cdot\|_{a_{u}} ∥ ⋅ ∥ a u
Let H ⊂ H 0 1 ( Ω ) H \subset H_{0}^{1}(\Omega) H ⊂ H 0 1 ( Ω ) u H ∈ H ∩ M u_{H} \in H \cap\mathcal{M} u H ∈ H ∩ M E E E H ∩ M H \cap\mathcal{M} H ∩ M
E ( u H ) = min v ∈ H ∩ M E ( v ) . E(u_{H})=\min_{v\in H\cap\mathcal{M}}E(v). E ( u H ) = v ∈ H ∩ M min E ( v ) . Then u H u_{H} u H
a u H ( u H , v ) = λ H ( u H , v ) L 2 ( Ω ) ∀ v ∈ H , (10) a_{u_{H}}(u_{H},v)=\lambda_{H}(u_{H},v)_{L^{2}(\Omega)} \qquad\forall v\in H, \tag*{(10)} a u H ( u H , v ) = λ H ( u H , v ) L 2 ( Ω ) ∀ v ∈ H , (10) where λ H : = a u H ( u H , u H ) \lambda_{H}:=a_{u_{H}}(u_{H},u_{H}) λ H := a u H ( u H , u H )
Lemma 2.9. For every u ∈ H ∩ M u \in H \cap\mathcal{M} u ∈ H ∩ M
E ( u ) − E ( u H ) = 1 2 ( a u H ( u , u ) − λ H ) + β 4 ∥ u 2 − u H 2 ∥ L 2 ( Ω ) 2 . (11) E(u)-E(u_H)=\frac{1}{2}\left(a_{u_H}(u,u)-\lambda_H\right)+\frac{\beta}{4}\lVert u^2-u_H^2\rVert_{L^2(\Omega)}^2. \tag*{(11)} E ( u ) − E ( u H ) = 2 1 ( a u H ( u , u ) − λ H ) + 4 β ∥ u 2 − u H 2 ∥ L 2 ( Ω ) 2 . (11) Proof. By the definition of λ H \lambda_H λ H
λ H = ∫ Ω ( ∣ ∇ u H ∣ 2 + V u H 2 + β u H 4 ) d x . \lambda_H=\int_{\Omega}\left(\lvert\nabla u_H\rvert^2+Vu_H^2+\beta u_H^4\right)\,\mathrm{d}x. λ H = ∫ Ω ( ∣ ∇ u H ∣ 2 + V u H 2 + β u H 4 ) d x . and hence
E ( u H ) = λ H 2 − β 4 ∫ Ω u H 4 d x . E(u_H)=\frac{\lambda_H}{2}-\frac{\beta}{4}\int_{\Omega}u_H^4\,\mathrm{d}x. E ( u H ) = 2 λ H − 4 β ∫ Ω u H 4 d x . Adding and subtracting β 2 ∫ Ω u H 2 u 2 d x \frac{\beta}{2}\int_{\Omega}u_H^2u^2\,\mathrm{d}x 2 β ∫ Ω u H 2 u 2 d x
E ( u ) − E ( u H ) = ∫ Ω ( 1 2 ∣ ∇ u ∣ 2 + 1 2 V u 2 + β 2 u H 2 u 2 ) d x + ∫ Ω β 4 u 4 − β 2 u H 2 u 2 + β 4 u H 4 d x − λ H 2 = 1 2 ( a u H ( u , u ) − λ H ) + β 4 ∫ Ω ( u 2 − u H 2 ) 2 d x , \begin{aligned}
E(u)-E(u_H)
&=\int_{\Omega}\left(\frac{1}{2}\lvert\nabla u\rvert^2+\frac{1}{2}Vu^2+\frac{\beta}{2}u_H^2u^2\right)\,\mathrm{d}x\\
&\quad+\int_{\Omega}\frac{\beta}{4}u^4-\frac{\beta}{2}u_H^2u^2+\frac{\beta}{4}u_H^4\,\mathrm{d}x-\frac{\lambda_H}{2}\\
&=\frac{1}{2}\left(a_{u_H}(u,u)-\lambda_H\right)+\frac{\beta}{4}\int_{\Omega}\left(u^2-u_H^2\right)^2\,\mathrm{d}x,
\end{aligned} E ( u ) − E ( u H ) = ∫ Ω ( 2 1 ∣ ∇ u ∣ 2 + 2 1 V u 2 + 2 β u H 2 u 2 ) d x + ∫ Ω 4 β u 4 − 2 β u H 2 u 2 + 4 β u H 4 d x − 2 λ H = 2 1 ( a u H ( u , u ) − λ H ) + 4 β ∫ Ω ( u 2 − u H 2 ) 2 d x , which completes the proof.
Consider the following linear eigenvalue problem on H H H ( w , μ ) ∈ H × R (w,\mu) \in H \times\mathbb{R} ( w , μ ) ∈ H × R
( ∇ w , ∇ v ) L 2 ( Ω ) + ( V w , v ) L 2 ( Ω ) + β ( ∣ u H ∣ 2 w , v ) L 2 ( Ω ) = μ ( w , v ) L 2 ( Ω ) ∀ v ∈ H . (12) (\nabla w,\nabla v)_{L^2(\Omega)}+(Vw,v)_{L^2(\Omega)}+\beta(\lvert u_H\rvert^2w,v)_{L^2(\Omega)}=\mu(w,v)_{L^2(\Omega)} \qquad\forall v \in H. \tag*{(12)} ( ∇ w , ∇ v ) L 2 ( Ω ) + ( V w , v ) L 2 ( Ω ) + β (∣ u H ∣ 2 w , v ) L 2 ( Ω ) = μ ( w , v ) L 2 ( Ω ) ∀ v ∈ H . (12) It follows from (10) that λ H \lambda_H λ H (12) with corresponding eigenfunction u H u_H u H
Lemma 2.10. Assume that λ H \lambda_H λ H (12) and that it is simple, and let μ 2 > λ H \mu_2>\lambda_H μ 2 > λ H (12) . Then for every u ∈ H ∩ M u \in H \cap\mathcal{M} u ∈ H ∩ M
min s ∈ { 1 , − 1 } ∥ u − s u H ∥ a u H 2 ≤ 2 ( μ 2 + λ H ) μ 2 − λ H ( E ( u ) − E ( u H ) ) . (13) \min_{s\in\{1,-1\}}\lVert u-su_H\rVert_{a_{u_H}}^2\leq\frac{2(\mu_2+\lambda_H)}{\mu_2-\lambda_H}\left(E(u)-E(u_H)\right). \tag*{(13)} s ∈ { 1 , − 1 } min ∥ u − s u H ∥ a u H 2 ≤ μ 2 − λ H 2 ( μ 2 + λ H ) ( E ( u ) − E ( u H ) ) . (13) Proof. Fix u ∈ H ∩ M u \in H \cap\mathcal{M} u ∈ H ∩ M t : = ∣ ( u , u H ) L 2 ( Ω ) ∣ t:=\lvert(u,u_H)_{L^2(\Omega)}\rvert t := ∣( u , u H ) L 2 ( Ω ) ∣ s ∈ { − 1 , 1 } s\in\{-1,1\} s ∈ { − 1 , 1 } ( s u , u H ) L 2 ( Ω ) = t (su,u_H)_{L^2(\Omega)}=t ( s u , u H ) L 2 ( Ω ) = t u ~ : = s u \widetilde{u}:=su u := s u w : = u ~ − t u H ∈ H w:=\widetilde{u}-tu_H\in H w := u − t u H ∈ H ∥ u ∥ L 2 ( Ω ) = ∥ u H ∥ L 2 ( Ω ) = 1 \lVert u\rVert_{L^2(\Omega)}=\lVert u_H\rVert_{L^2(\Omega)}=1 ∥ u ∥ L 2 ( Ω ) = ∥ u H ∥ L 2 ( Ω ) = 1 0 ≤ t ≤ 1 0\leq t\leq1 0 ≤ t ≤ 1
( w , u H ) L 2 ( Ω ) = ( u ~ , u H ) L 2 ( Ω ) − t ∥ u H ∥ L 2 ( Ω ) 2 = t − t = 0 , (w,u_H)_{L^2(\Omega)}=(\widetilde{u},u_H)_{L^2(\Omega)}-t\lVert u_H\rVert_{L^2(\Omega)}^2=t-t=0, ( w , u H ) L 2 ( Ω ) = ( u , u H ) L 2 ( Ω ) − t ∥ u H ∥ L 2 ( Ω ) 2 = t − t = 0 , and hence
1 = ∥ u ~ ∥ L 2 ( Ω ) 2 = ∥ w + t u H ∥ L 2 ( Ω ) 2 = ∥ w ∥ L 2 ( Ω ) 2 + t 2 . 1=\lVert\widetilde{u}\rVert_{L^2(\Omega)}^2=\lVert w+tu_H\rVert_{L^2(\Omega)}^2=\lVert w\rVert_{L^2(\Omega)}^2+t^2. 1 = ∥ u ∥ L 2 ( Ω ) 2 = ∥ w + t u H ∥ L 2 ( Ω ) 2 = ∥ w ∥ L 2 ( Ω ) 2 + t 2 . Taking v = w ∈ H v=w\in H v = w ∈ H (10) and using ( w , u H ) L 2 ( Ω ) = 0 (w,u_H)_{L^2(\Omega)}=0 ( w , u H ) L 2 ( Ω ) = 0
a u H ( u H , w ) = λ H ( u H , w ) L 2 ( Ω ) = 0. a_{u_H}(u_H,w)=\lambda_H(u_H,w)_{L^2(\Omega)}=0. a u H ( u H , w ) = λ H ( u H , w ) L 2 ( Ω ) = 0. Together with a u H ( u H , u H ) = λ H a_{u_H}(u_H,u_H)=\lambda_H a u H ( u H , u H ) = λ H ∥ w ∥ L 2 ( Ω ) 2 = 1 − t 2 \|w\|_{L^2(\Omega)}^2=1-t^2 ∥ w ∥ L 2 ( Ω ) 2 = 1 − t 2
a u H ( u ~ , u ~ ) − λ H = a u H ( w + t u H , w + t u H ) − λ H = a u H ( w , w ) + 2 t a u H ( w , u H ) + t 2 a u H ( u H , u H ) − λ H = a u H ( w , w ) − λ H ( 1 − t 2 ) = a u H ( w , w ) − λ H ∥ w ∥ L 2 ( Ω ) 2 . \begin{aligned}
a_{u_H}(\tilde u,\tilde u)-\lambda_H
&=a_{u_H}(w+tu_H,w+tu_H)-\lambda_H\\
&=a_{u_H}(w,w)+2t\,a_{u_H}(w,u_H)+t^2a_{u_H}(u_H,u_H)-\lambda_H\\
&=a_{u_H}(w,w)-\lambda_H(1-t^2)=a_{u_H}(w,w)-\lambda_H\|w\|_{L^2(\Omega)}^2.
\end{aligned} a u H ( u ~ , u ~ ) − λ H = a u H ( w + t u H , w + t u H ) − λ H = a u H ( w , w ) + 2 t a u H ( w , u H ) + t 2 a u H ( u H , u H ) − λ H = a u H ( w , w ) − λ H ( 1 − t 2 ) = a u H ( w , w ) − λ H ∥ w ∥ L 2 ( Ω ) 2 . Since λ H \lambda_H λ H (12) with eigenfunction u H u_H u H [5 ] yields
μ 2 = inf 0 ≠ z ∈ H , ( z , u H ) L 2 ( Ω ) = 0 a u H ( z , z ) ∥ z ∥ L 2 ( Ω ) 2 . \mu_2=\inf_{\substack{0\ne z\in H,\\(z,u_H)_{L^2(\Omega)}=0}}\frac{a_{u_H}(z,z)}{\|z\|_{L^2(\Omega)}^2}. μ 2 = 0 = z ∈ H , ( z , u H ) L 2 ( Ω ) = 0 inf ∥ z ∥ L 2 ( Ω ) 2 a u H ( z , z ) . and hence ∥ w ∥ L 2 ( Ω ) 2 ≤ μ 2 − 1 a u H ( w , w ) \|w\|_{L^2(\Omega)}^2\le\mu_2^{-1}a_{u_H}(w,w) ∥ w ∥ L 2 ( Ω ) 2 ≤ μ 2 − 1 a u H ( w , w )
a u H ( u ~ , u ~ ) − λ H ≥ ( 1 − λ H μ 2 ) a u H ( w , w ) . a_{u_H}(\tilde u,\tilde u)-\lambda_H\ge\left(1-\frac{\lambda_H}{\mu_2}\right)a_{u_H}(w,w). a u H ( u ~ , u ~ ) − λ H ≥ ( 1 − μ 2 λ H ) a u H ( w , w ) . Since E E E E ( u ) = E ( u ~ ) E(u)=E(\tilde u) E ( u ) = E ( u ~ ) Lemma 2.9 together with the above estimate yields
E ( u ) − E ( u H ) = E ( u ~ ) − E ( u H ) ≥ 1 2 ( a u H ( u ~ , u ~ ) − λ H ) ≥ 1 2 ( 1 − λ H μ 2 ) a u H ( w , w ) . E(u)-E(u_H)=E(\tilde u)-E(u_H)\ge\frac{1}{2}\left(a_{u_H}(\tilde u,\tilde u)-\lambda_H\right)\ge\frac{1}{2}\left(1-\frac{\lambda_H}{\mu_2}\right)a_{u_H}(w,w). E ( u ) − E ( u H ) = E ( u ~ ) − E ( u H ) ≥ 2 1 ( a u H ( u ~ , u ~ ) − λ H ) ≥ 2 1 ( 1 − μ 2 λ H ) a u H ( w , w ) . On the other hand, using a u H ( u H , w ) = 0 a_{u_H}(u_H,w)=0 a u H ( u H , w ) = 0 a u H ( u H , u H ) = λ H a_{u_H}(u_H,u_H)=\lambda_H a u H ( u H , u H ) = λ H
∥ u ~ − u H ∥ a u H 2 = a u H ( w , w ) + λ H ( 1 − t ) 2 . \|\tilde u-u_H\|_{a_{u_H}}^2=a_{u_H}(w,w)+\lambda_H(1-t)^2. ∥ u ~ − u H ∥ a u H 2 = a u H ( w , w ) + λ H ( 1 − t ) 2 . Since 0 ≤ t ≤ 1 0\le t\le1 0 ≤ t ≤ 1 ( 1 − t ) 2 ≤ 1 − t 2 = ∥ w ∥ L 2 ( Ω ) 2 (1-t)^2\le1-t^2=\|w\|_{L^2(\Omega)}^2 ( 1 − t ) 2 ≤ 1 − t 2 = ∥ w ∥ L 2 ( Ω ) 2 λ H > 0 \lambda_H>0 λ H > 0 ∥ w ∥ L 2 ( Ω ) 2 ≤ μ 2 − 1 a u H ( w , w ) \|w\|_{L^2(\Omega)}^2\le\mu_2^{-1}a_{u_H}(w,w) ∥ w ∥ L 2 ( Ω ) 2 ≤ μ 2 − 1 a u H ( w , w )
∥ u ~ − u H ∥ a u H 2 = a u H ( w , w ) + λ H ( 1 − t ) 2 ≤ ( 1 + λ H μ 2 ) a u H ( w , w ) . \|\tilde u-u_H\|_{a_{u_H}}^2=a_{u_H}(w,w)+\lambda_H(1-t)^2\le\left(1+\frac{\lambda_H}{\mu_2}\right)a_{u_H}(w,w). ∥ u ~ − u H ∥ a u H 2 = a u H ( w , w ) + λ H ( 1 − t ) 2 ≤ ( 1 + μ 2 λ H ) a u H ( w , w ) . Combining this with the lower bound for E ( u ) − E ( u H ) E(u)-E(u_H) E ( u ) − E ( u H )
∥ u ~ − u H ∥ a u H 2 ≤ 2 ( μ 2 + λ H ) μ 2 − λ H ( E ( u ) − E ( u H ) ) . \|\tilde u-u_H\|_{a_{u_H}}^2\le\frac{2(\mu_2+\lambda_H)}{\mu_2-\lambda_H}\left(E(u)-E(u_H)\right). ∥ u ~ − u H ∥ a u H 2 ≤ μ 2 − λ H 2 ( μ 2 + λ H ) ( E ( u ) − E ( u H ) ) . Since ∥ u ~ − u H ∥ a u H = ∥ u − s u H ∥ a u H \|\tilde u-u_H\|_{a_{u_H}}=\|u-su_H\|_{a_{u_H}} ∥ u ~ − u H ∥ a u H = ∥ u − s u H ∥ a u H
Corollary 2.11. Under the assumptions of Lemma 2.10 , the minimizer u H u_H u H E E E H ∩ M H\cap\mathcal{M} H ∩ M u u u E E E H ∩ M H\cap\mathcal{M} H ∩ M u = u H u=u_H u = u H u = − u H u=-u_H u = − u H
The standard finite element method
A discrete ground state of the standard FEM is defined as
u h ∗ ∈ a r g m i n v h ∈ V h 0 ∩ M E ( v h ) with ( u h ∗ , u ∗ ) L 2 ( Ω ) ≥ 0. u_h^*\in\mathop{\rm arg\,min}_{v_h\in V_h^0\cap\mathcal{M}}E(v_h)
\qquad\text{with}\qquad
(u_h^*,u^*)_{L^2(\Omega)}\ge0. u h ∗ ∈ arg min v h ∈ V h 0 ∩ M E ( v h ) with ( u h ∗ , u ∗ ) L 2 ( Ω ) ≥ 0. Taking H = V h 0 H=V_h^0 H = V h 0 (10) , we see that u h ∗ u_h^* u h ∗
a u h ∗ ( u h ∗ , v h ) = λ h ∗ ( u h ∗ , v h ) L 2 ( Ω ) ∀ v h ∈ V h 0 (14) a_{u_h^*}(u_h^*,v_h)=\lambda_h^*(u_h^*,v_h)_{L^2(\Omega)}
\qquad\forall v_h\in V_h^0 \tag*{(14)} a u h ∗ ( u h ∗ , v h ) = λ h ∗ ( u h ∗ , v h ) L 2 ( Ω ) ∀ v h ∈ V h 0 (14) where λ h ∗ : = a u h ∗ ( u h ∗ , u h ∗ ) \lambda_h^*:=a_{u_h^*}(u_h^*,u_h^*) λ h ∗ := a u h ∗ ( u h ∗ , u h ∗ )
Uniqueness of the discrete ground state
Since V h 0 ∩ M V_h^0 \cap\mathcal{M} V h 0 ∩ M V h 0 V_h^0 V h 0 E E E u h ∗ u_h^* u h ∗ u ∗ u^* u ∗ H 1 ( Ω ) H^1(\Omega) H 1 ( Ω ) [10 , 16 , 46 ] :
∥ u h ∗ − u ∗ ∥ H 1 ( Ω ) → 0 as h → 0. (15) \lVert u_h^* - u^* \rVert_{H^1(\Omega)} \to0 \qquad\text{as } h \to0. \tag*{(15)} ∥ u h ∗ − u ∗ ∥ H 1 ( Ω ) → 0 as h → 0. (15) As a consequence, the bilinear forms a u h ∗ ( ⋅ , ⋅ ) a_{u_h^*}(\cdot,\cdot) a u h ∗ ( ⋅ , ⋅ ) a u ∗ ( ⋅ , ⋅ ) a_{u^*}(\cdot,\cdot) a u ∗ ( ⋅ , ⋅ ) h h h h h h
Lemma 3.1. There exists a sequence ϵ h ≥ 0 \epsilon_h \ge0 ϵ h ≥ 0 ϵ h → 0 \epsilon_h \to0 ϵ h → 0 h → 0 h \to0 h → 0 v ∈ H 0 1 ( Ω ) v \in H_0^1(\Omega) v ∈ H 0 1 ( Ω )
( 1 − ϵ h ) a u ∗ ( v , v ) ≤ a u h ∗ ( v , v ) ≤ ( 1 + ϵ h ) a u ∗ ( v , v ) . (1-\epsilon_h)a_{u^*}(v,v) \le a_{u_h^*}(v,v) \le(1+\epsilon_h)a_{u^*}(v,v). ( 1 − ϵ h ) a u ∗ ( v , v ) ≤ a u h ∗ ( v , v ) ≤ ( 1 + ϵ h ) a u ∗ ( v , v ) . Proof. For any v ∈ H 0 1 ( Ω ) v \in H_0^1(\Omega) v ∈ H 0 1 ( Ω ) H 0 1 ( Ω ) ↪ L 6 ( Ω ) H_0^1(\Omega) \hookrightarrow L^6(\Omega) H 0 1 ( Ω ) ↪ L 6 ( Ω )
∣ a u h ∗ ( v , v ) − a u ∗ ( v , v ) ∣ = β ∣ ∫ Ω ( u h ∗ + u ∗ ) ( u h ∗ − u ∗ ) v 2 d x ∣ \left|a_{u_h^*}(v,v)-a_{u^*}(v,v)\right|
= \beta\left|\int_\Omega(u_h^*+u^*)(u_h^*-u^*)v^2\,\mathrm{d}x\right| a u h ∗ ( v , v ) − a u ∗ ( v , v ) = β ∫ Ω ( u h ∗ + u ∗ ) ( u h ∗ − u ∗ ) v 2 d x ≤ C β ∥ u h ∗ + u ∗ ∥ H 1 ( Ω ) ∥ u h ∗ − u ∗ ∥ H 1 ( Ω ) ∥ ∇ v ∥ L 2 ( Ω ) 2 . \le C\beta\lVert u_h^*+u^*\rVert_{H^1(\Omega)}\lVert u_h^*-u^*\rVert_{H^1(\Omega)}\lVert\nabla v\rVert_{L^2(\Omega)}^2. ≤ C β ∥ u h ∗ + u ∗ ∥ H 1 ( Ω ) ∥ u h ∗ − u ∗ ∥ H 1 ( Ω ) ∥ ∇ v ∥ L 2 ( Ω ) 2 . Setting ϵ h : = C β ∥ u h ∗ + u ∗ ∥ H 1 ( Ω ) ∥ u h ∗ − u ∗ ∥ H 1 ( Ω ) \epsilon_h := C\beta\lVert u_h^*+u^*\rVert_{H^1(\Omega)}\lVert u_h^*-u^*\rVert_{H^1(\Omega)} ϵ h := C β ∥ u h ∗ + u ∗ ∥ H 1 ( Ω ) ∥ u h ∗ − u ∗ ∥ H 1 ( Ω ) ∥ ∇ v ∥ L 2 ( Ω ) 2 ≤ a u ∗ ( v , v ) \lVert\nabla v\rVert_{L^2(\Omega)}^2 \le a_{u^*}(v,v) ∥ ∇ v ∥ L 2 ( Ω ) 2 ≤ a u ∗ ( v , v )
∣ a u h ∗ ( v , v ) − a u ∗ ( v , v ) ∣ ≤ ϵ h a u ∗ ( v , v ) . \left|a_{u_h^*}(v,v)-a_{u^*}(v,v)\right| \le\epsilon_h a_{u^*}(v,v). a u h ∗ ( v , v ) − a u ∗ ( v , v ) ≤ ϵ h a u ∗ ( v , v ) . Since ∥ u h ∗ ∥ H 1 ( Ω ) \lVert u_h^*\rVert_{H^1(\Omega)} ∥ u h ∗ ∥ H 1 ( Ω ) (15) implies ϵ h → 0 \epsilon_h \to0 ϵ h → 0 h → 0 h \to0 h → 0
Let Λ 1 , h ≤ Λ 2 , h ≤ ⋯ \Lambda_{1,h} \le\Lambda_{2,h} \le\cdots Λ 1 , h ≤ Λ 2 , h ≤ ⋯ ( w h , Λ h ) ∈ V h 0 × R (w_h,\Lambda_h) \in V_h^0 \times\mathbb{R} ( w h , Λ h ) ∈ V h 0 × R w h ≠ 0 w_h \ne0 w h = 0
a u h ∗ ( w h , v h ) = Λ h ( w h , v h ) L 2 ( Ω ) ∀ v h ∈ V h 0 . a_{u_h^*}(w_h,v_h) = \Lambda_h(w_h,v_h)_{L^2(\Omega)} \qquad\forall v_h \in V_h^0. a u h ∗ ( w h , v h ) = Λ h ( w h , v h ) L 2 ( Ω ) ∀ v h ∈ V h 0 . It follows from (14) that ( λ h ∗ , u h ∗ ) (\lambda_h^*,u_h^*) ( λ h ∗ , u h ∗ )
Lemma 3.2. There exists h 0 > 0 h_0>0 h 0 > 0 0 < h ≤ h 0 0<h\le h_0 0 < h ≤ h 0
Λ 1 , h = λ h ∗ < Λ 2 , h . \Lambda_{1,h}=\lambda_h^*<\Lambda_{2,h}. Λ 1 , h = λ h ∗ < Λ 2 , h . Proof. For v ∈ H 0 1 ( Ω ) ∖ { 0 } v \in H_0^1(\Omega)\setminus\{0\} v ∈ H 0 1 ( Ω ) ∖ { 0 }
R u ∗ ( v ) : = a u ∗ ( v , v ) ∥ v ∥ L 2 ( Ω ) 2 , R u h ∗ ( v ) : = a u h ∗ ( v , v ) ∥ v ∥ L 2 ( Ω ) 2 . R_{u^*}(v):=\frac{a_{u^*}(v,v)}{\lVert v\rVert_{L^2(\Omega)}^2}, \qquad R_{u_h^*}(v):=\frac{a_{u_h^*}(v,v)}{\lVert v\rVert_{L^2(\Omega)}^2}. R u ∗ ( v ) := ∥ v ∥ L 2 ( Ω ) 2 a u ∗ ( v , v ) , R u h ∗ ( v ) := ∥ v ∥ L 2 ( Ω ) 2 a u h ∗ ( v , v ) . For a subspace X ⊂ H 0 1 ( Ω ) X\subset H_0^1(\Omega) X ⊂ H 0 1 ( Ω ) S 2 ( X ) S_2(X) S 2 ( X ) X X X μ 2 \mu_2 μ 2 (3) . By Proposition 2.2, we have μ 2 > μ 1 = λ ∗ \mu_2>\mu_1=\lambda^* μ 2 > μ 1 = λ ∗
μ 2 = inf Y ∈ S 2 ( H 0 1 ( Ω ) ) max 0 ≠ v ∈ Y R u ∗ ( v ) . \mu_2=\inf_{Y\in S_2(H_0^1(\Omega))}\ \max_{0\ne v\in Y} R_{u^*}(v). μ 2 = Y ∈ S 2 ( H 0 1 ( Ω )) inf 0 = v ∈ Y max R u ∗ ( v ) . Fix S h ∈ S 2 ( V h 0 ) S_h \in\mathcal{S}_2(V_h^0) S h ∈ S 2 ( V h 0 ) S h S_h S h v ^ h ∈ S h ∖ { 0 } \widehat{v}_h \in S_h \setminus\{0\} v h ∈ S h ∖ { 0 }
R u ∗ ( v ^ h ) = max 0 ≠ v h ∈ S h R u ∗ ( v h ) . R_{u^*}(\widehat{v}_h)=\max_{0\ne v_h\in S_h}R_{u^*}(v_h). R u ∗ ( v h ) = 0 = v h ∈ S h max R u ∗ ( v h ) . By Lemma 3.1, we obtain
max 0 ≠ v h ∈ S h R u h ∗ ( v h ) ≥ R u h ∗ ( v ^ h ) ≥ ( 1 − ϵ h ) R u ∗ ( v ^ h ) = ( 1 − ϵ h ) max 0 ≠ v h ∈ S h R u ∗ ( v h ) . \max_{0\ne v_h\in S_h}R_{u_h^*}(v_h)\ge R_{u_h^*}(\widehat{v}_h)\ge(1-\epsilon_h)R_{u^*}(\widehat{v}_h)=(1-\epsilon_h)\max_{0\ne v_h\in S_h}R_{u^*}(v_h). 0 = v h ∈ S h max R u h ∗ ( v h ) ≥ R u h ∗ ( v h ) ≥ ( 1 − ϵ h ) R u ∗ ( v h ) = ( 1 − ϵ h ) 0 = v h ∈ S h max R u ∗ ( v h ) . Taking the infimum over all S h ∈ S 2 ( V h 0 ) S_h \in\mathcal{S}_2(V_h^0) S h ∈ S 2 ( V h 0 )
Λ 2 , h = inf S h ∈ S 2 ( V h 0 ) max 0 ≠ v h ∈ S h R u h ∗ ( v h ) ≥ ( 1 − ϵ h ) inf S h ∈ S 2 ( V h 0 ) max 0 ≠ v h ∈ S h R u ∗ ( v h ) ⏟ = : Λ ^ 2 , h . \Lambda_{2,h}=\inf_{S_h\in\mathcal{S}_2(V_h^0)}\max_{0\ne v_h\in S_h}R_{u_h^*}(v_h)\ge(1-\epsilon_h)\underbrace{\inf_{S_h\in\mathcal{S}_2(V_h^0)}\max_{0\ne v_h\in S_h}R_{u^*}(v_h)}_{=:\widehat{\Lambda}_{2,h}}. Λ 2 , h = S h ∈ S 2 ( V h 0 ) inf 0 = v h ∈ S h max R u h ∗ ( v h ) ≥ ( 1 − ϵ h ) =: Λ 2 , h S h ∈ S 2 ( V h 0 ) inf 0 = v h ∈ S h max R u ∗ ( v h ) . Since V h 0 ⊂ H 0 1 ( Ω ) V_h^0\subset H_0^1(\Omega) V h 0 ⊂ H 0 1 ( Ω ) S 2 ( V h 0 ) ⊂ S 2 ( H 0 1 ( Ω ) ) \mathcal{S}_2(V_h^0)\subset\mathcal{S}_2(H_0^1(\Omega)) S 2 ( V h 0 ) ⊂ S 2 ( H 0 1 ( Ω ))
Λ ^ 2 , h = inf S h ∈ S 2 ( V h 0 ) max 0 ≠ v h ∈ S h R u ∗ ( v h ) ≥ inf Y ∈ S 2 ( H 0 1 ( Ω ) ) max 0 ≠ v ∈ Y R u ∗ ( v ) = μ 2 . \widehat{\Lambda}_{2,h}=\inf_{S_h\in\mathcal{S}_2(V_h^0)}\max_{0\ne v_h\in S_h}R_{u^*}(v_h)\ge\inf_{Y\in\mathcal{S}_2(H_0^1(\Omega))}\max_{0\ne v\in Y}R_{u^*}(v)=\mu_2. Λ 2 , h = S h ∈ S 2 ( V h 0 ) inf 0 = v h ∈ S h max R u ∗ ( v h ) ≥ Y ∈ S 2 ( H 0 1 ( Ω )) inf 0 = v ∈ Y max R u ∗ ( v ) = μ 2 . Consequently,
Λ 2 , h ≥ ( 1 − ϵ h ) μ 2 . \Lambda_{2,h}\ge(1-\epsilon_h)\mu_2. Λ 2 , h ≥ ( 1 − ϵ h ) μ 2 . Set δ : = μ 2 − λ ∗ > 0 \delta:=\mu_2-\lambda^*>0 δ := μ 2 − λ ∗ > 0 λ h ∗ = a u h ∗ ( u h ∗ , u h ∗ ) \lambda_h^*=a_{u_h^*}(u_h^*,u_h^*) λ h ∗ = a u h ∗ ( u h ∗ , u h ∗ ) λ ∗ = a u ∗ ( u ∗ , u ∗ ) \lambda^*=a_{u^*}(u^*,u^*) λ ∗ = a u ∗ ( u ∗ , u ∗ ) (15) that λ h ∗ → λ ∗ \lambda_h^*\to\lambda^* λ h ∗ → λ ∗ h → 0 h\to0 h → 0 ϵ h → 0 \epsilon_h\to0 ϵ h → 0 h 0 > 0 h_0>0 h 0 > 0 h ≤ h 0 h\le h_0 h ≤ h 0
ϵ h μ 2 < δ 4 , ∣ λ h ∗ − λ ∗ ∣ < δ 4 . \epsilon_h\mu_2<\frac{\delta}{4},\qquad|\lambda_h^*-\lambda^*|<\frac{\delta}{4}. ϵ h μ 2 < 4 δ , ∣ λ h ∗ − λ ∗ ∣ < 4 δ . For such h h h
Λ 2 , h − λ h ∗ ≥ ( μ 2 − δ 4 ) − ( λ ∗ + δ 4 ) = δ 2 > 0. \Lambda_{2,h}-\lambda_h^*\ge\left(\mu_2-\frac{\delta}{4}\right)-\left(\lambda^*+\frac{\delta}{4}\right)=\frac{\delta}{2}>0. Λ 2 , h − λ h ∗ ≥ ( μ 2 − 4 δ ) − ( λ ∗ + 4 δ ) = 2 δ > 0. Since λ h ∗ \lambda_h^* λ h ∗ λ h ∗ < Λ 2 , h \lambda_h^*<\Lambda_{2,h} λ h ∗ < Λ 2 , h Λ 1 , h = λ h ∗ < Λ 2 , h \Lambda_{1,h}=\lambda_h^*<\Lambda_{2,h} Λ 1 , h = λ h ∗ < Λ 2 , h
Corollary 3.3. It holds that
lim inf h → 0 ( Λ 2 , h − λ h ∗ ) ≥ μ 2 − λ ∗ > 0. \liminf_{h\to0}(\Lambda_{2,h}-\lambda_h^*)\ge\mu_2-\lambda^*>0. h → 0 lim inf ( Λ 2 , h − λ h ∗ ) ≥ μ 2 − λ ∗ > 0. This follows directly from the proof of Lemma 3.2, since Λ 2 , h − λ h ∗ ≥ ( 1 − ϵ h ) μ 2 − λ h ∗ \Lambda_{2,h}-\lambda_h^*\ge(1-\epsilon_h)\mu_2-\lambda_h^* Λ 2 , h − λ h ∗ ≥ ( 1 − ϵ h ) μ 2 − λ h ∗ ϵ h → 0 \epsilon_h\to0 ϵ h → 0 λ h ∗ → λ ∗ \lambda_h^*\to\lambda^* λ h ∗ → λ ∗
Remark 3.4. In [29 Section 2] , it is pointed out that, if the discrete ground states converge to u ∗ u^* u ∗ L ∞ ( Ω ) L^\infty(\Omega) L ∞ ( Ω ) λ h ∗ \lambda_h^* λ h ∗ h h h H 1 ( Ω ) H^1(\Omega) H 1 ( Ω )
Combining Lemma 3.2 with Corollary 2.11 for H = V h 0 H=V_h^0 H = V h 0
Theorem 3.5. There exists h 0 > 0 h_0>0 h 0 > 0 0 < h ≤ h 0 0<h\le h_0 0 < h ≤ h 0 E E E V h 0 ∩ M V_h^0\cap\mathcal{M} V h 0 ∩ M u h ∗ u_h^* u h ∗ − u h ∗ -u_h^* − u h ∗
L p L^p L p W 1 , p W^{1,p} W 1 , p We first recall the following a priori error estimates in the L 2 L^2 L 2 H 1 H^1 H 1 [10 , 16 , 46 ] .
Theorem 3.6. There exists h 0 > 0 h_0 > 0 h 0 > 0 0 < h ≤ h 0 0 < h \le h_0 0 < h ≤ h 0
∣ λ h ∗ − λ ∗ ∣ + ∥ u h ∗ − u ∗ ∥ L 2 ( Ω ) + h ∥ ∇ ( u h ∗ − u ∗ ) ∥ L 2 ( Ω ) ≤ C h 2 . |\lambda_h^* - \lambda^*| + \|u_h^* - u^*\|_{L^2(\Omega)} + h\|\nabla(u_h^* - u^*)\|_{L^2(\Omega)} \le Ch^2. ∣ λ h ∗ − λ ∗ ∣ + ∥ u h ∗ − u ∗ ∥ L 2 ( Ω ) + h ∥∇ ( u h ∗ − u ∗ ) ∥ L 2 ( Ω ) ≤ C h 2 . As a consequence, we obtain the following uniform bounds.
Lemma 3.7. There exists h 0 > 0 h_0 > 0 h 0 > 0 0 < h ≤ h 0 0 < h \le h_0 0 < h ≤ h 0
∥ u h ∗ ∥ L ∞ ( Ω ) + ∣ λ h ∗ ∣ ≤ C . \|u_h^*\|_{L^\infty(\Omega)} + |\lambda_h^*| \le C. ∥ u h ∗ ∥ L ∞ ( Ω ) + ∣ λ h ∗ ∣ ≤ C . Proof. The bound for λ h ∗ \lambda_h^* λ h ∗ Theorem 3.6 , so it remains to bound u h ∗ u_h^* u h ∗ [8 ] and the interpolation estimate, we obtain
∥ u h ∗ ∥ L ∞ ( Ω ) ≤ ∥ u h ∗ − I h u ∗ ∥ L ∞ ( Ω ) + ∥ I h u ∗ ∥ L ∞ ( Ω ) ≤ C h − d 2 ( ∥ u h ∗ − u ∗ ∥ L 2 ( Ω ) + ∥ u ∗ − I h u ∗ ∥ L 2 ( Ω ) ) + ∥ u ∗ ∥ L ∞ ( Ω ) ≤ C h 2 − d 2 + C ≤ C , \begin{aligned}
\|u_h^*\|_{L^\infty(\Omega)}
&\le\|u_h^* - I_hu^*\|_{L^\infty(\Omega)} + \|I_hu^*\|_{L^\infty(\Omega)} \\
&\le Ch^{-\frac{d}{2}}\left(\|u_h^* - u^*\|_{L^2(\Omega)} + \|u^* - I_hu^*\|_{L^2(\Omega)}\right) + \|u^*\|_{L^\infty(\Omega)} \\
&\le Ch^{2-\frac{d}{2}} + C \le C,
\end{aligned} ∥ u h ∗ ∥ L ∞ ( Ω ) ≤ ∥ u h ∗ − I h u ∗ ∥ L ∞ ( Ω ) + ∥ I h u ∗ ∥ L ∞ ( Ω ) ≤ C h − 2 d ( ∥ u h ∗ − u ∗ ∥ L 2 ( Ω ) + ∥ u ∗ − I h u ∗ ∥ L 2 ( Ω ) ) + ∥ u ∗ ∥ L ∞ ( Ω ) ≤ C h 2 − 2 d + C ≤ C , where we used d ≤ 3 d \le3 d ≤ 3
Since Ω \Omega Ω Theorem 3.6 imply that for all p ∈ [ 1 , 2 ] p \in[1,2] p ∈ [ 1 , 2 ]
∥ u h ∗ − u ∗ ∥ L p ( Ω ) + h ∥ ∇ ( u h ∗ − u ∗ ) ∥ L p ( Ω ) ≤ C h 2 . (16) \|u_h^* - u^*\|_{L^p(\Omega)} + h\|\nabla(u_h^* - u^*)\|_{L^p(\Omega)} \le Ch^2. \tag*{(16)} ∥ u h ∗ − u ∗ ∥ L p ( Ω ) + h ∥∇ ( u h ∗ − u ∗ ) ∥ L p ( Ω ) ≤ C h 2 . (16) It therefore remains to consider the case 2 < p ≤ ∞ 2 < p \le\infty 2 < p ≤ ∞
We prove the convergence in L p L^p L p W 1 , p W^{1,p} W 1 , p [20 , 16 , 31 ] . The key observation is that both u ∗ u^* u ∗ u h ∗ u_h^* u h ∗ (1) , the ground state u ∗ ∈ H 0 1 ( Ω ) u^* \in H^1_0(\Omega) u ∗ ∈ H 0 1 ( Ω )
− Δ u ∗ = f : = λ ∗ u ∗ − V u ∗ − β ( u ∗ ) 3 . (17) -\Delta u^* = f := \lambda^*u^* - Vu^* - \beta(u^*)^3. \tag*{(17)} − Δ u ∗ = f := λ ∗ u ∗ − V u ∗ − β ( u ∗ ) 3 . (17) and, by (14) , the discrete ground state u h ∗ u_h^* u h ∗
( ∇ u h ∗ , ∇ v h ) L 2 ( Ω ) = ( f h , v h ) L 2 ( Ω ) ∀ v h ∈ V h 0 , (\nabla u_h^*, \nabla v_h)_{L^2(\Omega)} = (f_h, v_h)_{L^2(\Omega)} \qquad\forall v_h \in V_h^0, ( ∇ u h ∗ , ∇ v h ) L 2 ( Ω ) = ( f h , v h ) L 2 ( Ω ) ∀ v h ∈ V h 0 , where f h : = λ h ∗ u h ∗ − V u h ∗ − β ( u h ∗ ) 3 f_h := \lambda_h^*u_h^* - Vu_h^* - \beta(u_h^*)^3 f h := λ h ∗ u h ∗ − V u h ∗ − β ( u h ∗ ) 3 f − f h f - f_h f − f h
Lemma 3.8. There exists h 0 > 0 h_0 > 0 h 0 > 0 0 < h ≤ h 0 0 < h \le h_0 0 < h ≤ h 0
∥ f − f h ∥ L 2 ( Ω ) ≤ C h 2 . \|f - f_h\|_{L^2(\Omega)} \le Ch^2. ∥ f − f h ∥ L 2 ( Ω ) ≤ C h 2 . Proof. We decompose f − f h f - f_h f − f h
f − f h = ( λ ∗ − λ h ∗ ) u ∗ + λ h ∗ ( u ∗ − u h ∗ ) − V ( u ∗ − u h ∗ ) − β ( ( u ∗ ) 3 − ( u h ∗ ) 3 ) = ( λ ∗ − λ h ∗ ) u ∗ + ( λ h ∗ − V − β ( ( u ∗ ) 2 + u ∗ u h ∗ + ( u h ∗ ) 2 ) ) ( u ∗ − u h ∗ ) . \begin{aligned}
f - f_h
&= (\lambda^* - \lambda_h^*)u^* + \lambda_h^*(u^* - u_h^*) - V(u^* - u_h^*) - \beta((u^*)^3 - (u_h^*)^3) \\
&= (\lambda^* - \lambda_h^*)u^* + \left(\lambda_h^* - V - \beta\left((u^*)^2 + u^*u_h^* + (u_h^*)^2\right)\right)(u^* - u_h^*).
\end{aligned} f − f h = ( λ ∗ − λ h ∗ ) u ∗ + λ h ∗ ( u ∗ − u h ∗ ) − V ( u ∗ − u h ∗ ) − β (( u ∗ ) 3 − ( u h ∗ ) 3 ) = ( λ ∗ − λ h ∗ ) u ∗ + ( λ h ∗ − V − β ( ( u ∗ ) 2 + u ∗ u h ∗ + ( u h ∗ ) 2 ) ) ( u ∗ − u h ∗ ) . By Theorem 3.6 , Lemma 3.7 , and ∥ u ∗ ∥ L 2 ( Ω ) = 1 \|u^*\|_{L^2(\Omega)} = 1 ∥ u ∗ ∥ L 2 ( Ω ) = 1
∥ f − f h ∥ L 2 ( Ω ) ≤ ∥ λ h ∗ − V − β ( ( u ∗ ) 2 + u ∗ u h ∗ + ( u h ∗ ) 2 ) ∥ L ∞ ( Ω ) × ∥ u ∗ − u h ∗ ∥ L 2 ( Ω ) + ∣ λ ∗ − λ h ∗ ∣ ≤ C h 2 , \|f - f_h\|_{L^2(\Omega)}
\le\left\|\lambda_h^* - V - \beta\left((u^*)^2 + u^*u_h^* + (u_h^*)^2\right)\right\|_{L^\infty(\Omega)}
\times\|u^* - u_h^*\|_{L^2(\Omega)} + |\lambda^* - \lambda_h^*| \le Ch^2, ∥ f − f h ∥ L 2 ( Ω ) ≤ λ h ∗ − V − β ( ( u ∗ ) 2 + u ∗ u h ∗ + ( u h ∗ ) 2 ) L ∞ ( Ω ) × ∥ u ∗ − u h ∗ ∥ L 2 ( Ω ) + ∣ λ ∗ − λ h ∗ ∣ ≤ C h 2 , which completes the proof. □
The error admits the decomposition
u h ∗ − u ∗ = ( u h ∗ − R h u ∗ ) + ( R h u ∗ − u ∗ ) = : θ h + ρ h . (18) u_h^* - u^* = (u_h^* - R_hu^*) + (R_hu^* - u^*) =: \theta_h + \rho_h. \tag*{(18)} u h ∗ − u ∗ = ( u h ∗ − R h u ∗ ) + ( R h u ∗ − u ∗ ) =: θ h + ρ h . (18) By the definition of R h R_h R h (17) , θ h ∈ V h 0 \theta_h \in V_h^0 θ h ∈ V h 0
( ∇ θ h , ∇ v h ) L 2 ( Ω ) = ( f h − f , v h ) L 2 ( Ω ) ∀ v h ∈ V h 0 . (\nabla\theta_h, \nabla v_h)_{L^2(\Omega)} = (f_h - f, v_h)_{L^2(\Omega)} \qquad\forall v_h \in V_h^0. ( ∇ θ h , ∇ v h ) L 2 ( Ω ) = ( f h − f , v h ) L 2 ( Ω ) ∀ v h ∈ V h 0 . Lemmas 2.8 and 3.8 then yield the following superconvergence result.
Lemma 3.9. There exists h 0 > 0 h_0 > 0 h 0 > 0 0 < h ≤ h 0 0 < h \le h_0 0 < h ≤ h 0
∥ ∇ θ h ∥ L 6 ( Ω ) ≤ C h 2 . \|\nabla\theta_h\|_{L^6(\Omega)} \le Ch^2. ∥∇ θ h ∥ L 6 ( Ω ) ≤ C h 2 . Remark 3.10. For the linear finite element method, it is shown in [29 Theorem 2] that
∥ Π h u ∗ − u h ∗ ∥ H 1 ( Ω ) ≤ C h 2 , \|\Pi_hu^* - u_h^*\|_{H^1(\Omega)} \le Ch^2, ∥ Π h u ∗ − u h ∗ ∥ H 1 ( Ω ) ≤ C h 2 , where Π h \Pi_h Π h H 1 ( Ω ) H^1(\Omega) H 1 ( Ω ) V h 0 V_h^0 V h 0 W 1 , 6 ( Ω ) W^{1,6}(\Omega) W 1 , 6 ( Ω )
Combining Lemma 3.9 with the Sobolev embedding W 1 , 6 ( Ω ) ↪ L ∞ ( Ω ) W^{1,6}(\Omega) \hookrightarrow L^\infty(\Omega) W 1 , 6 ( Ω ) ↪ L ∞ ( Ω ) d ≤ 3 d \le3 d ≤ 3
Lemma 3.11. There exists h 0 > 0 h_0 > 0 h 0 > 0 0 < h ≤ h 0 0 < h \le h_0 0 < h ≤ h 0
∥ θ h ∥ L ∞ ( Ω ) ≤ C h 2 , ∥ ∇ θ h ∥ L ∞ ( Ω ) ≤ C h 2 − d 6 . \|\theta_h\|_{L^\infty(\Omega)} \le Ch^2,\qquad\|\nabla\theta_h\|_{L^\infty(\Omega)} \le Ch^{2-\frac{d}{6}}. ∥ θ h ∥ L ∞ ( Ω ) ≤ C h 2 , ∥∇ θ h ∥ L ∞ ( Ω ) ≤ C h 2 − 6 d . To establish convergence in L p ( Ω ) L^p(\Omega) L p ( Ω ) W 1 , p ( Ω ) W^{1,p}(\Omega) W 1 , p ( Ω ) p = ∞ p = \infty p = ∞ u ∗ u^* u ∗ V ∈ L ∞ ( Ω ) V \in L^\infty(\Omega) V ∈ L ∞ ( Ω ) u ∗ ∈ H 2 ( Ω ) ↪ L ∞ ( Ω ) u^* \in H^2(\Omega) \hookrightarrow L^\infty(\Omega) u ∗ ∈ H 2 ( Ω ) ↪ L ∞ ( Ω ) d ≤ 3 d \le3 d ≤ 3 f f f (17) belongs to L ∞ ( Ω ) L^\infty(\Omega) L ∞ ( Ω ) [23 , 28 ] then yields u ∗ ∈ W 2 , p ( Ω ) ∩ H 0 1 ( Ω ) u^* \in W^{2,p}(\Omega) \cap H_0^1(\Omega) u ∗ ∈ W 2 , p ( Ω ) ∩ H 0 1 ( Ω ) 2 ≤ p < ∞ 2 \le p < \infty 2 ≤ p < ∞
Theorem 3.12. For every 1 ≤ p < ∞ 1 \le p < \infty 1 ≤ p < ∞ h 0 > 0 h_0 > 0 h 0 > 0 0 < h ≤ h 0 0 < h \le h_0 0 < h ≤ h 0
∥ u ∗ − u h ∗ ∥ L p ( Ω ) + h ∥ ∇ ( u ∗ − u h ∗ ) ∥ L p ( Ω ) ≤ C p h 2 . \|u^* - u_h^*\|_{L^p(\Omega)} + h\|\nabla(u^* - u_h^*)\|_{L^p(\Omega)} \le C_ph^2. ∥ u ∗ − u h ∗ ∥ L p ( Ω ) + h ∥∇ ( u ∗ − u h ∗ ) ∥ L p ( Ω ) ≤ C p h 2 . Proof. In view of (16) , it suffices to consider 2 < p < ∞ 2 < p < \infty 2 < p < ∞ u ∗ ∈ W 2 , p ( Ω ) u^* \in W^{2,p}(\Omega) u ∗ ∈ W 2 , p ( Ω )
∥ u ∗ − u h ∗ ∥ L p ( Ω ) ≤ ∥ θ h ∥ L p ( Ω ) + ∥ ρ h ∥ L p ( Ω ) ≤ C ∥ θ h ∥ L ∞ ( Ω ) + C p h 2 ≤ C p h 2 . \|u^* - u_h^*\|_{L^p(\Omega)} \le\|\theta_h\|_{L^p(\Omega)} + \|\rho_h\|_{L^p(\Omega)} \le C\|\theta_h\|_{L^\infty(\Omega)} + C_ph^2 \le C_ph^2. ∥ u ∗ − u h ∗ ∥ L p ( Ω ) ≤ ∥ θ h ∥ L p ( Ω ) + ∥ ρ h ∥ L p ( Ω ) ≤ C ∥ θ h ∥ L ∞ ( Ω ) + C p h 2 ≤ C p h 2 . Similarly, using 2 − d 6 ≥ 1 2 - \frac{d}{6} \ge1 2 − 6 d ≥ 1 d ≤ 3 d \le3 d ≤ 3
∥ ∇ ( u ∗ − u h ∗ ) ∥ L p ( Ω ) ≤ C ∥ ∇ θ h ∥ L ∞ ( Ω ) + ∥ ∇ ρ h ∥ L p ( Ω ) ≤ C h 2 − d 6 + C p h ≤ C p h , \|\nabla(u^* - u_h^*)\|_{L^p(\Omega)} \le C\|\nabla\theta_h\|_{L^\infty(\Omega)} + \|\nabla\rho_h\|_{L^p(\Omega)} \le Ch^{2-\frac{d}{6}} + C_ph \le C_ph, ∥∇ ( u ∗ − u h ∗ ) ∥ L p ( Ω ) ≤ C ∥∇ θ h ∥ L ∞ ( Ω ) + ∥∇ ρ h ∥ L p ( Ω ) ≤ C h 2 − 6 d + C p h ≤ C p h , which completes the proof.
Remark 3.13. The superconvergence result in [29 ] implies ∥ ∇ θ h ∥ L 2 ( Ω ) ≤ C h 2 \|\nabla\theta_h\|_{L^2(\Omega)} \le Ch^2 ∥∇ θ h ∥ L 2 ( Ω ) ≤ C h 2 d = 2 d = 2 d = 2 H 0 1 ( Ω ) ↪ L p ( Ω ) H_0^1(\Omega) \hookrightarrow L^p(\Omega) H 0 1 ( Ω ) ↪ L p ( Ω ) 1 ≤ p < ∞ 1 \le p < \infty 1 ≤ p < ∞
∥ θ h ∥ L p ( Ω ) ≤ C p ∥ ∇ θ h ∥ L 2 ( Ω ) ≤ C p h 2 , ∥ ∇ θ h ∥ L ∞ ( Ω ) ≤ C h − 1 ∥ ∇ θ h ∥ L 2 ( Ω ) ≤ C h . \|\theta_h\|_{L^p(\Omega)} \le C_p\|\nabla\theta_h\|_{L^2(\Omega)} \le C_ph^2,\qquad\|\nabla\theta_h\|_{L^\infty(\Omega)} \le Ch^{-1}\|\nabla\theta_h\|_{L^2(\Omega)} \le Ch. ∥ θ h ∥ L p ( Ω ) ≤ C p ∥∇ θ h ∥ L 2 ( Ω ) ≤ C p h 2 , ∥∇ θ h ∥ L ∞ ( Ω ) ≤ C h − 1 ∥∇ θ h ∥ L 2 ( Ω ) ≤ C h . Combined with Proposition 2.6 , this recovers Theorem 3.12 for d = 2 d=2 d = 2 d = 3 d=3 d = 3 p ≤ 6 p\le6 p ≤ 6 H 0 1 ( Ω ) ↪ L p ( Ω ) H^1_0(\Omega)\hookrightarrow L^p(\Omega) H 0 1 ( Ω ) ↪ L p ( Ω ) p ≤ 6 p\le6 p ≤ 6 ∥ ∇ θ h ∥ L p ( Ω ) ≤ C h 1 2 + 3 p \|\nabla\theta_h\|_{L^p(\Omega)}\le Ch^{\frac{1}{2}+\frac{3}{p}} ∥∇ θ h ∥ L p ( Ω ) ≤ C h 2 1 + p 3 p > 6 p>6 p > 6
To establish convergence in L ∞ ( Ω ) L^\infty(\Omega) L ∞ ( Ω ) W 1 , ∞ ( Ω ) W^{1,\infty}(\Omega) W 1 , ∞ ( Ω ) W 2 , ∞ ( Ω ) W^{2,\infty}(\Omega) W 2 , ∞ ( Ω ) u ∗ u^* u ∗ u ∗ ∈ H 2 ( Ω ) ↪ C ( Ω ‾ ) u^*\in H^2(\Omega)\hookrightarrow C(\overline{\Omega}) u ∗ ∈ H 2 ( Ω ) ↪ C ( Ω ) u ∗ u^* u ∗ ∂ Ω \partial\Omega ∂ Ω f f f (17) vanishes on ∂ Ω \partial\Omega ∂ Ω d = 3 d=3 d = 3 Ω \Omega Ω V V V [23 , 28 , 32 ] .
Proposition 3.14. Assume, in addition, that V ∈ C 0 , α ( Ω ‾ ) V\in C^{0,\alpha}(\overline{\Omega}) V ∈ C 0 , α ( Ω ) α ∈ ( 0 , 1 ) \alpha\in(0,1) α ∈ ( 0 , 1 ) u ∗ ∈ W 2 , ∞ ( Ω ) u^*\in W^{2,\infty}(\Omega) u ∗ ∈ W 2 , ∞ ( Ω )
Proceeding as in the proof of Theorem 3.12 , we obtain the following result.
Theorem 3.15. Assume that V ∈ C 0 , α ( Ω ‾ ) V\in C^{0,\alpha}(\overline{\Omega}) V ∈ C 0 , α ( Ω ) α ∈ ( 0 , 1 ) \alpha\in(0,1) α ∈ ( 0 , 1 ) h 0 > 0 h_0>0 h 0 > 0 0 < h ≤ h 0 0<h\le h_0 0 < h ≤ h 0
∥ u ∗ − u h ∗ ∥ L ∞ ( Ω ) ≤ C h 2 ∣ ln h ∣ , ∥ ∇ ( u ∗ − u h ∗ ) ∥ L ∞ ( Ω ) ≤ C h . \|u^*-u_h^*\|_{L^\infty(\Omega)}\le Ch^2|\ln h|,\qquad\|\nabla(u^*-u_h^*)\|_{L^\infty(\Omega)}\le Ch. ∥ u ∗ − u h ∗ ∥ L ∞ ( Ω ) ≤ C h 2 ∣ ln h ∣ , ∥∇ ( u ∗ − u h ∗ ) ∥ L ∞ ( Ω ) ≤ C h . Superconvergence
Lemma 3.9 shows that θ h \theta_h θ h W 1 , p ( Ω ) W^{1,p}(\Omega) W 1 , p ( Ω ) 1 ≤ p ≤ 6 1\le p\le6 1 ≤ p ≤ 6 Theorem 3.12 , this result can be extended to 1 ≤ p ≤ ∞ 1\le p\le\infty 1 ≤ p ≤ ∞
Theorem 3.16. For every 1 ≤ p ≤ ∞ 1\le p\le\infty 1 ≤ p ≤ ∞ h 0 > 0 h_0>0 h 0 > 0 0 < h ≤ h 0 0<h\le h_0 0 < h ≤ h 0
∥ ∇ ( R h u ∗ − u h ∗ ) ∥ L p ( Ω ) ≤ C p h 2 . \|\nabla(R_hu^*-u_h^*)\|_{L^p(\Omega)}\le C_ph^2. ∥∇ ( R h u ∗ − u h ∗ ) ∥ L p ( Ω ) ≤ C p h 2 . Proof. It suffices to consider p = ∞ p=\infty p = ∞ Lemma 3.8 , combined with the L 4 L^4 L 4 Theorem 3.12 , yields
∥ f − f h ∥ L 4 ( Ω ) ≤ C h 2 . \|f-f_h\|_{L^4(\Omega)}\le Ch^2. ∥ f − f h ∥ L 4 ( Ω ) ≤ C h 2 . Let w ∈ H 0 1 ( Ω ) w\in H^1_0(\Omega) w ∈ H 0 1 ( Ω ) − Δ w = f h − f -\Delta w=f_h-f − Δ w = f h − f w ∈ W 2 , 4 ( Ω ) w\in W^{2,4}(\Omega) w ∈ W 2 , 4 ( Ω ) ∥ w ∥ W 2 , 4 ( Ω ) ≤ C ∥ f − f h ∥ L 4 ( Ω ) \|w\|_{W^{2,4}(\Omega)}\le C\|f-f_h\|_{L^4(\Omega)} ∥ w ∥ W 2 , 4 ( Ω ) ≤ C ∥ f − f h ∥ L 4 ( Ω ) θ h = R h w \theta_h=R_hw θ h = R h w θ h \theta_h θ h R h R_h R h Proposition 2.5 and the Sobolev embedding W 2 , 4 ( Ω ) ↪ W 1 , ∞ ( Ω ) W^{2,4}(\Omega)\hookrightarrow W^{1,\infty}(\Omega) W 2 , 4 ( Ω ) ↪ W 1 , ∞ ( Ω ) d ≤ 3 d\le3 d ≤ 3
∥ ∇ θ h ∥ L ∞ ( Ω ) ≤ C ∥ ∇ w ∥ L ∞ ( Ω ) ≤ C ∥ w ∥ W 2 , 4 ( Ω ) ≤ C ∥ f − f h ∥ L 4 ( Ω ) ≤ C h 2 , \|\nabla\theta_h\|_{L^\infty(\Omega)}\le C\|\nabla w\|_{L^\infty(\Omega)}\le C\|w\|_{W^{2,4}(\Omega)}\le C\|f-f_h\|_{L^4(\Omega)}\le Ch^2, ∥∇ θ h ∥ L ∞ ( Ω ) ≤ C ∥∇ w ∥ L ∞ ( Ω ) ≤ C ∥ w ∥ W 2 , 4 ( Ω ) ≤ C ∥ f − f h ∥ L 4 ( Ω ) ≤ C h 2 , which completes the proof.
The mass-lumped finite element method
The mass-lumped FEM differs from the standard FEM in that the L 2 L^2 L 2 K ∈ T h K\in\mathcal{T}_h K ∈ T h N K = { x K , i } i = 0 d N_K=\{x_{K,i}\}_{i=0}^d N K = { x K , i } i = 0 d ϕ ∈ C ( K ‾ ) \phi\in C(\overline{K}) ϕ ∈ C ( K ) K K K
Q K ( ϕ ) : = ∣ K ∣ d + 1 ∑ i = 0 d ϕ ( x K , i ) . Q_K(\phi):=\frac{|K|}{d+1}\sum_{i=0}^d\phi(x_{K,i}). Q K ( ϕ ) := d + 1 ∣ K ∣ i = 0 ∑ d ϕ ( x K , i ) . This quadrature rule is exact for affine functions on K K K ϕ , ψ ∈ C ( K ‾ ) \phi,\psi\in C(\overline{K}) ϕ , ψ ∈ C ( K )
( ϕ , ψ ) ℓ 2 , K : = Q K ( ϕ ψ ) = ∣ K ∣ d + 1 ∑ i = 0 d ϕ ( x K , i ) ψ ( x K , i ) . (\phi,\psi)_{\ell^2,K}:=Q_K(\phi\psi)=\frac{|K|}{d+1}\sum_{i=0}^d\phi(x_{K,i})\psi(x_{K,i}). ( ϕ , ψ ) ℓ 2 , K := Q K ( ϕ ψ ) = d + 1 ∣ K ∣ i = 0 ∑ d ϕ ( x K , i ) ψ ( x K , i ) . This leads to the global discrete L 2 L^2 L 2 L 2 L^2 L 2
( ϕ , ψ ) ℓ 2 : = ∑ K ∈ T h ( ϕ , ψ ) ℓ 2 , K = ∑ z ∈ N h ω z ϕ ( z ) ψ ( z ) , ∥ ϕ ∥ ℓ 2 2 : = ( ϕ , ϕ ) ℓ 2 , (\phi,\psi)_{\ell^2}:=\sum_{K\in\mathcal{T}_h}(\phi,\psi)_{\ell^2,K}=\sum_{z\in\mathcal{N}_h}\omega_z\phi(z)\psi(z),\qquad\|\phi\|_{\ell^2}^2:=(\phi,\phi)_{\ell^2}, ( ϕ , ψ ) ℓ 2 := K ∈ T h ∑ ( ϕ , ψ ) ℓ 2 , K = z ∈ N h ∑ ω z ϕ ( z ) ψ ( z ) , ∥ ϕ ∥ ℓ 2 2 := ( ϕ , ϕ ) ℓ 2 , where ω z : = 1 d + 1 ∑ K ∈ T h , z ∈ N K ∣ K ∣ \omega_z:=\frac{1}{d+1}\sum_{\substack{K\in\mathcal{T}_h,\,z\in\mathcal{N}_K}}|K| ω z := d + 1 1 ∑ K ∈ T h , z ∈ N K ∣ K ∣
Throughout this section, we additionally assume that V ∈ W 2 , ∞ ( Ω ) V\in W^{2,\infty}(\Omega) V ∈ W 2 , ∞ ( Ω ) v h ∈ V h 0 v_h\in V_h^0 v h ∈ V h 0
E h ( v h ) : = 1 2 ∥ ∇ v h ∥ L 2 ( Ω ) 2 + 1 2 ( V v h , v h ) ℓ 2 + β 4 ( v h 2 , v h 2 ) ℓ 2 . E_h(v_h):=\frac{1}{2}\|\nabla v_h\|_{L^2(\Omega)}^2+\frac{1}{2}(Vv_h,v_h)_{\ell^2}+\frac{\beta}{4}(v_h^2,v_h^2)_{\ell^2}. E h ( v h ) := 2 1 ∥∇ v h ∥ L 2 ( Ω ) 2 + 2 1 ( V v h , v h ) ℓ 2 + 4 β ( v h 2 , v h 2 ) ℓ 2 . The discrete ground state of the mass-lumped FEM is then defined as
u h ∗ , ℓ ∈ arg min v h ∈ V h 0 , ∥ v h ∥ ℓ 2 = 1 E h ( v h ) with ( u h ∗ , ℓ , u ∗ ) L 2 ( Ω ) ≥ 0. (19) u_h^{*,\ell}\in\underset{v_h\in V_h^0,\,\|v_h\|_{\ell^2}=1}{\arg\min}\ E_h(v_h)\quad\text{with}\quad(u_h^{*,\ell},u^*)_{L^2(\Omega)}\geq0. \tag*{(19)} u h ∗ , ℓ ∈ v h ∈ V h 0 , ∥ v h ∥ ℓ 2 = 1 arg min E h ( v h ) with ( u h ∗ , ℓ , u ∗ ) L 2 ( Ω ) ≥ 0. (19) It satisfies the discrete Euler–Lagrange equation
( ∇ u h ∗ , ℓ , ∇ v h ) L 2 ( Ω ) + ( V u h ∗ , ℓ , v h ) ℓ 2 + β ( ( u h ∗ , ℓ ) 3 , v h ) ℓ 2 = λ h ∗ , ℓ ( u h ∗ , ℓ , v h ) ℓ 2 ∀ v h ∈ V h 0 , (20) (\nabla u_h^{*,\ell},\nabla v_h)_{L^2(\Omega)}+(Vu_h^{*,\ell},v_h)_{\ell^2}+\beta((u_h^{*,\ell})^3,v_h)_{\ell^2}=\lambda_h^{*,\ell}(u_h^{*,\ell},v_h)_{\ell^2}\qquad\forall\,v_h\in V_h^0, \tag*{(20)} ( ∇ u h ∗ , ℓ , ∇ v h ) L 2 ( Ω ) + ( V u h ∗ , ℓ , v h ) ℓ 2 + β (( u h ∗ , ℓ ) 3 , v h ) ℓ 2 = λ h ∗ , ℓ ( u h ∗ , ℓ , v h ) ℓ 2 ∀ v h ∈ V h 0 , (20) where λ h ∗ , ℓ : = 2 E h ( u h ∗ , ℓ ) + β 2 ( ( u h ∗ , ℓ ) 2 , ( u h ∗ , ℓ ) 2 ) ℓ 2 \lambda_h^{*,\ell}:=2E_h(u_h^{*,\ell})+\frac{\beta}{2}((u_h^{*,\ell})^2,(u_h^{*,\ell})^2)_{\ell^2} λ h ∗ , ℓ := 2 E h ( u h ∗ , ℓ ) + 2 β (( u h ∗ , ℓ ) 2 , ( u h ∗ , ℓ ) 2 ) ℓ 2 [30 ] .
Proposition 4.1. Assume that the stiffness matrix associated with V h 0 V_h^0 V h 0 M M M u h ∗ , ℓ u_h^{*,\ell} u h ∗ , ℓ Ω \Omega Ω h 0 > 0 h_0>0 h 0 > 0 0 < h ≤ h 0 0<h\leq h_0 0 < h ≤ h 0
∣ λ ∗ − λ h ∗ , ℓ ∣ + ∥ u h ∗ , ℓ − u ∗ ∥ L 2 ( Ω ) + h ∥ ∇ ( u h ∗ , ℓ − u ∗ ) ∥ L 2 ( Ω ) ≤ C h 2 |\lambda^*-\lambda_h^{*,\ell}|+\|u_h^{*,\ell}-u^*\|_{L^2(\Omega)}+h\|\nabla(u_h^{*,\ell}-u^*)\|_{L^2(\Omega)}\leq Ch^2 ∣ λ ∗ − λ h ∗ , ℓ ∣ + ∥ u h ∗ , ℓ − u ∗ ∥ L 2 ( Ω ) + h ∥∇ ( u h ∗ , ℓ − u ∗ ) ∥ L 2 ( Ω ) ≤ C h 2 and
∥ u h ∗ , ℓ ∥ L ∞ ( Ω ) + ∣ λ h ∗ , ℓ ∣ ≤ C . \|u_h^{*,\ell}\|_{L^\infty(\Omega)}+|\lambda_h^{*,\ell}|\leq C. ∥ u h ∗ , ℓ ∥ L ∞ ( Ω ) + ∣ λ h ∗ , ℓ ∣ ≤ C . Some technical results
For an element K ∈ T h K\in\mathcal{T}_h K ∈ T h φ K , 0 , … , φ K , d \varphi_{K,0},\ldots,\varphi_{K,d} φ K , 0 , … , φ K , d K K K K K K φ K , i ( x K , j ) = δ i j \varphi_{K,i}(x_{K,j})=\delta_{ij} φ K , i ( x K , j ) = δ ij
∫ K φ K , i φ K , j d x = ∣ K ∣ ( d + 1 ) ( d + 2 ) ( 1 + δ i j ) , i , j = 0 , … , d . \int_K\varphi_{K,i}\varphi_{K,j}\,\mathrm{d}x=\frac{|K|}{(d+1)(d+2)}(1+\delta_{ij}),\qquad i,j=0,\ldots,d. ∫ K φ K , i φ K , j d x = ( d + 1 ) ( d + 2 ) ∣ K ∣ ( 1 + δ ij ) , i , j = 0 , … , d . Moreover, for 1 ≤ q < ∞ 1\leq q<\infty 1 ≤ q < ∞ V h V_h V h
∥ v h ∥ ℓ q : = ( ∑ z ∈ N h ω z ∣ v h ( z ) ∣ q ) 1 q , v h ∈ V h . \|v_h\|_{\ell^q}:=\left(\sum_{z\in\mathcal{N}_h}\omega_z|v_h(z)|^q\right)^{\frac{1}{q}},\qquad v_h\in V_h. ∥ v h ∥ ℓ q := ( z ∈ N h ∑ ω z ∣ v h ( z ) ∣ q ) q 1 , v h ∈ V h . Lemma 4.2. Let 1 ≤ q < ∞ 1\leq q<\infty 1 ≤ q < ∞ 0 < c q ≤ C q 0<c_q\leq C_q 0 < c q ≤ C q v h ∈ V h v_h\in V_h v h ∈ V h
c q ∥ v h ∥ L q ( Ω ) ≤ ∥ v h ∥ ℓ q ≤ C q ∥ v h ∥ L q ( Ω ) . c_q\|v_h\|_{L^q(\Omega)}\leq\|v_h\|_{\ell^q}\leq C_q\|v_h\|_{L^q(\Omega)}. c q ∥ v h ∥ L q ( Ω ) ≤ ∥ v h ∥ ℓ q ≤ C q ∥ v h ∥ L q ( Ω ) . In particular, for q = 2 q=2 q = 2
∥ v h ∥ L 2 ( Ω ) 2 ≤ ∥ v h ∥ ℓ 2 2 ≤ ( d + 2 ) ∥ v h ∥ L 2 ( Ω ) 2 . \|v_h\|_{L^2(\Omega)}^2\leq\|v_h\|_{\ell^2}^2\leq(d+2)\|v_h\|_{L^2(\Omega)}^2. ∥ v h ∥ L 2 ( Ω ) 2 ≤ ∥ v h ∥ ℓ 2 2 ≤ ( d + 2 ) ∥ v h ∥ L 2 ( Ω ) 2 . Proof. The general case follows from the equivalence of norms on finite-dimensional spaces and a standard scaling argument. It remains to derive the explicit constants for q = 2 q=2 q = 2 v h ∈ V h v_h \in V_h v h ∈ V h K ∈ T h K \in\mathcal{T}_h K ∈ T h v h ∣ K = ∑ i = 0 d v i φ K , i v_h|_K=\sum_{i=0}^{d}v_i\varphi_{K,i} v h ∣ K = ∑ i = 0 d v i φ K , i v i : = v h ( x K , i ) v_i:=v_h(x_{K,i}) v i := v h ( x K , i )
∥ v h ∥ L 2 ( K ) 2 = ∣ K ∣ ( d + 1 ) ( d + 2 ) [ ( ∑ i = 0 d v i ) 2 + ∑ i = 0 d v i 2 ] , Q K ( v h 2 ) = ∣ K ∣ d + 1 ∑ i = 0 d v i 2 . \|v_h\|_{L^2(K)}^2=\frac{|K|}{(d+1)(d+2)}\left[\left(\sum_{i=0}^{d}v_i\right)^2+\sum_{i=0}^{d}v_i^2\right],\qquad Q_K(v_h^2)=\frac{|K|}{d+1}\sum_{i=0}^{d}v_i^2. ∥ v h ∥ L 2 ( K ) 2 = ( d + 1 ) ( d + 2 ) ∣ K ∣ ( i = 0 ∑ d v i ) 2 + i = 0 ∑ d v i 2 , Q K ( v h 2 ) = d + 1 ∣ K ∣ i = 0 ∑ d v i 2 . Since 0 ≤ ( ∑ i = 0 d v i ) 2 ≤ ( d + 1 ) ∑ i = 0 d v i 2 0\leq\left(\sum_{i=0}^{d}v_i\right)^2\leq(d+1)\sum_{i=0}^{d}v_i^2 0 ≤ ( ∑ i = 0 d v i ) 2 ≤ ( d + 1 ) ∑ i = 0 d v i 2
∥ v h ∥ L 2 ( K ) 2 ≤ Q K ( v h 2 ) ≤ ( d + 2 ) ∥ v h ∥ L 2 ( K ) 2 . \|v_h\|_{L^2(K)}^2\leq Q_K(v_h^2)\leq(d+2)\|v_h\|_{L^2(K)}^2. ∥ v h ∥ L 2 ( K ) 2 ≤ Q K ( v h 2 ) ≤ ( d + 2 ) ∥ v h ∥ L 2 ( K ) 2 . Summing over all K ∈ T h K\in\mathcal{T}_h K ∈ T h q = 2 q=2 q = 2
For K ∈ T h K\in\mathcal{T}_h K ∈ T h
D K : = 1 ( d + 1 ) ( d + 2 ) ∑ 0 ≤ i < j ≤ d ( x K , i − x K , j ) ( x K , i − x K , j ) T ∈ R d × d . D_K:=\frac{1}{(d+1)(d+2)}\sum_{0\leq i<j\leq d}(x_{K,i}-x_{K,j})(x_{K,i}-x_{K,j})^{T}\in\mathbb{R}^{d\times d}. D K := ( d + 1 ) ( d + 2 ) 1 0 ≤ i < j ≤ d ∑ ( x K , i − x K , j ) ( x K , i − x K , j ) T ∈ R d × d . Lemma 4.3. Let K ∈ T h K\in\mathcal{T}_h K ∈ T h v h , w h ∈ P 1 ( K ) v_h,w_h\in\mathbb{P}_1(K) v h , w h ∈ P 1 ( K ) v i : = v h ( x K , i ) v_i:=v_h(x_{K,i}) v i := v h ( x K , i ) w i : = w h ( x K , i ) w_i:=w_h(x_{K,i}) w i := w h ( x K , i ) i = 0 , … , d i=0,\ldots,d i = 0 , … , d
Q K ( v h w h ) − ∫ K v h w h d x = ∣ K ∣ ( d + 1 ) ( d + 2 ) ∑ 0 ≤ i < j ≤ d ( v i − v j ) ( w i − w j ) = ∫ K ( D K ∇ v h ) ⋅ ∇ w h d x . (21) Q_K(v_hw_h)-\int_K v_hw_h\,\mathrm{d}x=\frac{|K|}{(d+1)(d+2)}\sum_{0\leq i<j\leq d}(v_i-v_j)(w_i-w_j)
=\int_K(D_K\nabla v_h)\cdot\nabla w_h\,\mathrm{d}x. \tag*{(21)} Q K ( v h w h ) − ∫ K v h w h d x = ( d + 1 ) ( d + 2 ) ∣ K ∣ 0 ≤ i < j ≤ d ∑ ( v i − v j ) ( w i − w j ) = ∫ K ( D K ∇ v h ) ⋅ ∇ w h d x . (21) Moreover, D K D_K D K
0 ≤ ξ T D K ξ ≤ d 2 ( d + 2 ) h K 2 ∣ ξ ∣ 2 ∀ ξ ∈ R d . 0\leq\xi^{T}D_K\xi\leq\frac{d}{2(d+2)}h_K^2|\xi|^2\qquad\forall\,\xi\in\mathbb{R}^{d}. 0 ≤ ξ T D K ξ ≤ 2 ( d + 2 ) d h K 2 ∣ ξ ∣ 2 ∀ ξ ∈ R d . Proof. Since v h ∣ K = ∑ i = 0 d v i φ K , i v_h|_K=\sum_{i=0}^{d}v_i\varphi_{K,i} v h ∣ K = ∑ i = 0 d v i φ K , i w h ∣ K = ∑ j = 0 d w j φ K , j w_h|_K=\sum_{j=0}^{d}w_j\varphi_{K,j} w h ∣ K = ∑ j = 0 d w j φ K , j ∫ K φ K , i φ K , j d x \int_K\varphi_{K,i}\varphi_{K,j}\,\mathrm{d}x ∫ K φ K , i φ K , j d x
∫ K v h w h d x = ∣ K ∣ ( d + 1 ) ( d + 2 ) ∑ i , j = 0 d v i w j ( 1 + δ i j ) = ∣ K ∣ ( d + 1 ) ( d + 2 ) [ ∑ i , j = 0 d v i w j + ∑ i = 0 d v i w i ] . \begin{aligned}
\int_Kv_hw_h\,\mathrm{d}x=\frac{|K|}{(d+1)(d+2)}\sum_{i,j=0}^{d}v_iw_j(1+\delta_{ij}) \\
&=\frac{|K|}{(d+1)(d+2)}\left[\sum_{i,j=0}^{d}v_iw_j+\sum_{i=0}^{d}v_iw_i\right].
\end{aligned} ∫ K v h w h d x = ( d + 1 ) ( d + 2 ) ∣ K ∣ i , j = 0 ∑ d v i w j ( 1 + δ ij ) = ( d + 1 ) ( d + 2 ) ∣ K ∣ [ i , j = 0 ∑ d v i w j + i = 0 ∑ d v i w i ] . Together with Q K ( v h w h ) = ∣ K ∣ d + 1 ∑ i = 0 d v i w i Q_K(v_hw_h)=\frac{|K|}{d+1}\sum_{i=0}^{d}v_iw_i Q K ( v h w h ) = d + 1 ∣ K ∣ ∑ i = 0 d v i w i
Q K ( v h w h ) − ∫ K v h w h d x = ∣ K ∣ ( d + 1 ) ( d + 2 ) [ ( d + 1 ) ∑ i = 0 d v i w i − ∑ i , j = 0 d v i w j ] . Q_K(v_hw_h)-\int_Kv_hw_h\,\mathrm{d}x=\frac{|K|}{(d+1)(d+2)}\left[(d+1)\sum_{i=0}^{d}v_iw_i-\sum_{i,j=0}^{d}v_iw_j\right]. Q K ( v h w h ) − ∫ K v h w h d x = ( d + 1 ) ( d + 2 ) ∣ K ∣ [ ( d + 1 ) i = 0 ∑ d v i w i − i , j = 0 ∑ d v i w j ] . On the other hand, a direct computation shows that
∑ 0 ≤ i < j ≤ d ( v i − v j ) ( w i − w j ) = 1 2 ∑ i , j = 0 d ( v i − v j ) ( w i − w j ) = ( d + 1 ) ∑ i = 0 d v i w i − ∑ i , j = 0 d v i w j . \sum_{0\leq i<j\leq d}(v_i-v_j)(w_i-w_j)=\frac{1}{2}\sum_{i,j=0}^{d}(v_i-v_j)(w_i-w_j)=(d+1)\sum_{i=0}^{d}v_iw_i-\sum_{i,j=0}^{d}v_iw_j. 0 ≤ i < j ≤ d ∑ ( v i − v j ) ( w i − w j ) = 2 1 i , j = 0 ∑ d ( v i − v j ) ( w i − w j ) = ( d + 1 ) i = 0 ∑ d v i w i − i , j = 0 ∑ d v i w j . Since v h , w h ∈ P 1 ( K ) v_h,w_h\in P_1(K) v h , w h ∈ P 1 ( K ) K K K
v i − v j = ( x K , i − x K , j ) T ∇ v h , w i − w j = ( x K , i − x K , j ) T ∇ w h . v_i-v_j=(x_{K,i}-x_{K,j})^T\nabla v_h,\qquad w_i-w_j=(x_{K,i}-x_{K,j})^T\nabla w_h. v i − v j = ( x K , i − x K , j ) T ∇ v h , w i − w j = ( x K , i − x K , j ) T ∇ w h . Hence,
∑ 0 ≤ i < j ≤ d ( v i − v j ) ( w i − w j ) = ∑ 0 ≤ i < j ≤ d ( ∇ v h ) T ( x K , i − x K , j ) ( x K , i − x K , j ) T ∇ w h = ( d + 1 ) ( d + 2 ) ( ∇ v h ) T D K ∇ w h . \begin{aligned}
\sum_{0\leq i<j\leq d}(v_i-v_j)(w_i-w_j)=\sum_{0\leq i<j\leq d}(\nabla v_h)^T(x_{K,i}-x_{K,j})(x_{K,i}-x_{K,j})^T\nabla w_h \\
&=(d+1)(d+2)(\nabla v_h)^T D_K\nabla w_h.
\end{aligned} 0 ≤ i < j ≤ d ∑ ( v i − v j ) ( w i − w j ) = 0 ≤ i < j ≤ d ∑ ( ∇ v h ) T ( x K , i − x K , j ) ( x K , i − x K , j ) T ∇ w h = ( d + 1 ) ( d + 2 ) ( ∇ v h ) T D K ∇ w h . Multiplying by ∣ K ∣ ( d + 1 ) ( d + 2 ) \frac{|K|}{(d+1)(d+2)} ( d + 1 ) ( d + 2 ) ∣ K ∣ ∇ v h \nabla v_h ∇ v h ∇ w h \nabla w_h ∇ w h K K K
Q K ( v h w h ) − ∫ K v h w h d x = ∣ K ∣ ( ∇ v h ) T D K ∇ w h = ∫ K ( D K ∇ v h ) ⋅ ∇ w h d x . Q_K(v_hw_h)-\int_Kv_hw_h\,\mathrm{d}x=|K|(\nabla v_h)^TD_K\nabla w_h=\int_K(D_K\nabla v_h)\cdot\nabla w_h\,\mathrm{d}x. Q K ( v h w h ) − ∫ K v h w h d x = ∣ K ∣ ( ∇ v h ) T D K ∇ w h = ∫ K ( D K ∇ v h ) ⋅ ∇ w h d x . Finally, for any ξ ∈ R d \xi\in\mathbb{R}^d ξ ∈ R d D K D_K D K
ξ T D K ξ = 1 ( d + 1 ) ( d + 2 ) ∑ 0 ≤ i < j ≤ d ( ξ ⋅ ( x K , i − x K , j ) ) 2 . \xi^TD_K\xi=\frac{1}{(d+1)(d+2)}\sum_{0\leq i<j\leq d}\left(\xi\cdot(x_{K,i}-x_{K,j})\right)^2. ξ T D K ξ = ( d + 1 ) ( d + 2 ) 1 0 ≤ i < j ≤ d ∑ ( ξ ⋅ ( x K , i − x K , j ) ) 2 . Since ∣ x K , i − x K , j ∣ ≤ h K \lvert x_{K,i}-x_{K,j}\rvert\leq h_K ∣ x K , i − x K , j ∣ ≤ h K d ( d + 1 ) 2 \frac{d(d+1)}{2} 2 d ( d + 1 ) ( i , j ) (i,j) ( i , j ) 0 ≤ i < j ≤ d 0\leq i<j\leq d 0 ≤ i < j ≤ d
0 ≤ ξ T D K ξ ≤ 1 ( d + 1 ) ( d + 2 ) ⋅ d ( d + 1 ) 2 h K 2 ∣ ξ ∣ 2 = d 2 ( d + 2 ) h K 2 ∣ ξ ∣ 2 , 0\leq\xi^TD_K\xi\leq\frac{1}{(d+1)(d+2)}\cdot\frac{d(d+1)}{2}h_K^2\lvert\xi\rvert^2=\frac{d}{2(d+2)}h_K^2\lvert\xi\rvert^2, 0 ≤ ξ T D K ξ ≤ ( d + 1 ) ( d + 2 ) 1 ⋅ 2 d ( d + 1 ) h K 2 ∣ ξ ∣ 2 = 2 ( d + 2 ) d h K 2 ∣ ξ ∣ 2 , which completes the proof. □ \square □
L p L^p L p W 1 , p W^{1,p} W 1 , p Recall from (17) that − Δ u ∗ = f -\Delta u^*=f − Δ u ∗ = f f = λ ∗ u ∗ − V u ∗ − β ( u ∗ ) 3 f=\lambda^*u^*-Vu^*-\beta(u^*)^3 f = λ ∗ u ∗ − V u ∗ − β ( u ∗ ) 3 (20) , the discrete ground state u h ∗ , ℓ u_h^{*,\ell} u h ∗ , ℓ
( ∇ u h ∗ , ℓ , ∇ v h ) L 2 ( Ω ) = ( f h ℓ , v h ) ℓ 2 ∀ v h ∈ V h 0 , (\nabla u_h^{*,\ell},\nabla v_h)_{L^2(\Omega)}=(f_h^\ell,v_h)_{\ell^2}\qquad\forall v_h\in V_h^0, ( ∇ u h ∗ , ℓ , ∇ v h ) L 2 ( Ω ) = ( f h ℓ , v h ) ℓ 2 ∀ v h ∈ V h 0 , where f h ℓ : = λ h ∗ , ℓ u h ∗ , ℓ − V u h ∗ , ℓ − β ( u h ∗ , ℓ ) 3 f_h^\ell:=\lambda_h^{*,\ell}u_h^{*,\ell}-Vu_h^{*,\ell}-\beta(u_h^{*,\ell})^3 f h ℓ := λ h ∗ , ℓ u h ∗ , ℓ − V u h ∗ , ℓ − β ( u h ∗ , ℓ ) 3 δ h : = I h f h ℓ − I h f ∈ V h \delta_h:=I_hf_h^\ell-I_hf\in V_h δ h := I h f h ℓ − I h f ∈ V h
Lemma 4.4. There exists h 0 > 0 h_0>0 h 0 > 0 0 < h ≤ h 0 0<h\leq h_0 0 < h ≤ h 0
∥ δ h ∥ L 2 ( Ω ) ≤ C h 2 . \|\delta_h\|_{L^2(\Omega)}\leq Ch^2. ∥ δ h ∥ L 2 ( Ω ) ≤ C h 2 . Proof. By Lemma 4.2 , we have ∥ δ h ∥ L 2 ( Ω ) ≤ ∥ δ h ∥ ℓ 2 \|\delta_h\|_{L^2(\Omega)}\leq\|\delta_h\|_{\ell^2} ∥ δ h ∥ L 2 ( Ω ) ≤ ∥ δ h ∥ ℓ 2 δ h \delta_h δ h f h ℓ − f f_h^\ell-f f h ℓ − f
∥ δ h ∥ ℓ 2 = ∥ f h ℓ − f ∥ ℓ 2 . \|\delta_h\|_{\ell^2}=\|f_h^\ell-f\|_{\ell^2}. ∥ δ h ∥ ℓ 2 = ∥ f h ℓ − f ∥ ℓ 2 . Here, ∥ ⋅ ∥ ℓ 2 \|\cdot\|_{\ell^2} ∥ ⋅ ∥ ℓ 2 C ( Ω ‾ ) C(\overline{\Omega}) C ( Ω ) ∥ u h ∗ , ℓ ∥ ℓ 2 = 1 \|u_h^{*,\ell}\|_{\ell^2}=1 ∥ u h ∗ , ℓ ∥ ℓ 2 = 1 Proposition 4.1 , we obtain
∥ f h ℓ − f ∥ ℓ 2 ≤ ∥ λ ∗ ( u ∗ − u h ∗ , ℓ ) ∥ ℓ 2 + ∣ λ ∗ − λ h ∗ , ℓ ∣ ∥ u h ∗ , ℓ ∥ ℓ 2 + ∥ V ( u ∗ − u h ∗ , ℓ ) ∥ ℓ 2 \|f_h^\ell-f\|_{\ell^2}\leq\|\lambda^*(u^*-u_h^{*,\ell})\|_{\ell^2}+|\lambda^*-\lambda_h^{*,\ell}|\|u_h^{*,\ell}\|_{\ell^2}+\|V(u^*-u_h^{*,\ell})\|_{\ell^2} ∥ f h ℓ − f ∥ ℓ 2 ≤ ∥ λ ∗ ( u ∗ − u h ∗ , ℓ ) ∥ ℓ 2 + ∣ λ ∗ − λ h ∗ , ℓ ∣∥ u h ∗ , ℓ ∥ ℓ 2 + ∥ V ( u ∗ − u h ∗ , ℓ ) ∥ ℓ 2 + β ∥ ( u ∗ ) 2 + u ∗ u h ∗ , ℓ + ( u h ∗ , ℓ ) 2 ) ( u ∗ − u h ∗ , ℓ ) ∥ ℓ 2 ≤ C ∥ u ∗ − u h ∗ , ℓ ∥ ℓ 2 + C h 2 + ∥ V ∥ L ∞ ( Ω ) ∥ u ∗ − u h ∗ , ℓ ∥ ℓ 2 \begin{aligned}
\qquad+\beta\|(u^*)^2+u^*u_h^{*,\ell}+(u_h^{*,\ell})^2)(u^*-u_h^{*,\ell})\|_{\ell^2} \\
&\leq C\|u^*-u_h^{*,\ell}\|_{\ell^2}+Ch^2+\|V\|_{L^\infty(\Omega)}\|u^*-u_h^{*,\ell}\|_{\ell^2}
\end{aligned} + β ∥ ( u ∗ ) 2 + u ∗ u h ∗ , ℓ + ( u h ∗ , ℓ ) 2 ) ( u ∗ − u h ∗ , ℓ ) ∥ ℓ 2 ≤ C ∥ u ∗ − u h ∗ , ℓ ∥ ℓ 2 + C h 2 + ∥ V ∥ L ∞ ( Ω ) ∥ u ∗ − u h ∗ , ℓ ∥ ℓ 2 + C ∥ ( u ∗ ) 2 + u ∗ u h ∗ , ℓ + ( u h ∗ , ℓ ) 2 ∥ L ∞ ( Ω ) ∥ u ∗ − u h ∗ , ℓ ∥ ℓ 2 \qquad+C\|(u^*)^2+u^*u_h^{*,\ell}+(u_h^{*,\ell})^2\|_{L^\infty(\Omega)}\|u^*-u_h^{*,\ell}\|_{\ell^2} + C ∥ ( u ∗ ) 2 + u ∗ u h ∗ , ℓ + ( u h ∗ , ℓ ) 2 ∥ L ∞ ( Ω ) ∥ u ∗ − u h ∗ , ℓ ∥ ℓ 2 ≤ C ∥ u ∗ − u h ∗ , ℓ ∥ ℓ 2 2 + C h 2 \le C\lVert u^* - u_h^{*,\ell}\rVert_{\ell^2}^2 + Ch^2 ≤ C ∥ u ∗ − u h ∗ , ℓ ∥ ℓ 2 2 + C h 2
Since u ∗ u^* u ∗ I h u ∗ I_hu^* I h u ∗ Lemma 4.2 , the interpolation estimate, and Proposition 4.1 yield
∥ u ∗ − u h ∗ , ℓ ∥ ℓ 2 2 = ∥ I h u ∗ − u h ∗ , ℓ ∥ ℓ 2 2 ≤ C ∥ I h u ∗ − u h ∗ , ℓ ∥ L 2 ( Ω ) ≤ C ( ∥ u ∗ − I h u ∗ ∥ L 2 ( Ω ) + ∥ u ∗ − u h ∗ , ℓ ∥ L 2 ( Ω ) ) ≤ C h 2 . \begin{aligned}
\lVert u^* - u_h^{*,\ell}\rVert_{\ell^2}^2 = \lVert I_hu^* - u_h^{*,\ell}\rVert_{\ell^2}^2 \le C\lVert I_hu^* - u_h^{*,\ell}\rVert_{L^2(\Omega)} \\
&\le C\left(\lVert u^* - I_hu^*\rVert_{L^2(\Omega)} + \lVert u^* - u_h^{*,\ell}\rVert_{L^2(\Omega)}\right) \le Ch^2.
\end{aligned} ∥ u ∗ − u h ∗ , ℓ ∥ ℓ 2 2 = ∥ I h u ∗ − u h ∗ , ℓ ∥ ℓ 2 2 ≤ C ∥ I h u ∗ − u h ∗ , ℓ ∥ L 2 ( Ω ) ≤ C ( ∥ u ∗ − I h u ∗ ∥ L 2 ( Ω ) + ∥ u ∗ − u h ∗ , ℓ ∥ L 2 ( Ω ) ) ≤ C h 2 . Combining the above estimates completes the proof. □
Let θ h ℓ : = u h ∗ , ℓ − R h u ∗ ∈ V h 0 \theta_h^\ell:= u_h^{*,\ell} - R_hu^* \in V_h^0 θ h ℓ := u h ∗ , ℓ − R h u ∗ ∈ V h 0 R h R_h R h (17) , we have for all v h ∈ V h 0 v_h \in V_h^0 v h ∈ V h 0
( ∇ θ h ℓ , ∇ v h ) L 2 ( Ω ) = ( f h ℓ , v h ) ℓ 2 − ( f , v h ) L 2 ( Ω ) = ( δ h , v h ) ℓ 2 − ( f − I h f , v h ) L 2 ( Ω ) + [ ( I h f , v h ) ℓ 2 − ( I h f , v h ) L 2 ( Ω ) ] . \begin{aligned}
(\nabla\theta_h^\ell,\nabla v_h)_{L^2(\Omega)} = (f_h^\ell,v_h)_{\ell^2} - (f,v_h)_{L^2(\Omega)} \\
&= (\delta_h,v_h)_{\ell^2} - (f-I_hf,v_h)_{L^2(\Omega)} + \left[(I_hf,v_h)_{\ell^2} - (I_hf,v_h)_{L^2(\Omega)}\right].
\end{aligned} ( ∇ θ h ℓ , ∇ v h ) L 2 ( Ω ) = ( f h ℓ , v h ) ℓ 2 − ( f , v h ) L 2 ( Ω ) = ( δ h , v h ) ℓ 2 − ( f − I h f , v h ) L 2 ( Ω ) + [ ( I h f , v h ) ℓ 2 − ( I h f , v h ) L 2 ( Ω ) ] . For the first term, by the Riesz representation theorem on V h 0 V_h^0 V h 0 q h ∈ V h 0 q_h \in V_h^0 q h ∈ V h 0
( q h , v h ) L 2 ( Ω ) = ( δ h , v h ) ℓ 2 ∀ v h ∈ V h 0 . (q_h,v_h)_{L^2(\Omega)} = (\delta_h,v_h)_{\ell^2} \qquad\forall v_h \in V_h^0. ( q h , v h ) L 2 ( Ω ) = ( δ h , v h ) ℓ 2 ∀ v h ∈ V h 0 . Taking v h = q h v_h=q_h v h = q h Lemma 4.2 , we obtain
∥ q h ∥ L 2 ( Ω ) 2 ≤ ∥ δ h ∥ ℓ 2 ∥ q h ∥ ℓ 2 ≤ ( d + 2 ) ∥ δ h ∥ L 2 ( Ω ) ∥ q h ∥ L 2 ( Ω ) . \lVert q_h\rVert_{L^2(\Omega)}^2 \le\lVert\delta_h\rVert_{\ell^2}\lVert q_h\rVert_{\ell^2} \le(d+2)\lVert\delta_h\rVert_{L^2(\Omega)}\lVert q_h\rVert_{L^2(\Omega)}. ∥ q h ∥ L 2 ( Ω ) 2 ≤ ∥ δ h ∥ ℓ 2 ∥ q h ∥ ℓ 2 ≤ ( d + 2 ) ∥ δ h ∥ L 2 ( Ω ) ∥ q h ∥ L 2 ( Ω ) . Hence, Lemma 4.4 implies ∥ q h ∥ L 2 ( Ω ) ≤ C h 2 \lVert q_h\rVert_{L^2(\Omega)} \le Ch^2 ∥ q h ∥ L 2 ( Ω ) ≤ C h 2 Lemma 4.3 on each element K ∈ T h K\in\mathcal{T}_h K ∈ T h
( I h f , v h ) ℓ 2 − ( I h f , v h ) L 2 ( Ω ) = ∑ K ∈ T h ∫ K ( D K ∇ I h f ) ⋅ ∇ v h d x = ( F h , ∇ v h ) L 2 ( Ω ) , (I_hf,v_h)_{\ell^2} - (I_hf,v_h)_{L^2(\Omega)} = \sum_{K\in\mathcal{T}_h}\int_K (D_K\nabla I_hf)\cdot\nabla v_h\,dx = (F_h,\nabla v_h)_{L^2(\Omega)}, ( I h f , v h ) ℓ 2 − ( I h f , v h ) L 2 ( Ω ) = K ∈ T h ∑ ∫ K ( D K ∇ I h f ) ⋅ ∇ v h d x = ( F h , ∇ v h ) L 2 ( Ω ) , where F h F_h F h F h ∣ K : = D K ∇ I h f ∣ K F_h|_K:=D_K\nabla I_hf|_K F h ∣ K := D K ∇ I h f ∣ K K ∈ T h K\in\mathcal{T}_h K ∈ T h D K D_K D K Lemma 4.3 yields
∥ F h ∥ L ∞ ( Ω ) = max K ∈ T h ∥ D K ∇ I h f ∥ L ∞ ( K ) ≤ C h 2 ∥ ∇ I h f ∥ L ∞ ( Ω ) . \lVert F_h\rVert_{L^\infty(\Omega)} = \max_{K\in\mathcal{T}_h}\lVert D_K\nabla I_hf\rVert_{L^\infty(K)} \le Ch^2\lVert\nabla I_hf\rVert_{L^\infty(\Omega)}. ∥ F h ∥ L ∞ ( Ω ) = K ∈ T h max ∥ D K ∇ I h f ∥ L ∞ ( K ) ≤ C h 2 ∥ ∇ I h f ∥ L ∞ ( Ω ) . Since V ∈ W 2 , ∞ ( Ω ) V\in W^{2,\infty}(\Omega) V ∈ W 2 , ∞ ( Ω ) u ∗ ∈ W 2 , ∞ ( Ω ) u^*\in W^{2,\infty}(\Omega) u ∗ ∈ W 2 , ∞ ( Ω ) Proposition 3.14 , we have f ∈ W 2 , ∞ ( Ω ) f\in W^{2,\infty}(\Omega) f ∈ W 2 , ∞ ( Ω )
∥ ∇ I h f ∥ L ∞ ( Ω ) ≤ ∥ ∇ f ∥ L ∞ ( Ω ) + ∥ ∇ ( f − I h f ) ∥ L ∞ ( Ω ) ≤ ∥ ∇ f ∥ L ∞ ( Ω ) + C h ∥ f ∥ W 2 , ∞ ( Ω ) ≤ C , \begin{aligned}
\lVert\nabla I_hf\rVert_{L^\infty(\Omega)} \le\lVert\nabla f\rVert_{L^\infty(\Omega)} + \lVert\nabla(f-I_hf)\rVert_{L^\infty(\Omega)} \\
&\le\lVert\nabla f\rVert_{L^\infty(\Omega)} + Ch\lVert f\rVert_{W^{2,\infty}(\Omega)} \le C,
\end{aligned} ∥ ∇ I h f ∥ L ∞ ( Ω ) ≤ ∥ ∇ f ∥ L ∞ ( Ω ) + ∥ ∇ ( f − I h f ) ∥ L ∞ ( Ω ) ≤ ∥ ∇ f ∥ L ∞ ( Ω ) + C h ∥ f ∥ W 2 , ∞ ( Ω ) ≤ C , and hence ∥ F h ∥ L ∞ ( Ω ) ≤ C h 2 \lVert F_h\rVert_{L^\infty(\Omega)}\le Ch^2 ∥ F h ∥ L ∞ ( Ω ) ≤ C h 2 Lemma 2.8 , and the inverse inequality, we obtain the following result.
Lemma 4.5. There exists h 0 > 0 h_0>0 h 0 > 0 0 < h ≤ h 0 0<h\le h_0 0 < h ≤ h 0
∥ ∇ θ h ℓ ∥ L 6 ( Ω ) + ∥ θ h ℓ ∥ L ∞ ( Ω ) ≤ C h 2 , ∥ ∇ θ h ℓ ∥ L ∞ ( Ω ) ≤ C h 2 − d 6 . \lVert\nabla\theta_h^\ell\rVert_{L^6(\Omega)} + \lVert\theta_h^\ell\rVert_{L^\infty(\Omega)} \le Ch^2,\qquad\lVert\nabla\theta_h^\ell\rVert_{L^\infty(\Omega)} \le Ch^{2-\frac{d}{6}}. ∥ ∇ θ h ℓ ∥ L 6 ( Ω ) + ∥ θ h ℓ ∥ L ∞ ( Ω ) ≤ C h 2 , ∥ ∇ θ h ℓ ∥ L ∞ ( Ω ) ≤ C h 2 − 6 d . Proceeding as in the proofs of Theorems 3.12 and 3.15 , with Lemma 4.5 in place of Lemmas 3.9 and 3.11 , we obtain the following result.
Theorem 4.6. There exists h 0 > 0 h_0>0 h 0 > 0 0 < h ≤ h 0 0<h\le h_0 0 < h ≤ h 0
∥ u ∗ − u h ∗ , ℓ ∥ L p ( Ω ) + h ∥ ∇ ( u ∗ − u h ∗ , ℓ ) ∥ L p ( Ω ) ≤ C p h 2 for 1 ≤ p < ∞ , \lVert u^* - u_h^{*,\ell}\rVert_{L^p(\Omega)} + h\lVert\nabla(u^* - u_h^{*,\ell})\rVert_{L^p(\Omega)} \le C_ph^2 \qquad\text{for } 1\le p<\infty, ∥ u ∗ − u h ∗ , ℓ ∥ L p ( Ω ) + h ∥ ∇ ( u ∗ − u h ∗ , ℓ ) ∥ L p ( Ω ) ≤ C p h 2 for 1 ≤ p < ∞ , and
∥ u ∗ − u h ∗ , ℓ ∥ L ∞ ( Ω ) ≤ C h 2 ∣ ln h ∣ , ∥ ∇ ( u ∗ − u h ∗ , ℓ ) ∥ L ∞ ( Ω ) ≤ C h . \lVert u^* - u_h^{*,\ell}\rVert_{L^\infty(\Omega)} \le Ch^2|\ln h|,\qquad\lVert\nabla(u^* - u_h^{*,\ell})\rVert_{L^\infty(\Omega)} \le Ch. ∥ u ∗ − u h ∗ , ℓ ∥ L ∞ ( Ω ) ≤ C h 2 ∣ ln h ∣ , ∥ ∇ ( u ∗ − u h ∗ , ℓ ) ∥ L ∞ ( Ω ) ≤ C h . Superconvergence
With the help of Theorem 4.6 , the estimate of δ h \delta_h δ h Lemma 4.4 can be improved as follows.
Lemma 4.7. For every 2 ≤ q < ∞ 2 \le q < \infty 2 ≤ q < ∞ h 0 > 0 h_0 > 0 h 0 > 0 0 < h ≤ h 0 0 < h \le h_0 0 < h ≤ h 0
∥ δ h ∥ L q ( Ω ) ≤ C q h 2 . \lVert\delta_h\rVert_{L^q(\Omega)} \le C_q h^2. ∥ δ h ∥ L q ( Ω ) ≤ C q h 2 . Proof. As in the proof of Lemma 4.4 , using Lemma 4.2 and the fact that u ∗ u^* u ∗ I h u ∗ I_hu^* I h u ∗
∥ δ h ∥ L q ( Ω ) ≤ C q ∥ δ h ∥ ℓ q ≤ C q ∣ λ h ∗ , ℓ − λ ∗ ∣ ∥ u h ∗ , ℓ ∥ ℓ q + C q ∥ u h ∗ , ℓ − I h u ∗ ∥ ℓ q ≤ C q h 2 ∥ u h ∗ , ℓ ∥ L q ( Ω ) + C q ∥ u h ∗ , ℓ − I h u ∗ ∥ L q ( Ω ) ≤ C q h 2 , \begin{aligned}
\lVert\delta_h\rVert_{L^q(\Omega)} \le C_q\lVert\delta_h\rVert_{\ell^q} \le C_q\lvert\lambda_h^{*,\ell}-\lambda^*\rvert\lVert u_h^{*,\ell}\rVert_{\ell^q}+C_q\lVert u_h^{*,\ell}-I_hu^*\rVert_{\ell^q} \\
&\le C_qh^2\lVert u_h^{*,\ell}\rVert_{L^q(\Omega)}+C_q\lVert u_h^{*,\ell}-I_hu^*\rVert_{L^q(\Omega)}\le C_qh^2,
\end{aligned} ∥ δ h ∥ L q ( Ω ) ≤ C q ∥ δ h ∥ ℓ q ≤ C q ∣ λ h ∗ , ℓ − λ ∗ ∣ ∥ u h ∗ , ℓ ∥ ℓ q + C q ∥ u h ∗ , ℓ − I h u ∗ ∥ ℓ q ≤ C q h 2 ∥ u h ∗ , ℓ ∥ L q ( Ω ) + C q ∥ u h ∗ , ℓ − I h u ∗ ∥ L q ( Ω ) ≤ C q h 2 , where the last step follows from Proposition 4.1 , the interpolation estimate, and Theorem 4.6 .
To exploit Lemma 4.7 , we rewrite the equation for θ h ℓ \theta_h^\ell θ h ℓ v h ∈ V h 0 v_h\in V_h^0 v h ∈ V h 0
( ∇ θ h ℓ , ∇ v h ) L 2 ( Ω ) = ( f h ℓ , v h ) ℓ 2 − ( f , v h ) L 2 ( Ω ) = ( δ h , v h ) L 2 ( Ω ) + [ ( δ h , v h ) ℓ 2 − ( δ h , v h ) L 2 ( Ω ) ] \begin{aligned}
(\nabla\theta_h^\ell,\nabla v_h)_{L^2(\Omega)}=(f_h^\ell,v_h)_{\ell^2}-(f,v_h)_{L^2(\Omega)} \\
&=(\delta_h,v_h)_{L^2(\Omega)}+\left[(\delta_h,v_h)_{\ell^2}-(\delta_h,v_h)_{L^2(\Omega)}\right]
\end{aligned} ( ∇ θ h ℓ , ∇ v h ) L 2 ( Ω ) = ( f h ℓ , v h ) ℓ 2 − ( f , v h ) L 2 ( Ω ) = ( δ h , v h ) L 2 ( Ω ) + [ ( δ h , v h ) ℓ 2 − ( δ h , v h ) L 2 ( Ω ) ] − ( f − I h f , v h ) L 2 ( Ω ) + [ ( I h f , v h ) ℓ 2 − ( I h f , v h ) L 2 ( Ω ) ] . -(f-I_hf,v_h)_{L^2(\Omega)}+\left[(I_hf,v_h)_{\ell^2}-(I_hf,v_h)_{L^2(\Omega)}\right]. − ( f − I h f , v h ) L 2 ( Ω ) + [ ( I h f , v h ) ℓ 2 − ( I h f , v h ) L 2 ( Ω ) ] . For the second term, Lemma 4.3 yields
( δ h , v h ) ℓ 2 − ( δ h , v h ) L 2 ( Ω ) = ∑ K ∈ T h ∫ K ( D K ∇ δ h ) ⋅ ∇ v h d x = ( G h , ∇ v h ) L 2 ( Ω ) , (\delta_h,v_h)_{\ell^2}-(\delta_h,v_h)_{L^2(\Omega)}
=\sum_{K\in\mathcal{T}_h}\int_K(D_K\nabla\delta_h)\cdot\nabla v_h\,\mathrm{d}x=(G_h,\nabla v_h)_{L^2(\Omega)}, ( δ h , v h ) ℓ 2 − ( δ h , v h ) L 2 ( Ω ) = K ∈ T h ∑ ∫ K ( D K ∇ δ h ) ⋅ ∇ v h d x = ( G h , ∇ v h ) L 2 ( Ω ) , where G h ∣ K : = D K ∇ δ h ∣ K G_h|_K:=D_K\nabla\delta_h|_K G h ∣ K := D K ∇ δ h ∣ K K ∈ T h K\in\mathcal{T}_h K ∈ T h G h G_h G h
Lemma 4.8. For every 2 ≤ q < ∞ 2 \le q < \infty 2 ≤ q < ∞ h 0 > 0 h_0 > 0 h 0 > 0 0 < h ≤ h 0 0 < h \le h_0 0 < h ≤ h 0
∥ G h ∥ L q ( Ω ) ≤ C q h 3 . \lVert G_h\rVert_{L^q(\Omega)}\le C_qh^3. ∥ G h ∥ L q ( Ω ) ≤ C q h 3 . Proof. Since D K D_K D K Lemma 4.3 together with the inverse inequality and Lemma 4.7 yields
∥ G h ∥ L q ( Ω ) = ( ∑ K ∈ T h ∫ K ∣ D K ∇ δ h ∣ q d x ) 1 q ≤ C h 2 ∥ ∇ δ h ∥ L q ( Ω ) ≤ C q h 3 , \lVert G_h\rVert_{L^q(\Omega)}
=\left(\sum_{K\in\mathcal{T}_h}\int_K\lvert D_K\nabla\delta_h\rvert^q\,\mathrm{d}x\right)^{\frac{1}{q}}
\le Ch^2\lVert\nabla\delta_h\rVert_{L^q(\Omega)}\le C_qh^3, ∥ G h ∥ L q ( Ω ) = ( K ∈ T h ∑ ∫ K ∣ D K ∇ δ h ∣ q d x ) q 1 ≤ C h 2 ∥ ∇ δ h ∥ L q ( Ω ) ≤ C q h 3 , which completes the proof.
Let g h : = δ h + ( I h f − f ) g_h:=\delta_h+(I_hf-f) g h := δ h + ( I h f − f ) F ~ h : = F h + G h \widetilde{F}_h:=F_h+G_h F h := F h + G h Lemmas 4.7 and 4.8 and the interpolation estimate, for every 2 ≤ q < ∞ 2\le q<\infty 2 ≤ q < ∞
∥ g h ∥ L q ( Ω ) + ∥ F ~ h ∥ L q ( Ω ) ≤ C q h 2 . \lVert g_h\rVert_{L^q(\Omega)}+\lVert\widetilde{F}_h\rVert_{L^q(\Omega)}\le C_qh^2. ∥ g h ∥ L q ( Ω ) + ∥ F h ∥ L q ( Ω ) ≤ C q h 2 . We are now in a position to state the superconvergence result for the mass-lumped FEM.
Theorem 4.9. For every 1 ≤ q < ∞ 1\le q<\infty 1 ≤ q < ∞ h 0 > 0 h_0>0 h 0 > 0 0 < h ≤ h 0 0<h\le h_0 0 < h ≤ h 0
∥ ∇ ( u h ∗ , ℓ − R h u ∗ ) ∥ L q ( Ω ) ≤ C q h 2 . \lVert\nabla(u_h^{*,\ell}-R_hu^*)\rVert_{L^q(\Omega)}\le C_qh^2. ∥ ∇ ( u h ∗ , ℓ − R h u ∗ ) ∥ L q ( Ω ) ≤ C q h 2 . Proof. It suffices to consider 2 ≤ q < ∞ 2 \le q < \infty 2 ≤ q < ∞ 1 ≤ q < 2 1 \le q < 2 1 ≤ q < 2 θ h ℓ \theta_h^\ell θ h ℓ
( ∇ θ h ℓ , ∇ v h ) L 2 ( Ω ) = ( g h , v h ) L 2 ( Ω ) + ( F ~ h , ∇ v h ) L 2 ( Ω ) ∀ v h ∈ V h 0 . (\nabla\theta_h^\ell,\nabla v_h)_{L^2(\Omega)}=(g_h,v_h)_{L^2(\Omega)}+(\widetilde{F}_h,\nabla v_h)_{L^2(\Omega)} \qquad\forall v_h\in V_h^0. ( ∇ θ h ℓ , ∇ v h ) L 2 ( Ω ) = ( g h , v h ) L 2 ( Ω ) + ( F h , ∇ v h ) L 2 ( Ω ) ∀ v h ∈ V h 0 . Let w g h , w F ~ h ∈ H 0 1 ( Ω ) w_{g_h},w_{\widetilde{F}_h}\in H_0^1(\Omega) w g h , w F h ∈ H 0 1 ( Ω )
( ∇ w g h , ∇ v ) L 2 ( Ω ) = ( g h , v ) L 2 ( Ω ) ∀ v ∈ H 0 1 ( Ω ) , (\nabla w_{g_h},\nabla v)_{L^2(\Omega)}=(g_h,v)_{L^2(\Omega)} \qquad\forall v\in H_0^1(\Omega), ( ∇ w g h , ∇ v ) L 2 ( Ω ) = ( g h , v ) L 2 ( Ω ) ∀ v ∈ H 0 1 ( Ω ) , ( ∇ w F ~ h , ∇ v ) L 2 ( Ω ) = ( F ~ h , ∇ v ) L 2 ( Ω ) ∀ v ∈ H 0 1 ( Ω ) , (\nabla w_{\widetilde{F}_h},\nabla v)_{L^2(\Omega)}=(\widetilde{F}_h,\nabla v)_{L^2(\Omega)} \qquad\forall v\in H_0^1(\Omega), ( ∇ w F h , ∇ v ) L 2 ( Ω ) = ( F h , ∇ v ) L 2 ( Ω ) ∀ v ∈ H 0 1 ( Ω ) , so that θ h ℓ = R h ( w g h + w F ~ h ) \theta_h^\ell=R_h(w_{g_h}+w_{\widetilde{F}_h}) θ h ℓ = R h ( w g h + w F h ) Lemma 2.7 , respectively, we have
∥ w g h ∥ W 2 , 4 ( Ω ) ≤ C ∥ g h ∥ L 4 ( Ω ) ≤ C h 2 , ∥ w F ~ h ∥ W 1 , q ( Ω ) ≤ C q ∥ F ~ h ∥ L q ( Ω ) ≤ C q h 2 . \|w_{g_h}\|_{W^{2,4}(\Omega)}\le C\|g_h\|_{L^4(\Omega)}\le Ch^2,\qquad\|w_{\widetilde{F}_h}\|_{W^{1,q}(\Omega)}\le C_q\|\widetilde{F}_h\|_{L^q(\Omega)}\le C_qh^2. ∥ w g h ∥ W 2 , 4 ( Ω ) ≤ C ∥ g h ∥ L 4 ( Ω ) ≤ C h 2 , ∥ w F h ∥ W 1 , q ( Ω ) ≤ C q ∥ F h ∥ L q ( Ω ) ≤ C q h 2 . Hence, Proposition 2.5 and the Sobolev embedding W 2 , 4 ( Ω ) ↪ W 1 , q ( Ω ) W^{2,4}(\Omega)\hookrightarrow W^{1,q}(\Omega) W 2 , 4 ( Ω ) ↪ W 1 , q ( Ω ) d ≤ 3 d\le3 d ≤ 3
∥ ∇ θ h ℓ ∥ L q ( Ω ) ≤ C ∥ ∇ ( w g h + w F ~ h ) ∥ L q ( Ω ) ≤ C q ∥ w g h ∥ W 2 , 4 ( Ω ) + C ∥ w F ~ h ∥ W 1 , q ( Ω ) ≤ C q h 2 , \begin{aligned}
\|\nabla\theta_h^\ell\|_{L^q(\Omega)}&\le C\|\nabla(w_{g_h}+w_{\widetilde{F}_h})\|_{L^q(\Omega)}\\
&\le C_q\|w_{g_h}\|_{W^{2,4}(\Omega)}+C\|w_{\widetilde{F}_h}\|_{W^{1,q}(\Omega)}\le C_qh^2,
\end{aligned} ∥∇ θ h ℓ ∥ L q ( Ω ) ≤ C ∥∇ ( w g h + w F h ) ∥ L q ( Ω ) ≤ C q ∥ w g h ∥ W 2 , 4 ( Ω ) + C ∥ w F h ∥ W 1 , q ( Ω ) ≤ C q h 2 , which completes the proof.
Remark 4.10. For d = 1 d=1 d = 1 q = ∞ q=\infty q = ∞ w F ~ h w_{\widetilde{F}_h} w F h w F ~ h = w F h + w G h w_{\widetilde{F}_h}=w_{F_h}+w_{G_h} w F h = w F h + w G h w F h w_{F_h} w F h w G h w_{G_h} w G h F h F_h F h G h G_h G h ∥ w F h ′ ∥ L ∞ ( Ω ) ≤ C h 2 \|w_{F_h}'\|_{L^\infty(\Omega)}\le Ch^2 ∥ w F h ′ ∥ L ∞ ( Ω ) ≤ C h 2 ∥ w G h ′ ∥ L q ( Ω ) ≤ C q h 3 \|w_{G_h}'\|_{L^q(\Omega)}\le C_qh^3 ∥ w G h ′ ∥ L q ( Ω ) ≤ C q h 3 θ h ℓ = R h ( w g h + w F h ) + R h w G h \theta_h^\ell=R_h(w_{g_h}+w_{F_h})+R_hw_{G_h} θ h ℓ = R h ( w g h + w F h ) + R h w G h ∥ R h ( w g h + w F h ) ∥ W 1 , ∞ ( Ω ) ≤ C h 2 \|R_h(w_{g_h}+w_{F_h})\|_{W^{1,\infty}(\Omega)}\le Ch^2 ∥ R h ( w g h + w F h ) ∥ W 1 , ∞ ( Ω ) ≤ C h 2 ∥ R h w G h ∥ W 1 , q ( Ω ) ≤ C q h 3 \|R_hw_{G_h}\|_{W^{1,q}(\Omega)}\le C_qh^3 ∥ R h w G h ∥ W 1 , q ( Ω ) ≤ C q h 3 ∥ R h w G h ∥ W 1 , ∞ ( Ω ) ≤ C q h 3 − 1 q ≤ C q h 2 \|R_hw_{G_h}\|_{W^{1,\infty}(\Omega)}\le C_qh^{3-\frac{1}{q}}\le C_qh^2 ∥ R h w G h ∥ W 1 , ∞ ( Ω ) ≤ C q h 3 − q 1 ≤ C q h 2
Remark 4.11. For d ∈ { 2 , 3 } d\in\{2,3\} d ∈ { 2 , 3 } q = ∞ q=\infty q = ∞ D K = D D_K=D D K = D K ∈ T h K\in\mathcal{T}_h K ∈ T h D D D K K K F h F_h F h
F h = D ∇ I h f = D ∇ f + D ∇ ( I h f − f ) = : F h 1 + F h 2 . F_h=D\nabla I_hf=D\nabla f+D\nabla(I_hf-f)=:F_h^1+F_h^2. F h = D ∇ I h f = D ∇ f + D ∇ ( I h f − f ) =: F h 1 + F h 2 . For the first term, integration by parts yields
( F h 1 , ∇ v ) L 2 ( Ω ) = ( − div ( D ∇ f ) , v ) L 2 ( Ω ) ∀ v ∈ H 0 1 ( Ω ) , (F_h^1,\nabla v)_{L^2(\Omega)}=(-\operatorname{div}(D\nabla f),v)_{L^2(\Omega)} \qquad\forall v\in H_0^1(\Omega), ( F h 1 , ∇ v ) L 2 ( Ω ) = ( − div ( D ∇ f ) , v ) L 2 ( Ω ) ∀ v ∈ H 0 1 ( Ω ) , where ∥ div ( D ∇ f ) ∥ L ∞ ( Ω ) ≤ C h 2 \|\operatorname{div}(D\nabla f)\|_{L^\infty(\Omega)}\le Ch^2 ∥ div ( D ∇ f ) ∥ L ∞ ( Ω ) ≤ C h 2 g h g_h g h ∥ F h 2 ∥ L q ( Ω ) + ∥ G h ∥ L q ( Ω ) ≤ C q h 3 \|F_h^2\|_{L^q(\Omega)}+\|G_h\|_{L^q(\Omega)}\le C_qh^3 ∥ F h 2 ∥ L q ( Ω ) + ∥ G h ∥ L q ( Ω ) ≤ C q h 3 F h 2 F_h^2 F h 2 G h G_h G h q = ∞ q=\infty q = ∞
Numerical experiments
In this section, we present two numerical examples in two dimensions to compare the standard FEM and the mass-lumped FEM. The errors are measured in the L p L^p L p W 1 , p W^{1,p} W 1 , p p = 1 , 2 , 4 , 8 , ∞ p=1,2,4,8,\infty p = 1 , 2 , 4 , 8 , ∞ V h 0 V_h^0 V h 0 D K = D D_K=D D K = D K ∈ T h K\in\mathcal{T}_h K ∈ T h L 2 L^2 L 2 ( ⋅ , ⋅ ) ℓ 2 (\cdot,\cdot)_{\ell^2} ( ⋅ , ⋅ ) ℓ 2 2 × 10 − 11 2\times10^{-11} 2 × 1 0 − 11 10 − 12 10^{-12} 1 0 − 12
Smooth ground state constructed from two sine modes
Let Ω = ( 0 , 1 ) 2 \Omega=(0,1)^2 Ω = ( 0 , 1 ) 2 β = 10 \beta=10 β = 10 V V V
u ∗ ( x , y ) = 8 17 ( sin π x + 1 4 sin 3 π x ) sin π y , λ ∗ = 120 , u^*(x,y)=\frac{8}{\sqrt{17}}\left(\sin\pi x+\frac{1}{4}\sin3\pi x\right)\sin\pi y,\qquad\lambda^*=120, u ∗ ( x , y ) = 17 8 ( sin π x + 4 1 sin 3 π x ) sin π y , λ ∗ = 120 , where the normalization factor ensures ∥ u ∗ ∥ L 2 ( Ω ) = 1 \lVert u^*\rVert_{L^2(\Omega)}=1 ∥ u ∗ ∥ L 2 ( Ω ) = 1
V ( x , y ) = 120 − q ( x ) − 10 u ∗ ( x , y ) 2 , q ( x ) = 2 π 2 9 + 10 cos 2 π x 5 + 2 cos 2 π x . V(x,y)=120-q(x)-10u^*(x,y)^2,\qquad q(x)=2\pi^2\frac{9+10\cos2\pi x}{5+2\cos2\pi x}. V ( x , y ) = 120 − q ( x ) − 10 u ∗ ( x , y ) 2 , q ( x ) = 2 π 2 5 + 2 cos 2 π x 9 + 10 cos 2 π x . A direct computation shows that − Δ u ∗ = q u ∗ -\Delta u^*=q u^* − Δ u ∗ = q u ∗ ( λ ∗ , u ∗ ) (\lambda^*,u^*) ( λ ∗ , u ∗ ) (1) . Moreover, V V V Ω ‾ \overline{\Omega} Ω
V ( x , y ) ≥ 120 − 38 π 2 7 − 1000 17 > 7.59. V(x,y)\ge120-\frac{38\pi^2}{7}-\frac{1000}{17}>7.59. V ( x , y ) ≥ 120 − 7 38 π 2 − 17 1000 > 7.59. Since u ∗ u^* u ∗ Ω \Omega Ω u ∗ u^* u ∗ [10 ] . The mesh size is h = 2 / N h=\sqrt{2}/N h = 2 / N N N N I h u ∗ I_hu^* I h u ∗
Figure 5.1 shows the errors in the L p L^p L p W 1 , p W^{1,p} W 1 , p p = 1 , 2 , 4 , 8 , ∞ p=1,2,4,8,\infty p = 1 , 2 , 4 , 8 , ∞ L p L^p L p W 1 , p W^{1,p} W 1 , p Theorems 3.12 , 3.15 , and 4.6 . Moreover, Table 5.1 reports the errors ∥ u h − R h u ∗ ∥ W 1 , p ( Ω ) \lVert u_h-R_hu^*\rVert_{W^{1,p}(\Omega)} ∥ u h − R h u ∗ ∥ W 1 , p ( Ω ) u h u_h u h u h ∗ u_h^* u h ∗ u h ∗ , ℓ u_h^{*,\ell} u h ∗ , ℓ p p p ∥ u ∗ − u h ∥ W 1 , p ( Ω ) \lVert u^*-u_h\rVert_{W^{1,p}(\Omega)} ∥ u ∗ − u h ∥ W 1 , p ( Ω ) Theorems 3.16 and 4.9 . For the mass-lumped FEM with p = ∞ p=\infty p = ∞
Figure 5.1. Errors of the standard FEM (top) and the mass-lumped FEM (bottom) in the L p L^p L p W 1 , p W^{1,p} W 1 , p p = 1 , 2 , 4 , 8 , ∞ p=1,2,4,8,\infty p = 1 , 2 , 4 , 8 , ∞ O ( N − 2 ) O(N^{-2}) O ( N − 2 ) O ( N − 1 ) O(N^{-1}) O ( N − 1 )
Table 5.1. Errors ∥ u h − R h u ∗ ∥ W 1 , p ( Ω ) \lVert u_h - R_hu^* \rVert_{W^{1,p}(\Omega)} ∥ u h − R h u ∗ ∥ W 1 , p ( Ω ) u h u_h u h u h ∗ u_h^* u h ∗ u h ∗ , ℓ u_h^{*,\ell} u h ∗ , ℓ
Ground state in a harmonic potential
This example concerns the ground state of a repulsively interacting Bose–Einstein condensate confined by an isotropic harmonic trap. Let Ω = ( − 6 , 6 ) 2 \Omega=(-6,6)^2 Ω = ( − 6 , 6 ) 2 β = 100 \beta=100 β = 100
V ( x , y ) = x 2 + y 2 2 . V(x,y)=\frac{x^2+y^2}{2}. V ( x , y ) = 2 x 2 + y 2 . The mesh size is h = 12 2 / N h=12\sqrt{2}/N h = 12 2 / N N N N
u 0 ( x , y ) = exp ( − x 2 + y 2 8 ) u_0(x,y)=\exp\left(-\frac{x^2+y^2}{8}\right) u 0 ( x , y ) = exp ( − 8 x 2 + y 2 ) and whose boundary nodal values vanish, normalized with respect to the corresponding inner product. Since the exact ground state is not available, the error on the mesh of size h h h u h − u h / 2 u_h-u_{h/2} u h − u h /2 u h u_h u h
Figure 5.2 shows the results for both methods. For p = 1 , 2 , 4 , 8 , ∞ p=1,2,4,8,\infty p = 1 , 2 , 4 , 8 , ∞ ∥ u h − u h / 2 ∥ L p ( Ω ) \lVert u_h-u_{h/2}\rVert_{L^p(\Omega)} ∥ u h − u h /2 ∥ L p ( Ω ) ∥ u h − u h / 2 ∥ W 1 , p ( Ω ) \lVert u_h-u_{h/2}\rVert_{W^{1,p}(\Omega)} ∥ u h − u h /2 ∥ W 1 , p ( Ω ) Theorems 3.12 , 3.15 , and 4.6 .
Figure 5.2. Differences u h − u h / 2 u_h-u_{h/2} u h − u h /2 L p L^p L p W 1 , p W^{1,p} W 1 , p p = 1 , 2 , 4 , 8 , ∞ p=1,2,4,8,\infty p = 1 , 2 , 4 , 8 , ∞ O ( N − 2 ) O(N^{-2}) O ( N − 2 ) O ( N − 1 ) O(N^{-1}) O ( N − 1 )
Concluding remarks
In this paper, we have analyzed the standard and the mass-lumped linear finite element methods for the Gross–Pitaevskii eigenvalue problem. For the standard FEM, we have proved that the discrete ground state is unique up to sign for sufficiently small mesh sizes. For both methods, we have derived optimal convergence rates in the L p L^p L p 1 ≤ p < ∞ 1 \leq p < \infty 1 ≤ p < ∞ W 1 , p W^{1,p} W 1 , p 1 ≤ p ≤ ∞ 1 \leq p \leq\infty 1 ≤ p ≤ ∞ L ∞ L^\infty L ∞ W 1 , p W^{1,p} W 1 , p 1 ≤ p ≤ ∞ 1 \leq p \leq\infty 1 ≤ p ≤ ∞ 1 ≤ p < ∞ 1 \leq p < \infty 1 ≤ p < ∞ p = ∞ p=\infty p = ∞ d = 1 d=1 d = 1 D K D_K D K K K K p = ∞ p=\infty p = ∞
Declaration of generative AI use. The research problem considered in this paper, the main ideas and proof strategies underlying the error analysis, and the overall structure of the manuscript were conceived and developed by the authors. Generative AI tools, ChatGPT (OpenAI) and Claude (Anthropic), were used to assist with the language editing of the manuscript, the verification of mathematical derivations, the design of numerical examples, and the review of related results in the literature. All AI-generated suggestions were critically examined and, where necessary, revised by the authors. The authors have verified all mathematical arguments, references, and numerical results, and take full responsibility for the content of this paper.
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