Introduction
In this paper, we consider the fully nonlinear Monge–Ampère equation with Dirichlet boundary conditions:
det ( D 2 u ) = f in Ω , (1) \det(D^2 u) = f \quad\text{in } \Omega, \tag*{(1)} det ( D 2 u ) = f in Ω , (1) u = g on ∂ Ω , (2) u = g \quad\text{on } \partial\Omega, \tag*{(2)} u = g on ∂ Ω , (2) where Ω ⊂ R 2 \Omega\subset\mathbb{R}^2 Ω ⊂ R 2 Γ = ∂ Ω \Gamma= \partial\Omega Γ = ∂ Ω D 2 u D^2u D 2 u u u u [23 , 29 ] , optical reflector design [43 ] , and the theory of optimal transport [3 , 41 ] . For the Monge–Ampère equation, alternative notions of solutions, such as Aleksandrov and viscosity solutions [9 , 23 , 26 , 29 ] , have been developed over the past century, as classical solutions may not exist even when the data f f f g g g
Due to its nonlinear nature and lack of classical variational structure, the development of numerical techniques for the Monge–Ampère equation is challenging. The first numerical methods can be traced back to the late 1980s when Oliker and Prussner [40 ] constructed the finite difference methods to compute the Aleksandrov solutions. In 2008, Oberman proposed a wide-stencil framework for solving the Monge–Ampère equation [39 ] . Thus far, a variety of numerical methods have been proposed for the approximation of Monge–Ampère equations, including filtered schemes [24 , 25 ] , L 2 L^2 L 2 C 0 C^0 C 0 [1 , 4 , 5 , 6 , 33 ] , least-squares approaches [8 , 16 ] , vanishing moment methods [12 , 18 , 19 , 20 , 21 , 32 ] , and two-scale finite element methods [37 , 38 ] . For a comprehensive overview, we refer the reader to the review articles [17 , 34 , 35 ] .
We concentrate on the finite element approximation of the following singularly perturbed fourth-order semilinear partial differential equation:
− ε Δ 2 u ε + det ( D 2 u ε ) = f in Ω , (3) -\varepsilon\Delta^2 u^\varepsilon+ \det(D^2u^\varepsilon) = f \quad\text{in } \Omega, \tag*{(3)} − ε Δ 2 u ε + det ( D 2 u ε ) = f in Ω , (3) u ε = g on ∂ Ω , (4) u^\varepsilon= g \quad\text{on } \partial\Omega, \tag*{(4)} u ε = g on ∂ Ω , (4) Δ u ε = ε on ∂ Ω . (5) \Delta u^\varepsilon= \varepsilon\quad\text{on } \partial\Omega. \tag*{(5)} Δ u ε = ε on ∂ Ω. (5) The solution of the model problem (1.2) has been shown to converge uniformly (in the viscosity sense) to the viscosity solution of the Monge–Ampère equation (1.1) as ε → 0 + \varepsilon\to0^+ ε → 0 + [20 , 22 ] . Thus, equation (1.2) offers a regularization framework for computing viscosity solutions of the Monge–Ampère equation. Following the assumptions in [12 , 18 ] , we further suppose that the following a priori estimates hold:
∥ u ε ∥ H j ( Ω ) = O ( ε 1 2 − j ) ( j = 2 , 3 ) , ∥ u ε ∥ W j , ∞ ( Ω ) = O ( ε 1 − j ) ( j = 1 , 2 ) , ∥ Φ ε ∥ L 2 ( Ω ) = O ( ε − 1 2 ) , ∥ Φ ε ∥ L ∞ ( Ω ) = O ( ε − 1 ) , (6) \begin{aligned}
\lVert u^\varepsilon\rVert_{H^j(\Omega)} &= O\left(\varepsilon^{\frac{1}{2}-j}\right) \quad(j=2,3), \qquad
\lVert u^\varepsilon\rVert_{W^{j,\infty}(\Omega)} = O\left(\varepsilon^{1-j}\right) \quad(j=1,2), \\
\lVert\Phi^\varepsilon\rVert_{L^2(\Omega)} &= O\left(\varepsilon^{-\frac{1}{2}}\right), \qquad
\lVert\Phi^\varepsilon\rVert_{L^\infty(\Omega)} = O\left(\varepsilon^{-1}\right), \tag*{(6)}
\end{aligned} ∥ u ε ∥ H j ( Ω ) ∥ Φ ε ∥ L 2 ( Ω ) = O ( ε 2 1 − j ) ( j = 2 , 3 ) , ∥ u ε ∥ W j , ∞ ( Ω ) = O ( ε 1 − j ) ( j = 1 , 2 ) , = O ( ε − 2 1 ) , ∥ Φ ε ∥ L ∞ ( Ω ) = O ( ε − 1 ) , (6) where Φ ε \Phi^\varepsilon Φ ε D 2 u ε D^2u^\varepsilon D 2 u ε u ε ∈ H 3 ( Ω ) ∩ W 2 , ∞ ( Ω ) u^\varepsilon\in H^3(\Omega) \cap W^{2,\infty}(\Omega) u ε ∈ H 3 ( Ω ) ∩ W 2 , ∞ ( Ω ) (6) .
For the semilinear equation (1.2), one of the primary numerical challenges lies in discretizing the biharmonic operator. Classical finite element methods for the biharmonic operator, such as conforming finite element methods [21 ] , mixed finite element methods [18 ] , nonconforming finite element methods [32 ] , and C 0 C^0 C 0 [15 , 35 ] , have been applied to solve (1.2). Most of these studies use at least quadratic elements. In recent years, there has been increasing interest in designing linear finite element methods for solving fourth-order PDEs; see, e.g., [11 , 13 , 28 , 30 , 31 , 44 ] . However, these approaches are not directly applicable to (1.2) due to difficulties in establishing stability conditions and the lack of consistency in the finite element functions. To the best of our knowledge, [12 ] is the first work devoted to the development of linear element methods for solving the model equation (1.2). However, certain assumptions are required in [12 ] to ensure stability.
The primary goal of this paper is to propose a new linear element-based method with rigorous mathematical theory. The main motivation comes from our recent work on the Hamilton–Jacobi–Bellman equation [14 ] , where a novel gradient recovery operator is proposed using the Morley finite element function as a bridge. Unlike existing gradient recovery operators [27 ] , the proposed operator is a true gradient of a Morley finite element function, which allows us to establish the consistency of the gradient recovery operator for finite element functions. In addition, a discrete Miranda–Talenti (MT) estimate for the gradient recovery operator is established in [14 ] . Unlike the analysis method in [12 ] , our analysis uses the discrete MT estimate to design the finite element scheme and analyze the stability of the method. Based on the Banach fixed-point theorem, we prove the existence and uniqueness of the finite element solution, as well as a discrete H 2 H^2 H 2 H 1 H^1 H 1 [12 ] , our method achieves O ( h 2 ) O(h^2) O ( h 2 ) O ( h ) O(h) O ( h ) L 2 L^2 L 2 H 1 H^1 H 1 [12 ] only yields O ( h ) O(h) O ( h ) O ( h 0.5 ) O(h^{0.5}) O ( h 0.5 ) L 2 L^2 L 2 H 1 H^1 H 1
The remainder of the paper is organized as follows. In § 2 , we introduce the necessary notation for finite element functions and review the gradient recovery operator developed in [14 ] . Section 3 is devoted to the design and analysis of a finite element scheme for the linearized model, with particular emphasis on the application of a discrete MT estimate. In Section 4, we present a recovery-based finite element method for the semilinear equation and establish error estimates in the discrete H 2 H^2 H 2 H 1 H^1 H 1
Preliminaries
Notation
Let ∣ D ∣ |D| ∣ D ∣ D ⊂ R 2 D \subset\mathbb{R}^2 D ⊂ R 2 s ≥ 0 s \ge0 s ≥ 0 p ∈ [ 1 , ∞ ] p \in[1,\infty] p ∈ [ 1 , ∞ ] W s , p ( D ) W^{s,p}(D) W s , p ( D ) D D D L 0 p ( D ) : = W 0 0 , p ( D ) L^p_0(D) := W^{0,p}_0(D) L 0 p ( D ) := W 0 0 , p ( D ) H s ( D ) : = W s , 2 ( D ) H^s(D) := W^{s,2}(D) H s ( D ) := W s , 2 ( D ) W s , p ( D ) W^{s,p}(D) W s , p ( D ) ∥ ⋅ ∥ W s , p ( D ) \|\cdot\|_{W^{s,p}(D)} ∥ ⋅ ∥ W s , p ( D ) ∣ ⋅ ∣ W s , p ( D ) |\cdot|_{W^{s,p}(D)} ∣ ⋅ ∣ W s , p ( D ) H 0 1 ( D ) H^1_0(D) H 0 1 ( D ) H 1 ( D ) H^1(D) H 1 ( D ) ∂ D \partial D ∂ D L 2 ( D ) L^2(D) L 2 ( D ) ( ⋅ , ⋅ ) L 2 ( D ) (\cdot,\cdot)_{L^2(D)} ( ⋅ , ⋅ ) L 2 ( D )
V : = H 2 ( Ω ) , V 0 : = H 2 ( Ω ) ∩ H 0 1 ( Ω ) , V g : = { v ∈ V : v ∣ ∂ Ω = g } . V := H^2(\Omega), \qquad V_0 := H^2(\Omega) \cap H^1_0(\Omega), \qquad V_g := \{v \in V : v|_{\partial\Omega} = g\}. V := H 2 ( Ω ) , V 0 := H 2 ( Ω ) ∩ H 0 1 ( Ω ) , V g := { v ∈ V : v ∣ ∂ Ω = g } . Let T h \mathcal{T}_h T h Ω \Omega Ω T ∈ T h T \in\mathcal{T}_h T ∈ T h h T : = diam ( T ) h_T := \operatorname{diam}(T) h T := diam ( T ) h : = max T ∈ T h h T h := \max_{T \in\mathcal{T}_h} h_T h := max T ∈ T h h T V h \mathcal{V}_h V h E h \mathcal{E}_h E h T h \mathcal{T}_h T h e ∈ E h e \in\mathcal{E}_h e ∈ E h h e h_e h e E h \mathcal{E}_h E h E h I \mathcal{E}^I_h E h I E h B \mathcal{E}^B_h E h B V h \mathcal{V}_h V h V h I \mathcal{V}^I_h V h I V h B \mathcal{V}^B_h V h B T ∈ T h T \in\mathcal{T}_h T ∈ T h V T \mathcal{V}_T V T M T \mathcal{M}_T M T T T T T p \mathcal{T}_p T p T h \mathcal{T}_h T h p p p T e \mathcal{T}_e T e e ∈ E h e \in\mathcal{E}_h e ∈ E h M h I \mathcal{M}^I_h M h I M h B \mathcal{M}^B_h M h B M h : = M h I ∪ M h B \mathcal{M}_h := \mathcal{M}^I_h \cup\mathcal{M}^B_h M h := M h I ∪ M h B
For any edge e ∈ E h e \in\mathcal{E}_h e ∈ E h n e ∈ R 2 n_e \in\mathbb{R}^2 n e ∈ R 2 e e e e ∈ E h B e \in\mathcal{E}^B_h e ∈ E h B n e n_e n e ∂ Ω \partial\Omega ∂ Ω e e e t e t_e t e e e e n e n_e n e π / 2 \pi/2 π /2 ⟦ ⋅ ⟧ \llbracket\cdot\rrbracket [ [ ⋅ ] ] e e e [36 ] :
⟦ v ⟧ : = v ∣ T o u t − v ∣ T i n if e = ∂ T o u t ∩ ∂ T i n ∈ E h I , ⟦ v ⟧ : = v ∣ T o u t if e = ∂ T o u t ∩ ∂ Ω ∈ E h B . \begin{aligned}
\llbracket v \rrbracket&:= v|_{T_{\mathrm{out}}} - v|_{T_{\mathrm{in}}} && \text{if } e = \partial T_{\mathrm{out}} \cap\partial T_{\mathrm{in}} \in\mathcal{E}^I_h, \\
\llbracket v \rrbracket&:= v|_{T_{\mathrm{out}}} && \text{if } e = \partial T_{\mathrm{out}} \cap\partial\Omega\in\mathcal{E}^B_h.
\end{aligned} [ [ v ] ] [ [ v ] ] := v ∣ T out − v ∣ T in := v ∣ T out if e = ∂ T out ∩ ∂ T in ∈ E h I , if e = ∂ T out ∩ ∂ Ω ∈ E h B . where v v v T o u t T_{\mathrm{out}} T out T i n T_{\mathrm{in}} T in T h \mathcal{T}_h T h e e e n e n_e n e T o u t T_{\mathrm{out}} T out T i n T_{\mathrm{in}} T in
Let P k ( D ) \mathcal{P}_k(D) P k ( D ) k k k D D D T h \mathcal{T}_h T h
P k ( Ω ; T h ) : = { v ∈ L 2 ( Ω ) : v ∣ T ∈ P k ( T ) for all T ∈ T h } . \mathcal{P}_k(\Omega;\mathcal{T}_h) := \{v \in L^2(\Omega) : v|_T \in\mathcal{P}_k(T)\ \text{for all } T \in\mathcal{T}_h\}. P k ( Ω ; T h ) := { v ∈ L 2 ( Ω ) : v ∣ T ∈ P k ( T ) for all T ∈ T h } . and we define the norm ∥ ⋅ ∥ L 2 ( T h ) \|\cdot\|_{L^2(\mathcal{T}_h)} ∥ ⋅ ∥ L 2 ( T h )
∥ v ∥ L 2 ( T h ) 2 = ∑ T ∈ T h ∥ v ∥ L 2 ( T ) 2 . \|v\|_{L^2(\mathcal{T}_h)}^2 = \sum_{T \in\mathcal{T}_h} \|v\|_{L^2(T)}^2. ∥ v ∥ L 2 ( T h ) 2 = T ∈ T h ∑ ∥ v ∥ L 2 ( T ) 2 . The following are definitions of some finite element spaces:
V h : = { v h ∈ P 1 ( Ω ; T h ) : v h is continuous at all p ∈ V h } , V h 0 : = V h ∩ H 0 1 ( Ω ) , V h g : = { v h ∈ V h : v h = g I on ∂ Ω } . \begin{aligned}
V_h &:= \{v_h \in\mathcal{P}_1(\Omega;\mathcal{T}_h): v_h\text{ is continuous at all }p \in\mathcal{V}_h\},\\
V_h^0 &:= V_h \cap H^1_0(\Omega),\\
V_h^g &:= \{v_h \in V_h: v_h = g_I\text{ on }\partial\Omega\}.
\end{aligned} V h V h 0 V h g := { v h ∈ P 1 ( Ω ; T h ) : v h is continuous at all p ∈ V h } , := V h ∩ H 0 1 ( Ω ) , := { v h ∈ V h : v h = g I on ∂ Ω } . V h C R : = { v h ∈ P 1 ( Ω ; T h ) : v h is continuous at all p ∈ M h I } . V_h^{CR} := \{v_h \in\mathcal{P}_1(\Omega;\mathcal{T}_h): v_h\text{ is continuous at all }p \in\mathcal{M}_h^I\}. V h C R := { v h ∈ P 1 ( Ω ; T h ) : v h is continuous at all p ∈ M h I } . M h : = { w h ∈ P 2 ( Ω ; T h ) : w h is continuous at all p ∈ V h and ∂ w h ∂ n e is continuous at the midpoint of e for all e ∈ E h I } . M_h := \left\{w_h \in\mathcal{P}_2(\Omega;\mathcal{T}_h): w_h\text{ is continuous at all }p \in\mathcal{V}_h\text{ and }\frac{\partial w_h}{\partial n_e}\text{ is continuous at the midpoint of }e\text{ for all }e \in\mathcal{E}_h^I\right\}. M h := { w h ∈ P 2 ( Ω ; T h ) : w h is continuous at all p ∈ V h and ∂ n e ∂ w h is continuous at the midpoint of e for all e ∈ E h I } . In addition, we denote by v I v_I v I I h v I_hv I h v v v v C 0 C^0 C 0 g I g_I g I g g g ∂ Ω \partial\Omega ∂ Ω C C C h h h ϵ \epsilon ϵ [7 ] .
Lemma 2.1 (Trace and inverse inequalities). For any T ∈ T h T \in\mathcal{T}_h T ∈ T h and e ∈ E h e \in\mathcal{E}_h e ∈ E h integers 0 ≤ n ≤ m 0 \le n \le m 0 ≤ n ≤ m and 1 ≤ p , q ≤ ∞ 1 \le p,q \le\infty 1 ≤ p , q ≤ ∞ there holds
∥ v ∥ L 2 ( ∂ T ) ≤ C ( h T 1 2 ∣ v ∣ H 1 ( T ) + h T − 1 2 ∥ v ∥ L 2 ( T ) ) ∀ v ∈ H 1 ( T ) , (7) \|v\|_{L^2(\partial T)} \le C\left(h_T^{\frac{1}{2}}|v|_{H^1(T)}+h_T^{-\frac{1}{2}}\|v\|_{L^2(T)}\right)\qquad\forall v \in H^1(T), \tag*{(7)} ∥ v ∥ L 2 ( ∂ T ) ≤ C ( h T 2 1 ∣ v ∣ H 1 ( T ) + h T − 2 1 ∥ v ∥ L 2 ( T ) ) ∀ v ∈ H 1 ( T ) , (7) ∥ v ∥ W m , q ( T ) ≤ C h T n − m + 2 min { 0 , 1 / q − 1 / p } ∥ v ∥ W n , p ( T ) ∀ v ∈ P k ( T ) , (8) \|v\|_{W^{m,q}(T)} \le Ch_T^{n-m+2\min\{0,1/q-1/p\}}\|v\|_{W^{n,p}(T)}\qquad\forall v \in\mathcal{P}_k(T), \tag*{(8)} ∥ v ∥ W m , q ( T ) ≤ C h T n − m + 2 m i n { 0 , 1/ q − 1/ p } ∥ v ∥ W n , p ( T ) ∀ v ∈ P k ( T ) , (8) ∥ v ∥ W m , q ( e ) ≤ C h e n − m + min { 0 , 1 / q − 1 / p } ∥ v ∥ W n , p ( e ) ∀ v ∈ P k ( e ) . (9) \|v\|_{W^{m,q}(e)} \le Ch_e^{n-m+\min\{0,1/q-1/p\}}\|v\|_{W^{n,p}(e)}\qquad\forall v \in\mathcal{P}_k(e). \tag*{(9)} ∥ v ∥ W m , q ( e ) ≤ C h e n − m + m i n { 0 , 1/ q − 1/ p } ∥ v ∥ W n , p ( e ) ∀ v ∈ P k ( e ) . (9) Lemma 2.2 (Bramble–Hilbert Lemma). Let B B B be a ball in Ω \Omega Ω such that Ω \Omega Ω is star-shaped with respect to B B B and such that its radius r > 1 2 r max r > \frac{1}{2}r_{\max} r > 2 1 r m a x where r max r_{\max} r m a x is the supremum of the radii of all balls with respect to which Ω \Omega Ω is star-shaped. Let Q m u Q^m u Q m u be the Taylor polynomial of order m m m of u u u averaged over B B B where u ∈ W m , p ( Ω ) u \in W^{m,p}(\Omega) u ∈ W m , p ( Ω ) and p ≥ 1 p \ge1 p ≥ 1 Then
∣ u − Q m u ∣ W k , p ( Ω ) ≤ C d m − k ∣ u ∣ W m , p ( Ω ) k = 0 , 1 , ⋯ , m , |u-Q^m u|_{W^{k,p}(\Omega)} \le Cd^{m-k}|u|_{W^{m,p}(\Omega)}\qquad k=0,1,\cdots,m, ∣ u − Q m u ∣ W k , p ( Ω ) ≤ C d m − k ∣ u ∣ W m , p ( Ω ) k = 0 , 1 , ⋯ , m , where d = diam ( Ω ) d=\operatorname{diam}(\Omega) d = diam ( Ω )
Recovery operator
We first recall the construction of the recovery operator G h G_h G h v h ∈ V h v_h \in V_h v h ∈ V h w h ∈ M h w_h \in M_h w h ∈ M h v h v_h v h
w h ( p ) = v h ( p ) . w_h(p)=v_h(p). w h ( p ) = v h ( p ) . ∂ w h ∂ n e ( m e ) = n e ⋅ 1 ∣ K e ∣ ∫ K e ∇ v h d x , \frac{\partial w_h}{\partial n_e}(m_e)=n_e\cdot\frac{1}{|\mathcal{K}_e|}\int_{\mathcal{K}_e}\nabla v_h\,\mathrm{d}x, ∂ n e ∂ w h ( m e ) = n e ⋅ ∣ K e ∣ 1 ∫ K e ∇ v h d x , where m e m_e m e e e e K e = ⋃ T ∈ T e T \mathcal{K}_e=\bigcup_{T\in\mathcal{T}_e}T K e = ⋃ T ∈ T e T
Then, we define the gradient recovery operator G h : V h → P 1 ( Ω ; T h ) × P 1 ( Ω ; T h ) G_h : V_h \to P_1(\Omega;\mathcal{T}_h) \times P_1(\Omega;\mathcal{T}_h) G h : V h → P 1 ( Ω ; T h ) × P 1 ( Ω ; T h ) G h v h ∣ T = ∇ w h ∣ T G_hv_h|_T = \nabla w_h|_T G h v h ∣ T = ∇ w h ∣ T T ∈ T h T \in\mathcal{T}_h T ∈ T h
Lemma 2.3. For any v h ∈ V h v_h \in V_h v h ∈ V h G h v h ∈ V h C R × V h C R G_hv_h \in V_h^{CR} \times V_h^{CR} G h v h ∈ V h C R × V h C R G h v h G_hv_h G h v h e ∈ E h e \in\mathcal{E}_h e ∈ E h
G h v h ( m e ) = 1 ∣ K e ∣ ∫ K e ∇ v h d x . G_hv_h(m_e) = \frac{1}{|\mathcal{K}_e|}\int_{\mathcal{K}_e} \nabla v_h\,\mathrm{d}x. G h v h ( m e ) = ∣ K e ∣ 1 ∫ K e ∇ v h d x . We assume that the recovery operator G h G_h G h
Assumption 2.4. There exists 0 < γ < 1 0 < \gamma< 1 0 < γ < 1
∥ ∇ v − G h v I ∥ L 2 ( Ω ) ≤ C h 1 + γ ∥ v ∥ H 3 ( Ω ) , ( ∑ T ∈ T h h T 2 ∥ ∇ v − G h I h v ∥ L 2 ( T ) 2 ) 1 / 2 ≤ C h γ ∥ v ∥ H 3 ( Ω ) , ∥ D 2 v − D G h v I ∥ L 2 ( Ω ) ≤ C h γ ∥ v ∥ H 3 ( Ω ) , \begin{aligned}
\|\nabla v-G_hv_I\|_{L^2(\Omega)} &\leq Ch^{1+\gamma}\|v\|_{H^3(\Omega)},\\
\left(\sum_{T\in\mathcal{T}_h}h_T^2\|\nabla v-G_hI_hv\|_{L^2(T)}^2\right)^{1/2} &\leq Ch^\gamma\|v\|_{H^3(\Omega)},\\
\|D^2v-DG_hv_I\|_{L^2(\Omega)} &\leq Ch^\gamma\|v\|_{H^3(\Omega)},
\end{aligned} ∥∇ v − G h v I ∥ L 2 ( Ω ) ( T ∈ T h ∑ h T 2 ∥∇ v − G h I h v ∥ L 2 ( T ) 2 ) 1/2 ∥ D 2 v − D G h v I ∥ L 2 ( Ω ) ≤ C h 1 + γ ∥ v ∥ H 3 ( Ω ) , ≤ C h γ ∥ v ∥ H 3 ( Ω ) , ≤ C h γ ∥ v ∥ H 3 ( Ω ) , hold for all v ∈ H 3 ( Ω ) v \in H^3(\Omega) v ∈ H 3 ( Ω )
Assumption 2.4 only depends on the mesh. It holds on most commonly used meshes, see, e.g. [14 Section 5.1] . We refer Appendix § 7 and [14 Section 3.3] for details on the relation between the superconvergence and mesh properties.
Some technical results
In this subsection, we will introduce some results used in the following sections. Let P e : L 1 ( e ) → P 0 ( e ) P_e : L^1(e) \to P_0(e) P e : L 1 ( e ) → P 0 ( e )
P e ( v ) = 1 h e ∫ e v d s . P_e(v) = \frac{1}{h_e}\int_e v\,\mathrm{d}s. P e ( v ) = h e 1 ∫ e v d s . Then we have the following result.
Lemma 2.5. Suppose { e i } i = 1 3 \{e_i\}_{i=1}^3 { e i } i = 1 3 T T T v ∈ H 1 ( T ) v \in H^1(T) v ∈ H 1 ( T )
∑ i = 1 3 ∥ v − P e i ( v ) ∥ L 2 ( e i ) ≤ C h T 1 / 2 ∣ v ∣ H 1 ( T ) . (10) \sum_{i=1}^{3}\|v-P_{e_i}(v)\|_{L^2(e_i)} \leq Ch_T^{1/2}|v|_{H^1(T)}. \tag*{(10)} i = 1 ∑ 3 ∥ v − P e i ( v ) ∥ L 2 ( e i ) ≤ C h T 1/2 ∣ v ∣ H 1 ( T ) . (10) Proof. The result follows from the Bramble–Hilbert lemma on the reference element and a standard scaling argument.
Lemma 2.6 (Lemma 4 and Lemma 5 in [42 ] ). For any v h ∈ V h v_h \in V_h v h ∈ V h
∫ e ⟦ G h v h ⟧ d s = 0 ∀ e ∈ E h I , \int_e \llbracket G_hv_h\rrbracket\,\mathrm{d}s=0 \qquad\forall e\in\mathcal{E}_h^I, ∫ e [ [ G h v h ] ] d s = 0 ∀ e ∈ E h I , and suppose e ∈ E h I e\in\mathcal{E}_h^I e ∈ E h I T ∩ T ′ = e T\cap T'=e T ∩ T ′ = e
∥ ⟦ G h v h ⟧ ∥ L 2 ( e ) ≤ C h T 1 / 2 ( ∥ D G h v h ∥ L 2 ( T ) + ∥ D G h v h ∥ L 2 ( T ′ ) ) . \|\llbracket G_hv_h\rrbracket\|_{L^2(e)}
\leq Ch_T^{1/2}\left(\|DG_hv_h\|_{L^2(T)}+\|DG_hv_h\|_{L^2(T')}\right). ∥ [ [ G h v h ] ] ∥ L 2 ( e ) ≤ C h T 1/2 ( ∥ D G h v h ∥ L 2 ( T ) + ∥ D G h v h ∥ L 2 ( T ′ ) ) . Moreover if v h ∈ V h 0 v_h\in V_h^0 v h ∈ V h 0
∥ G h v h ⋅ t e ∥ L 2 ( e ) ≤ C h T 1 / 2 ∥ D G h v h ∥ L 2 ( T ′ ) , \|G_hv_h\cdot t_e\|_{L^2(e)}
\leq Ch_T^{1/2}\|DG_hv_h\|_{L^2(T')}, ∥ G h v h ⋅ t e ∥ L 2 ( e ) ≤ C h T 1/2 ∥ D G h v h ∥ L 2 ( T ′ ) , if e ∈ E h B e\in\mathcal{E}_h^B e ∈ E h B e ⊂ T e\subset T e ⊂ T
Remark 2.7. A direct corollary of Lemma 2.6 is that, for every e ∈ E h I e \in\mathcal{E}^{I}_{h} e ∈ E h I A ∈ R 2 × 2 A \in\mathbb{R}^{2 \times2} A ∈ R 2 × 2
∫ e [ [ G h v h ] ] ⋅ ( A n e ) d s = 0 , ∀ v h ∈ V h . \int_{e} [\![G_{h}v_{h}]\!] \cdot(A n_{e}) \, \mathrm{d}s = 0, \qquad\forall v_{h} \in V_{h}. ∫ e [ [ G h v h ] ] ⋅ ( A n e ) d s = 0 , ∀ v h ∈ V h . Lemma 2.8. Suppose that A ∈ ( H 1 ( Ω ) ) 2 × 2 A \in\left(H^{1}(\Omega)\right)^{2 \times2} A ∈ ( H 1 ( Ω ) ) 2 × 2 v h ∈ V h v_{h} \in V_{h} v h ∈ V h
∣ ∑ e ∈ E h I ∫ e [ [ G h v h ] ] ⋅ ( A n e ) d s ∣ ≤ C h ( ∑ e ∈ E h I h e − 1 ∥ [ [ G h v h ] ] ∥ L 2 ( e ) 2 ) 1 / 2 ∣ A ∣ H 1 ( Ω ) . \left|\sum_{e \in\mathcal{E}^{I}_{h}} \int_{e} [\![G_{h}v_{h}]\!] \cdot(A n_{e}) \, \mathrm{d}s\right|
\leq C h \left(\sum_{e \in\mathcal{E}^{I}_{h}} h_{e}^{-1}\|[\![G_{h}v_{h}]\!]\|_{L^{2}(e)}^{2}\right)^{1/2}
|A|_{H^{1}(\Omega)}. e ∈ E h I ∑ ∫ e [ [ G h v h ] ] ⋅ ( A n e ) d s ≤ C h e ∈ E h I ∑ h e − 1 ∥ [ [ G h v h ] ] ∥ L 2 ( e ) 2 1/2 ∣ A ∣ H 1 ( Ω ) . Proof. We have the decomposition
∫ e [ [ G h v h ] ] ⋅ ( A n e ) d s = ∫ e [ [ G h v h ] ] ⋅ ( ( A − P e ( A ) ) n e ) d s + ∫ e [ [ G h v h ] ] ⋅ ( P e ( A ) n e ) d s = ∫ e [ [ G h v h ] ] ⋅ ( ( A − P e ( A ) ) n e ) d s , \begin{aligned}
\int_{e} [\![G_{h}v_{h}]\!] \cdot(A n_{e}) \, \mathrm{d}s
&= \int_{e} [\![G_{h}v_{h}]\!] \cdot\left((A-P_{e}(A))n_{e}\right) \, \mathrm{d}s
+\int_{e} [\![G_{h}v_{h}]\!] \cdot\left(P_{e}(A)n_{e}\right) \, \mathrm{d}s \\
&= \int_{e} [\![G_{h}v_{h}]\!] \cdot\left((A-P_{e}(A))n_{e}\right) \, \mathrm{d}s,
\end{aligned} ∫ e [ [ G h v h ] ] ⋅ ( A n e ) d s = ∫ e [ [ G h v h ] ] ⋅ ( ( A − P e ( A )) n e ) d s + ∫ e [ [ G h v h ] ] ⋅ ( P e ( A ) n e ) d s = ∫ e [ [ G h v h ] ] ⋅ ( ( A − P e ( A )) n e ) d s , where we have used Remark 2.7. Then we have
∣ ∑ e ∈ E h I ∫ e [ [ G h v h ] ] ⋅ ( A n e ) d s ∣ \left|\sum_{e \in\mathcal{E}^{I}_{h}} \int_{e} [\![G_{h}v_{h}]\!] \cdot(A n_{e}) \, \mathrm{d}s\right| e ∈ E h I ∑ ∫ e [ [ G h v h ] ] ⋅ ( A n e ) d s ≤ ∑ e ∈ E h I ∣ ∫ e [ [ G h v h ] ] ⋅ ( ( A − P e ( A ) ) n e ) d s ∣ \leq\sum_{e \in\mathcal{E}^{I}_{h}}
\left|\int_{e} [\![G_{h}v_{h}]\!] \cdot\left((A-P_{e}(A))n_{e}\right) \, \mathrm{d}s\right| ≤ e ∈ E h I ∑ ∫ e [ [ G h v h ] ] ⋅ ( ( A − P e ( A )) n e ) d s ≤ ∑ e ∈ E h I ∥ A − P e ( A ) ∥ L 2 ( e ) ∥ [ [ G h v h ] ] ∥ L 2 ( e ) \leq\sum_{e \in\mathcal{E}^{I}_{h}}
\|A-P_{e}(A)\|_{L^{2}(e)}\|[\![G_{h}v_{h}]\!]\|_{L^{2}(e)} ≤ e ∈ E h I ∑ ∥ A − P e ( A ) ∥ L 2 ( e ) ∥ [ [ G h v h ] ] ∥ L 2 ( e ) ≤ ( ∑ e ∈ E h I h e ∥ A − P e ( A ) ∥ L 2 ( e ) 2 ) 1 / 2 ( ∑ e ∈ E h I h e − 1 ∥ [ [ G h v h ] ] ∥ L 2 ( e ) 2 ) 1 / 2 \leq
\left(\sum_{e \in\mathcal{E}^{I}_{h}} h_{e}\|A-P_{e}(A)\|_{L^{2}(e)}^{2}\right)^{1/2}
\left(\sum_{e \in\mathcal{E}^{I}_{h}} h_{e}^{-1}\|[\![G_{h}v_{h}]\!]\|_{L^{2}(e)}^{2}\right)^{1/2} ≤ e ∈ E h I ∑ h e ∥ A − P e ( A ) ∥ L 2 ( e ) 2 1/2 e ∈ E h I ∑ h e − 1 ∥ [ [ G h v h ] ] ∥ L 2 ( e ) 2 1/2 ≤ C h ( ∑ e ∈ E h I h e − 1 ∥ [ [ G h v h ] ] ∥ L 2 ( e ) 2 ) 1 / 2 ∣ A ∣ H 1 ( Ω ) , \leq C h
\left(\sum_{e \in\mathcal{E}^{I}_{h}} h_{e}^{-1}\|[\![G_{h}v_{h}]\!]\|_{L^{2}(e)}^{2}\right)^{1/2}
|A|_{H^{1}(\Omega)}, ≤ C h e ∈ E h I ∑ h e − 1 ∥ [ [ G h v h ] ] ∥ L 2 ( e ) 2 1/2 ∣ A ∣ H 1 ( Ω ) , where we have used Lemma 2.5 and this completes the proof. □ \square □
Remark 2.9. For the term ( ∑ e ∈ E h I h e − 1 ∥ [ [ G h v h ] ] ∥ L 2 ( e ) 2 ) 1 / 2 \left(\sum_{e \in\mathcal{E}^{I}_{h} } h_{e}^{-1}\|[\![G_{h}v_{h}]\!]\|_{L^{2}(e)}^{2}\right)^{1/2} ( ∑ e ∈ E h I h e − 1 ∥ [ [ G h v h ] ] ∥ L 2 ( e ) 2 ) 1/2 Lemma 2.6 we have
( ∑ e ∈ E h I h e − 1 ∥ [ [ G h v h ] ] ∥ L 2 ( e ) 2 ) 1 / 2 ≤ C ∥ D G h v h ∥ L 2 ( T h ) . \left(\sum_{e \in\mathcal{E}^{I}_{h}} h_{e}^{-1}\|[\![G_{h}v_{h}]\!]\|_{L^{2}(e)}^{2}\right)^{1/2}
\leq C\|DG_{h}v_{h}\|_{L^{2}(\mathcal{T}_{h})}. e ∈ E h I ∑ h e − 1 ∥ [ [ G h v h ] ] ∥ L 2 ( e ) 2 1/2 ≤ C ∥ D G h v h ∥ L 2 ( T h ) . Moreover, if v h = v I v_{h}=v_{I} v h = v I v ∈ H 3 ( Ω ) v \in H^{3}(\Omega) v ∈ H 3 ( Ω ) Lemma 2.1 ,
( ∑ e ∈ E h I h e − 1 ∥ [ [ G h v I ] ] ∥ L 2 ( e ) 2 ) 1 / 2 \left(\sum_{e \in\mathcal{E}^{I}_{h}} h_{e}^{-1}\|[\![G_{h}v_{I}]\!]\|_{L^{2}(e)}^{2}\right)^{1/2} e ∈ E h I ∑ h e − 1 ∥ [ [ G h v I ] ] ∥ L 2 ( e ) 2 1/2 ( ∑ e ∈ E h h e − 1 ∥ ∇ v − G h v ∥ L 2 ( e ) 2 ) 1 / 2 ≤ C ( ∑ T ∈ T h h T − 2 ∥ ∇ v − G h v ∥ L 2 ( T ) 2 + ∑ T ∈ T h ∥ D 2 v − D G h v ∥ L 2 ( T ) 2 ) 1 / 2 ≤ C h γ ∥ v ∥ H 3 ( Ω ) . \begin{aligned}
\left(\sum_{e\in\mathcal{E}_h} h_e^{-1}\lVert\nabla v-G_hv\rVert_{L^2(e)}^2\right)^{1/2} \\
&\le C\left(\sum_{T\in\mathcal{T}_h}h_T^{-2}\lVert\nabla v-G_hv\rVert_{L^2(T)}^2+\sum_{T\in\mathcal{T}_h}\lVert D^2v-DG_hv\rVert_{L^2(T)}^2\right)^{1/2}\le Ch^\gamma\lVert v\rVert_{H^3(\Omega)}.
\end{aligned} ( e ∈ E h ∑ h e − 1 ∥ ∇ v − G h v ∥ L 2 ( e ) 2 ) 1/2 ≤ C ( T ∈ T h ∑ h T − 2 ∥ ∇ v − G h v ∥ L 2 ( T ) 2 + T ∈ T h ∑ ∥ D 2 v − D G h v ∥ L 2 ( T ) 2 ) 1/2 ≤ C h γ ∥ v ∥ H 3 ( Ω ) . The above two estimates will be used in the following sections. Similarly, for v h ∈ V h 0 v_h\in V_h^0 v h ∈ V h 0 v ∈ H 3 ( Ω ) ∩ H 0 1 ( Ω ) v\in H^3(\Omega)\cap H_0^1(\Omega) v ∈ H 3 ( Ω ) ∩ H 0 1 ( Ω )
( ∑ e ∈ E h B h e − 1 ∥ G h v h ⋅ t e ∥ L 2 ( e ) 2 ) 1 / 2 . \left(\sum_{e\in\mathcal{E}_h^B}h_e^{-1}\lVert G_hv_h\cdot t_e\rVert_{L^2(e)}^2\right)^{1/2}. e ∈ E h B ∑ h e − 1 ∥ G h v h ⋅ t e ∥ L 2 ( e ) 2 1/2 . Linearization and finite element approximation
In this section, we consider the linearization of (1.2) at the solution u ϵ u^\epsilon u ϵ
We note that for a given smooth function v v v t ∈ R t\in\mathbb{R} t ∈ R [10 ]
det ( D 2 ( u ϵ + t v ) ) = det ( D 2 u ϵ ) + t ( Φ ϵ : D 2 v ) + t 2 det ( D 2 v ) . \det(D^2(u^\epsilon+tv))=\det(D^2u^\epsilon)+t(\Phi^\epsilon:D^2v)+t^2\det(D^2v). det ( D 2 ( u ϵ + t v )) = det ( D 2 u ϵ ) + t ( Φ ϵ : D 2 v ) + t 2 det ( D 2 v ) . Thus, the linearization of the operator M ϵ ( u ϵ ) : = ϵ Δ 2 u ϵ − det ( D 2 u ϵ ) M^\epsilon(u^\epsilon):=\epsilon\Delta^2u^\epsilon-\det(D^2u^\epsilon) M ϵ ( u ϵ ) := ϵ Δ 2 u ϵ − det ( D 2 u ϵ ) u ϵ u^\epsilon u ϵ
L u ϵ ( v ) : = ϵ Δ 2 v − Φ ϵ : D 2 v = ϵ Δ 2 v − div ( Φ ϵ ∇ v ) . L_{u^\epsilon}(v):=\epsilon\Delta^2v-\Phi^\epsilon:D^2v=\epsilon\Delta^2v-\operatorname{div}(\Phi^\epsilon\nabla v). L u ϵ ( v ) := ϵ Δ 2 v − Φ ϵ : D 2 v = ϵ Δ 2 v − div ( Φ ϵ ∇ v ) . where we have used the divergence-free property of the cofactor matrix [32 Lemma 7] .
Let V 0 ∗ V_0^* V 0 ∗ V 0 V_0 V 0 φ ∈ V 0 ∗ \varphi\in V_0^* φ ∈ V 0 ∗ ψ ∈ H − 1 / 2 ( ∂ Ω ) \psi\in H^{-1/2}(\partial\Omega) ψ ∈ H − 1/2 ( ∂ Ω )
(3.1a)
L u ϵ ( v ) = φ in Ω , (11) L_{u^\epsilon}(v)=\varphi\quad\text{in }\Omega, \tag*{(11)} L u ϵ ( v ) = φ in Ω , (11) (3.1b)
v = 0 on ∂ Ω , (12) v=0\quad\text{on }\partial\Omega, \tag*{(12)} v = 0 on ∂ Ω , (12) (3.1c)
Δ v = ψ on ∂ Ω . (13) \Delta v=\psi\quad\text{on }\partial\Omega. \tag*{(13)} Δ v = ψ on ∂ Ω. (13) To define the finite element method for (3.1), we first consider its weak formulation. Multiplying by a test function w ∈ V 0 w\in V_0 w ∈ V 0 v ∈ V 0 v\in V_0 v ∈ V 0
(3.2)
a ( v , w ) = ⟨ φ , w ⟩ + ϵ ( ψ , ∇ w ⋅ n ) L 2 ( ∂ Ω ) ∀ w ∈ V 0 , (14) a(v,w)=\langle\varphi,w\rangle+\epsilon(\psi,\nabla w\cdot n)_{L^2(\partial\Omega)}\quad\forall w\in V_0, \tag*{(14)} a ( v , w ) = ⟨ φ , w ⟩ + ϵ ( ψ , ∇ w ⋅ n ) L 2 ( ∂ Ω ) ∀ w ∈ V 0 , (14) where
a ( v , w ) = ϵ ( Δ v , Δ w ) L 2 ( Ω ) + ( Φ ϵ ∇ v , ∇ w ) L 2 ( Ω ) . a(v,w)=\epsilon(\Delta v,\Delta w)_{L^2(\Omega)}+(\Phi^\epsilon\nabla v,\nabla w)_{L^2(\Omega)}. a ( v , w ) = ϵ ( Δ v , Δ w ) L 2 ( Ω ) + ( Φ ϵ ∇ v , ∇ w ) L 2 ( Ω ) . Since u ϵ u^\epsilon u ϵ Φ ϵ \Phi^\epsilon Φ ϵ
a ( v , v ) ≥ C ϵ ∥ v ∥ H 2 ( Ω ) 2 . a(v,v)\ge C\epsilon\lVert v\rVert_{H^2(\Omega)}^2. a ( v , v ) ≥ C ϵ ∥ v ∥ H 2 ( Ω ) 2 . Hence the well-posedness of the problem (14) follows from the Lax–Milgram Lemma. Likewise, to ensure the well-posedness of the discrete problem, the following lemma plays a crucial role. Its proof can be found in [14 ] .
Lemma 3.1 (Discrete MT estimate). For any v h ∈ V h 0 v_h\in V_h^0 v h ∈ V h 0
(3.3)
∥ D G h v h ∥ L 2 ( T h ) 2 ≤ C ( ∥ div ( G h v h ) ∥ L 2 ( T h ) 2 + b h ( v h , v h ) ) (15) \lVert DG_hv_h\rVert_{L^2(\mathcal{T}_h)}^2\le C\left(\lVert\operatorname{div}(G_hv_h)\rVert_{L^2(\mathcal{T}_h)}^2+b_h(v_h,v_h)\right) \tag*{(15)} ∥ D G h v h ∥ L 2 ( T h ) 2 ≤ C ( ∥ div ( G h v h ) ∥ L 2 ( T h ) 2 + b h ( v h , v h ) ) (15) where
b h ( u h , v h ) = ∑ e ∈ E h I h e − 1 ( [ [ G h u h ] ] , [ [ G h v h ] ] ) L 2 ( e ) + ∑ e ∈ E h B h e − 1 ( G h u h ⋅ t e , G h v h ⋅ t e ) L 2 ( e ) . b_h(u_h,v_h)=\sum_{e\in\mathcal{E}_h^I}h_e^{-1}\left([\![G_hu_h]\!],[\![G_hv_h]\!]\right)_{L^2(e)}+\sum_{e\in\mathcal{E}_h^B}h_e^{-1}\left(G_hu_h\cdot t_e,G_hv_h\cdot t_e\right)_{L^2(e)}. b h ( u h , v h ) = e ∈ E h I ∑ h e − 1 ( [ [ G h u h ] ] , [ [ G h v h ] ] ) L 2 ( e ) + e ∈ E h B ∑ h e − 1 ( G h u h ⋅ t e , G h v h ⋅ t e ) L 2 ( e ) . Inspired by the discrete MT estimate, we design the following finite element method for (14) : find v h ∈ V h 0 v_h\in V_h^0 v h ∈ V h 0
a h ( v h , w h ) + ε b h ( v h , w h ) = ⟨ φ , w h ⟩ + ε ( ψ , G h w h ⋅ n ) L 2 ( ∂ Ω ) ∀ w h ∈ V h 0 , (16) a_h(v_h,w_h)+\varepsilon b_h(v_h,w_h)=\langle\varphi,w_h\rangle+\varepsilon(\psi,G_hw_h\cdot n)_{L^2(\partial\Omega)}\quad\forall w_h\in V_h^0, \tag*{(16)} a h ( v h , w h ) + ε b h ( v h , w h ) = ⟨ φ , w h ⟩ + ε ( ψ , G h w h ⋅ n ) L 2 ( ∂ Ω ) ∀ w h ∈ V h 0 , (16) where
a h ( v h , w h ) = ε ( div ( G h v h ) , div ( G h w h ) ) L 2 ( Ω ) + ( Φ ε ∇ v h , ∇ w h ) L 2 ( Ω ) . a_h(v_h,w_h)=\varepsilon(\operatorname{div}(G_hv_h),\operatorname{div}(G_hw_h))_{L^2(\Omega)}+(\Phi^\varepsilon\nabla v_h,\nabla w_h)_{L^2(\Omega)}. a h ( v h , w h ) = ε ( div ( G h v h ) , div ( G h w h ) ) L 2 ( Ω ) + ( Φ ε ∇ v h , ∇ w h ) L 2 ( Ω ) . Then by Lemma 3.1 and the Lax–Milgram lemma we have the following result.
Theorem 3.2. Problem (3.4) has a unique solution v h ∈ V h 0 v_h\in V_h^0 v h ∈ V h 0
In addition, we can establish the following error estimate:
Theorem 3.3. Suppose v ∈ H 3 ( Ω ) v\in H^3(\Omega) v ∈ H 3 ( Ω )
∥ D 2 v − D G h v h ∥ L 2 ( T h ) ≤ C ( h r ∥ v ∥ H 3 ( Ω ) + ε − 2 h ∣ v ∣ H 2 ( Ω ) ) . (17) \lVert D^2v-DG_hv_h\rVert_{L^2(\mathcal{T}_h)}\leq C\left(h^r\lVert v\rVert_{H^3(\Omega)}+\varepsilon^{-2}h\lvert v\rvert_{H^2(\Omega)}\right). \tag*{(17)} ∥ D 2 v − D G h v h ∥ L 2 ( T h ) ≤ C ( h r ∥ v ∥ H 3 ( Ω ) + ε − 2 h ∣ v ∣ H 2 ( Ω ) ) . (17) Proof. Let w h = v h − v I w_h=v_h-v_I w h = v h − v I
a h ( w h , w h ) + ε b h ( w h , w h ) = a h ( v h , w h ) + ε b h ( v h , w h ) − ( a h ( v I , w h ) + ε b h ( v I , w h ) ) = ⟨ φ , w h ⟩ + ε ( ψ , G h w h ⋅ n ) L 2 ( ∂ Ω ) − ( a h ( v I , w h ) + ε b h ( v I , w h ) ) = ( ε Δ 2 v − div ( Φ ε ∇ v ) , w h ) L 2 ( Ω ) + ε ( ψ , G h w h ⋅ n ) L 2 ( ∂ Ω ) − ( a h ( v I , w h ) + ε b h ( v I , w h ) ) . (18) \begin{aligned}
a_h(w_h,w_h)+\varepsilon b_h(w_h,w_h)
={}&a_h(v_h,w_h)+\varepsilon b_h(v_h,w_h)-\left(a_h(v_I,w_h)+\varepsilon b_h(v_I,w_h)\right)\\
={}&\langle\varphi,w_h\rangle+\varepsilon(\psi,G_hw_h\cdot n)_{L^2(\partial\Omega)}\\
&-\left(a_h(v_I,w_h)+\varepsilon b_h(v_I,w_h)\right)\\
={}&(\varepsilon\Delta^2v-\operatorname{div}(\Phi^\varepsilon\nabla v),w_h)_{L^2(\Omega)}+\varepsilon(\psi,G_hw_h\cdot n)_{L^2(\partial\Omega)}\\
&-\left(a_h(v_I,w_h)+\varepsilon b_h(v_I,w_h)\right). \tag*{(18)}
\end{aligned} a h ( w h , w h ) + ε b h ( w h , w h ) = = = a h ( v h , w h ) + ε b h ( v h , w h ) − ( a h ( v I , w h ) + ε b h ( v I , w h ) ) ⟨ φ , w h ⟩ + ε ( ψ , G h w h ⋅ n ) L 2 ( ∂ Ω ) − ( a h ( v I , w h ) + ε b h ( v I , w h ) ) ( ε Δ 2 v − div ( Φ ε ∇ v ) , w h ) L 2 ( Ω ) + ε ( ψ , G h w h ⋅ n ) L 2 ( ∂ Ω ) − ( a h ( v I , w h ) + ε b h ( v I , w h ) ) . (18) For the first term, since w h ∈ V h 0 w_h\in V_h^0 w h ∈ V h 0
( ε Δ 2 v − div ( Φ ε ∇ v ) , w h ) L 2 ( Ω ) = − ε ( ∇ Δ v , ∇ w h ) L 2 ( Ω ) + ( Φ ε ∇ v , ∇ w h ) L 2 ( Ω ) = − ε ( ∇ Δ v , ∇ w h − G h w h ) L 2 ( Ω ) − ε ( ∇ Δ v , G h w h ) L 2 ( Ω ) + ( Φ ε ∇ v , ∇ w h ) L 2 ( Ω ) = − ε ( ∇ Δ v , ∇ w h − G h w h ) L 2 ( Ω ) + ε ( Δ v , div ( G h w h ) ) L 2 ( Ω ) − ε ( ψ , G h w h ⋅ n ) L 2 ( ∂ Ω ) − ε ∑ e ∈ E h I ( Δ v , [ [ G h w h ⋅ n e ] ] ) L 2 ( e ) + ( Φ ε ∇ v , ∇ w h ) L 2 ( Ω ) . \begin{aligned}
(\varepsilon\Delta^2v-\operatorname{div}(\Phi^\varepsilon\nabla v),w_h)_{L^2(\Omega)}
={}&-\varepsilon(\nabla\Delta v,\nabla w_h)_{L^2(\Omega)}+(\Phi^\varepsilon\nabla v,\nabla w_h)_{L^2(\Omega)}\\
={}&-\varepsilon(\nabla\Delta v,\nabla w_h-G_hw_h)_{L^2(\Omega)}-\varepsilon(\nabla\Delta v,G_hw_h)_{L^2(\Omega)}\\
&+(\Phi^\varepsilon\nabla v,\nabla w_h)_{L^2(\Omega)}\\
={}&-\varepsilon(\nabla\Delta v,\nabla w_h-G_hw_h)_{L^2(\Omega)}+\varepsilon(\Delta v,\operatorname{div}(G_hw_h))_{L^2(\Omega)}\\
&-\varepsilon(\psi,G_hw_h\cdot n)_{L^2(\partial\Omega)}\\
&-\varepsilon\sum_{e\in\mathcal{E}_h^I}(\Delta v,[\![G_hw_h\cdot n_e]\!])_{L^2(e)}+(\Phi^\varepsilon\nabla v,\nabla w_h)_{L^2(\Omega)}.
\end{aligned} ( ε Δ 2 v − div ( Φ ε ∇ v ) , w h ) L 2 ( Ω ) = = = − ε ( ∇Δ v , ∇ w h ) L 2 ( Ω ) + ( Φ ε ∇ v , ∇ w h ) L 2 ( Ω ) − ε ( ∇Δ v , ∇ w h − G h w h ) L 2 ( Ω ) − ε ( ∇Δ v , G h w h ) L 2 ( Ω ) + ( Φ ε ∇ v , ∇ w h ) L 2 ( Ω ) − ε ( ∇Δ v , ∇ w h − G h w h ) L 2 ( Ω ) + ε ( Δ v , div ( G h w h ) ) L 2 ( Ω ) − ε ( ψ , G h w h ⋅ n ) L 2 ( ∂ Ω ) − ε e ∈ E h I ∑ ( Δ v , [ [ G h w h ⋅ n e ] ] ) L 2 ( e ) + ( Φ ε ∇ v , ∇ w h ) L 2 ( Ω ) . This leads to
a h ( w h , w h ) + ε b h ( w h , w h ) = ε ( Δ v − div ( G h v I ) , div ( G h w h ) ) L 2 ( Ω ) + ( Φ ε ∇ v − Φ ε ∇ v I , ∇ w h ) L 2 ( Ω ) \begin{aligned}
a_h(w_h,w_h)+\varepsilon b_h(w_h,w_h)
={}&\varepsilon(\Delta v-\operatorname{div}(G_hv_I),\operatorname{div}(G_hw_h))_{L^2(\Omega)}\\
&+(\Phi^\varepsilon\nabla v-\Phi^\varepsilon\nabla v_I,\nabla w_h)_{L^2(\Omega)}
\end{aligned} a h ( w h , w h ) + ε b h ( w h , w h ) = ε ( Δ v − div ( G h v I ) , div ( G h w h ) ) L 2 ( Ω ) + ( Φ ε ∇ v − Φ ε ∇ v I , ∇ w h ) L 2 ( Ω ) − ϵ ( Δ v , ∇ w h − G h w h ) L 2 ( Ω ) − ϵ ∑ e ∈ E h I ( Δ v , [ [ G h w h ⋅ n e ] ] ) L 2 ( e ) − ϵ b h ( v I , w h ) -\epsilon(\Delta v,\nabla w_h-G_h w_h)_{L^2(\Omega)}-\epsilon\sum_{e\in\mathcal{E}_h^I}(\Delta v,[[G_h w_h\cdot n_e]])_{L^2(e)}-\epsilon b_h(v_I,w_h) − ϵ ( Δ v , ∇ w h − G h w h ) L 2 ( Ω ) − ϵ e ∈ E h I ∑ ( Δ v , [[ G h w h ⋅ n e ]] ) L 2 ( e ) − ϵ b h ( v I , w h ) We can deduce the following error bounds
∣ E 1 ∣ ≤ C ϵ h γ ∥ v ∥ H 3 ( Ω ) ∥ D G h w h ∥ L 2 ( T h ) + C ϵ − 1 h ∣ v ∣ H 2 ( Ω ) ∥ ∇ w h ∥ L 2 ( T h ) , |E_1|\leq C\epsilon h^\gamma\|v\|_{H^3(\Omega)}\|DG_h w_h\|_{L^2(\mathcal{T}_h)}+C\epsilon^{-1}h|v|_{H^2(\Omega)}\|\nabla w_h\|_{L^2(\mathcal{T}_h)}, ∣ E 1 ∣ ≤ C ϵ h γ ∥ v ∥ H 3 ( Ω ) ∥ D G h w h ∥ L 2 ( T h ) + C ϵ − 1 h ∣ v ∣ H 2 ( Ω ) ∥∇ w h ∥ L 2 ( T h ) , ∣ E 2 ∣ ≤ C ϵ h ∣ v ∣ H 3 ( Ω ) ∥ D G h w h ∥ L 2 ( T h ) , |E_2|\leq C\epsilon h|v|_{H^3(\Omega)}\|DG_h w_h\|_{L^2(\mathcal{T}_h)}, ∣ E 2 ∣ ≤ C ϵ h ∣ v ∣ H 3 ( Ω ) ∥ D G h w h ∥ L 2 ( T h ) , ∣ E 3 ∣ ≤ C ϵ h γ ∣ v ∣ H 3 ( Ω ) ∥ D G h w h ∥ L 2 ( T h ) , |E_3|\leq C\epsilon h^\gamma|v|_{H^3(\Omega)}\|DG_h w_h\|_{L^2(\mathcal{T}_h)}, ∣ E 3 ∣ ≤ C ϵ h γ ∣ v ∣ H 3 ( Ω ) ∥ D G h w h ∥ L 2 ( T h ) , ∣ E 4 ∣ ≤ C ϵ h γ ∥ v ∥ H 3 ( Ω ) ∥ D G h w h ∥ L 2 ( T h ) . |E_4|\leq C\epsilon h^\gamma\|v\|_{H^3(\Omega)}\|DG_h w_h\|_{L^2(\mathcal{T}_h)}. ∣ E 4 ∣ ≤ C ϵ h γ ∥ v ∥ H 3 ( Ω ) ∥ D G h w h ∥ L 2 ( T h ) . The estimate of E 3 E_3 E 3 Lemma 2.8 and Remark 2.9, while that of E 4 E_4 E 4 Lemma 2.6 and Remark 2.9. Note that we have
a h ( w h , w h ) + ϵ b h ( w h , w h ) ≥ C ϵ ∥ D G h w h ∥ L 2 ( T h ) 2 . a_h(w_h,w_h)+\epsilon b_h(w_h,w_h)\geq C\epsilon\|DG_h w_h\|_{L^2(\mathcal{T}_h)}^2. a h ( w h , w h ) + ϵ b h ( w h , w h ) ≥ C ϵ ∥ D G h w h ∥ L 2 ( T h ) 2 . Then we have the following estimate
∥ D G h w h ∥ L 2 ( T h ) ≤ C h γ ( ∥ v ∥ H 3 ( Ω ) + ϵ − 2 h 1 − γ ∣ v ∣ H 2 ( Ω ) ) . (19) \|DG_h w_h\|_{L^2(\mathcal{T}_h)}\leq Ch^\gamma\left(\|v\|_{H^3(\Omega)}+\epsilon^{-2}h^{1-\gamma}|v|_{H^2(\Omega)}\right). \tag*{(19)} ∥ D G h w h ∥ L 2 ( T h ) ≤ C h γ ( ∥ v ∥ H 3 ( Ω ) + ϵ − 2 h 1 − γ ∣ v ∣ H 2 ( Ω ) ) . (19) The triangle inequality implies that
∥ D 2 v − D G h v h ∥ L 2 ( T h ) ≤ ∥ D 2 v − D G h v I ∥ L 2 ( T h ) + ∥ D G h w h ∥ L 2 ( T h ) , \|D^2v-DG_hv_h\|_{L^2(\mathcal{T}_h)}\leq\|D^2v-DG_hv_I\|_{L^2(\mathcal{T}_h)}+\|DG_hw_h\|_{L^2(\mathcal{T}_h)}, ∥ D 2 v − D G h v h ∥ L 2 ( T h ) ≤ ∥ D 2 v − D G h v I ∥ L 2 ( T h ) + ∥ D G h w h ∥ L 2 ( T h ) , which concludes our proof.
C 0 C^0 C 0 In this section, we present the numerical scheme and its convergence analysis.
Numerical scheme
The weak form of the model equation (1.2) reads as follows: seek u ϵ ∈ V g h u^\epsilon\in V_g^h u ϵ ∈ V g h v ∈ V 0 h v\in V_0^h v ∈ V 0 h
− ϵ ( Δ u ϵ , Δ v ) L 2 ( Ω ) + ( det ( D 2 u ϵ ) , v ) L 2 ( Ω ) = ( f , v ) L 2 ( Ω ) − ϵ ( ϵ , ∇ v ⋅ n ) L 2 ( ∂ Ω ) . (20) -\epsilon(\Delta u^\epsilon,\Delta v)_{L^2(\Omega)}+\left(\det(D^2u^\epsilon),v\right)_{L^2(\Omega)}=(f,v)_{L^2(\Omega)}-\epsilon(\epsilon,\nabla v\cdot n)_{L^2(\partial\Omega)}. \tag*{(20)} − ϵ ( Δ u ϵ , Δ v ) L 2 ( Ω ) + ( det ( D 2 u ϵ ) , v ) L 2 ( Ω ) = ( f , v ) L 2 ( Ω ) − ϵ ( ϵ , ∇ v ⋅ n ) L 2 ( ∂ Ω ) . (20) Then the numerical scheme is designed by the idea of approximating the Hessian by derivatives of the recovered gradient.
Since we consider the nonhomogeneous boundary condition, the definition of b h ( ⋅ , ⋅ ) b_h(\cdot,\cdot) b h ( ⋅ , ⋅ ) V g h × V 0 h V_g^h\times V_0^h V g h × V 0 h b h b_h b h g g g e ∈ E h B e\in\mathcal{E}_h^B e ∈ E h B g I 2 g_I^2 g I 2 e e e v h ∈ V g h v_h\in V_g^h v h ∈ V g h w h ∈ V 0 h w_h\in V_0^h w h ∈ V 0 h
b h ( v h , w h ) = ∑ e ∈ E h I h e − 1 ( [ [ G h v h ] ] , [ [ G h w h ] ] ) L 2 ( e ) + ∑ e ∈ E h B h e − 1 ( G h v h ⋅ t e − ∂ t e g I 2 , G h w h ⋅ t e ) L 2 ( e ) . b_h(v_h,w_h)=\sum_{e\in\mathcal{E}_h^I}h_e^{-1}([[G_hv_h]],[[G_hw_h]])_{L^2(e)}
+\sum_{e\in\mathcal{E}_h^B}h_e^{-1}(G_hv_h\cdot t_e-\partial_{t_e}g_I^2,G_hw_h\cdot t_e)_{L^2(e)}. b h ( v h , w h ) = e ∈ E h I ∑ h e − 1 ([[ G h v h ]] , [[ G h w h ]] ) L 2 ( e ) + e ∈ E h B ∑ h e − 1 ( G h v h ⋅ t e − ∂ t e g I 2 , G h w h ⋅ t e ) L 2 ( e ) . Moreover, if v h = v I v_h=v_I v h = v I v ∈ V g ∩ H 3 ( Ω ) v\in V_g\cap H^3(\Omega) v ∈ V g ∩ H 3 ( Ω )
∣ b h ( v I , w h ) ∣ ≤ C h γ ∥ v ∥ H 3 ( Ω ) ∥ D G h w h ∥ L 2 ( T h ) . |b_h(v_I,w_h)|\leq Ch^\gamma\|v\|_{H^3(\Omega)}\|DG_h w_h\|_{L^2(\mathcal{T}_h)}. ∣ b h ( v I , w h ) ∣ ≤ C h γ ∥ v ∥ H 3 ( Ω ) ∥ D G h w h ∥ L 2 ( T h ) . Hence we design the following numerical scheme for the model problem: Find u h ϵ ∈ V h g u_h^\epsilon\in V_h^g u h ϵ ∈ V h g v h ∈ V h 0 v_h \in V_h^0 v h ∈ V h 0
− ϵ ( div ( G h u h ϵ ) , div ( G h v h ) ) L 2 ( Ω ) + ( det ( D G h u h ϵ ) , v h ) L 2 ( Ω ) − ϵ b h ( u h ϵ , v h ) = ( f , v h ) L 2 ( Ω ) − ϵ ( ϵ , G h v h ⋅ n ) L 2 ( ∂ Ω ) . (21) -\epsilon\left(\operatorname{div}(G_hu_h^\epsilon),\operatorname{div}(G_hv_h)\right)_{L^2(\Omega)}+\left(\det(DG_hu_h^\epsilon),v_h\right)_{L^2(\Omega)}-\epsilon b_h(u_h^\epsilon,v_h)
= (f,v_h)_{L^2(\Omega)}-\epsilon(\epsilon,G_hv_h\cdot n)_{L^2(\partial\Omega)}. \tag*{(21)} − ϵ ( div ( G h u h ϵ ) , div ( G h v h ) ) L 2 ( Ω ) + ( det ( D G h u h ϵ ) , v h ) L 2 ( Ω ) − ϵ b h ( u h ϵ , v h ) = ( f , v h ) L 2 ( Ω ) − ϵ ( ϵ , G h v h ⋅ n ) L 2 ( ∂ Ω ) . (21) The second term on the left-hand side is nonlinear; hence, the well-posedness of the numerical scheme must be justified, which will be discussed in the next subsection.
Existence, uniqueness and H 2 H^2 H 2
In this subsection, we prove the well-posedness of the numerical scheme. We first define an operator T : V h g → V h g T: V_h^g \to V_h^g T : V h g → V h g v h ∈ V h g v_h \in V_h^g v h ∈ V h g T ( v h ) ∈ V h g T(v_h) \in V_h^g T ( v h ) ∈ V h g
A h ϵ ( v h − T ( v h ) , w h ) = ϵ ( div ( G h v h ) , div ( G h w h ) ) L 2 ( Ω ) − ( det ( D G h v h ) , w h ) L 2 ( Ω ) + ϵ b h ( v h , w h ) + ( f , w h ) L 2 ( Ω ) − ( ϵ 2 , G h w h ⋅ n ) L 2 ( ∂ Ω ) , ∀ w h ∈ V h 0 . A_h^\epsilon(v_h-T(v_h),w_h)
=\epsilon\left(\operatorname{div}(G_hv_h),\operatorname{div}(G_hw_h)\right)_{L^2(\Omega)}-\left(\det(DG_hv_h),w_h\right)_{L^2(\Omega)}
+\epsilon b_h(v_h,w_h)+(f,w_h)_{L^2(\Omega)}-(\epsilon^2,G_hw_h\cdot n)_{L^2(\partial\Omega)},\qquad\forall w_h\in V_h^0. A h ϵ ( v h − T ( v h ) , w h ) = ϵ ( div ( G h v h ) , div ( G h w h ) ) L 2 ( Ω ) − ( det ( D G h v h ) , w h ) L 2 ( Ω ) + ϵ b h ( v h , w h ) + ( f , w h ) L 2 ( Ω ) − ( ϵ 2 , G h w h ⋅ n ) L 2 ( ∂ Ω ) , ∀ w h ∈ V h 0 . where
A h ϵ ( v h − T ( v h ) , w h ) = a h ( v h − T ( v h ) , w h ) + ϵ b h ( v h − T ( v h ) , w h ) . A_h^\epsilon(v_h-T(v_h),w_h)=a_h(v_h-T(v_h),w_h)+\epsilon b_h(v_h-T(v_h),w_h). A h ϵ ( v h − T ( v h ) , w h ) = a h ( v h − T ( v h ) , w h ) + ϵ b h ( v h − T ( v h ) , w h ) . The results of § 3 imply that T ( v h ) T(v_h) T ( v h ) T T T (21) . To facilitate the error analysis, we introduce the following notation:
B h ( ρ ) : = { v h ∈ V h g : ∥ D G h ( v h − u I ϵ ) ∥ L 2 ( T h ) ≤ ρ } . \mathbb{B}_h(\rho):=\left\{v_h\in V_h^g:\left\|DG_h(v_h-u_I^\epsilon)\right\|_{L^2(\mathcal{T}_h)}\leq\rho\right\}. B h ( ρ ) := { v h ∈ V h g : ∥ D G h ( v h − u I ϵ ) ∥ L 2 ( T h ) ≤ ρ } . Lemma 4.1. We have the following estimates for u I ϵ − T ( u I ϵ ) u_I^\epsilon-T(u_I^\epsilon) u I ϵ − T ( u I ϵ )
∥ D G h ( u I ϵ − T ( u I ϵ ) ) ∥ L 2 ( T h ) ≤ C h γ ( 1 + ϵ − 1 ∥ u ϵ ∥ W 2 , ∞ ( Ω ) ) ∥ u ϵ ∥ H 3 ( Ω ) ≤ C 1 ( ϵ − 1 + ϵ − 3 ) h γ , (22) \left\|DG_h(u_I^\epsilon-T(u_I^\epsilon))\right\|_{L^2(\mathcal{T}_h)}
\leq Ch^\gamma\left(1+\epsilon^{-1}\left\|u^\epsilon\right\|_{W^{2,\infty}(\Omega)}\right)\left\|u^\epsilon\right\|_{H^3(\Omega)}
\leq C_1(\epsilon^{-1}+\epsilon^{-3})h^\gamma, \tag*{(22)} ∥ D G h ( u I ϵ − T ( u I ϵ )) ∥ L 2 ( T h ) ≤ C h γ ( 1 + ϵ − 1 ∥ u ϵ ∥ W 2 , ∞ ( Ω ) ) ∥ u ϵ ∥ H 3 ( Ω ) ≤ C 1 ( ϵ − 1 + ϵ − 3 ) h γ , (22) for some constant C 1 > 0 C_1>0 C 1 > 0
Proof. Let w h ϵ = u I ϵ − T ( u I ϵ ) w_h^\epsilon=u_I^\epsilon-T(u_I^\epsilon) w h ϵ = u I ϵ − T ( u I ϵ )
A h ϵ ( w h ϵ , w h ϵ ) = ϵ ( div ( G h u I ϵ ) , div ( G h w h ϵ ) ) L 2 ( Ω ) − ( det ( D G h u I ϵ ) , w h ϵ ) L 2 ( Ω ) + ϵ b h ( u I ϵ , w h ϵ ) + ( f , w h ϵ ) L 2 ( Ω ) − ( ϵ 2 , G h w h ϵ ⋅ n ) L 2 ( ∂ Ω ) . A_h^\epsilon(w_h^\epsilon,w_h^\epsilon)=\epsilon\left(\operatorname{div}(G_hu_I^\epsilon),\operatorname{div}(G_hw_h^\epsilon)\right)_{L^2(\Omega)}-\left(\det(DG_hu_I^\epsilon),w_h^\epsilon\right)_{L^2(\Omega)}
+\epsilon b_h(u_I^\epsilon,w_h^\epsilon)+(f,w_h^\epsilon)_{L^2(\Omega)}-(\epsilon^2,G_hw_h^\epsilon\cdot n)_{L^2(\partial\Omega)}. A h ϵ ( w h ϵ , w h ϵ ) = ϵ ( div ( G h u I ϵ ) , div ( G h w h ϵ ) ) L 2 ( Ω ) − ( det ( D G h u I ϵ ) , w h ϵ ) L 2 ( Ω ) + ϵ b h ( u I ϵ , w h ϵ ) + ( f , w h ϵ ) L 2 ( Ω ) − ( ϵ 2 , G h w h ϵ ⋅ n ) L 2 ( ∂ Ω ) . It follows from (1.2) that
( f , w h ϵ ) L 2 ( Ω ) = − ϵ ( Δ 2 u ϵ , w h ϵ ) L 2 ( Ω ) + ( det ( D 2 u ϵ ) , w h ϵ ) L 2 ( Ω ) = ϵ ( ∇ Δ u ϵ , G h w h ϵ ) L 2 ( Ω ) + ϵ ( ∇ Δ u ϵ , ∇ w h ϵ − G h w h ϵ ) L 2 ( Ω ) + ( det ( D 2 u ϵ ) , w h ϵ ) L 2 ( Ω ) = − ϵ ( Δ u ϵ , div ( G h w h ϵ ) ) L 2 ( Ω ) + ϵ ∑ e ∈ E h I ( Δ u ϵ , [ [ G h w h ϵ ⋅ n e ] ] ) L 2 ( e ) + ( ϵ 2 , G h w h ϵ ⋅ n ) L 2 ( ∂ Ω ) . + ϵ ( ∇ Δ u ϵ , ∇ w h ϵ − G h w h ϵ ) L 2 ( Ω ) + ( det ( D 2 u ϵ ) , w h ϵ ) L 2 ( Ω ) . \begin{aligned}
(f,w_h^\epsilon)_{L^2(\Omega)}
=-\epsilon(\Delta^2u^\epsilon,w_h^\epsilon)_{L^2(\Omega)}+(\det(D^2u^\epsilon),w_h^\epsilon)_{L^2(\Omega)} \\
&=\epsilon(\nabla\Delta u^\epsilon,G_hw_h^\epsilon)_{L^2(\Omega)}+\epsilon(\nabla\Delta u^\epsilon,\nabla w_h^\epsilon-G_hw_h^\epsilon)_{L^2(\Omega)}+(\det(D^2u^\epsilon),w_h^\epsilon)_{L^2(\Omega)} \\
&=-\epsilon(\Delta u^\epsilon,\operatorname{div}(G_hw_h^\epsilon))_{L^2(\Omega)}
+\epsilon\sum_{e\in\mathcal{E}_h^I}(\Delta u^\epsilon,[[G_hw_h^\epsilon\cdot n_e]])_{L^2(e)}
+(\epsilon^2,G_hw_h^\epsilon\cdot n)_{L^2(\partial\Omega)}. \\
&+\epsilon(\nabla\Delta u^\epsilon,\nabla w_h^\epsilon-G_h w_h^\epsilon)_{L^2(\Omega)}+(\det(D^2u^\epsilon),w_h^\epsilon)_{L^2(\Omega)}.
\end{aligned} ( f , w h ϵ ) L 2 ( Ω ) = − ϵ ( Δ 2 u ϵ , w h ϵ ) L 2 ( Ω ) + ( det ( D 2 u ϵ ) , w h ϵ ) L 2 ( Ω ) = ϵ ( ∇Δ u ϵ , G h w h ϵ ) L 2 ( Ω ) + ϵ ( ∇Δ u ϵ , ∇ w h ϵ − G h w h ϵ ) L 2 ( Ω ) + ( det ( D 2 u ϵ ) , w h ϵ ) L 2 ( Ω ) = − ϵ ( Δ u ϵ , div ( G h w h ϵ ) ) L 2 ( Ω ) + ϵ e ∈ E h I ∑ ( Δ u ϵ , [[ G h w h ϵ ⋅ n e ]] ) L 2 ( e ) + ( ϵ 2 , G h w h ϵ ⋅ n ) L 2 ( ∂ Ω ) . + ϵ ( ∇Δ u ϵ , ∇ w h ϵ − G h w h ϵ ) L 2 ( Ω ) + ( det ( D 2 u ϵ ) , w h ϵ ) L 2 ( Ω ) . Then we have
A h ϵ ( w h ϵ , w h ϵ ) = ϵ ( div ( G h u I ϵ ) − Δ u ϵ , div ( G h w h ϵ ) ) L 2 ( Ω ) + ϵ ∑ e ∈ E h I ( Δ u ϵ , [ [ G h w h ϵ ⋅ n e ] ] ) L 2 ( e ) + ϵ ( ∇ Δ u ϵ , ∇ w h ϵ − G h w h ϵ ) L 2 ( Ω ) + ϵ b h ( u I ϵ , w h ϵ ) + ( det ( D 2 u ϵ ) − det ( D G h u I ϵ ) , w h ϵ ) L 2 ( Ω ) ≤ C ϵ h γ ∥ u ϵ ∥ H 3 ( Ω ) ∥ D G h w h ϵ ∥ L 2 ( T h ) + ( det ( D 2 u ϵ ) − det ( D G h u I ϵ ) , w h ϵ ) L 2 ( Ω ) . \begin{aligned}
A_h^\epsilon(w_h^\epsilon,w_h^\epsilon)
={}&\epsilon\left(\operatorname{div}(G_hu_I^\epsilon)-\Delta u^\epsilon,\operatorname{div}(G_hw_h^\epsilon)\right)_{L^2(\Omega)}
+\epsilon\sum_{e\in\mathcal{E}_h^I}(\Delta u^\epsilon,[[G_hw_h^\epsilon\cdot n_e]])_{L^2(e)}\\
&+\epsilon(\nabla\Delta u^\epsilon,\nabla w_h^\epsilon-G_hw_h^\epsilon)_{L^2(\Omega)}
+\epsilon b_h(u_I^\epsilon,w_h^\epsilon)\\
&+\left(\det(D^2u^\epsilon)-\det(D_Gh u_I^\epsilon),w_h^\epsilon\right)_{L^2(\Omega)}\\
\le{}&C\epsilon h^\gamma\lVert u^\epsilon\rVert_{H^3(\Omega)}\lVert D_Gh w_h^\epsilon\rVert_{L^2(\mathcal{T}_h)}
+\left(\det(D^2u^\epsilon)-\det(D_Gh u_I^\epsilon),w_h^\epsilon\right)_{L^2(\Omega)}.
\end{aligned} A h ϵ ( w h ϵ , w h ϵ ) = ≤ ϵ ( div ( G h u I ϵ ) − Δ u ϵ , div ( G h w h ϵ ) ) L 2 ( Ω ) + ϵ e ∈ E h I ∑ ( Δ u ϵ , [[ G h w h ϵ ⋅ n e ]] ) L 2 ( e ) + ϵ ( ∇Δ u ϵ , ∇ w h ϵ − G h w h ϵ ) L 2 ( Ω ) + ϵ b h ( u I ϵ , w h ϵ ) + ( det ( D 2 u ϵ ) − det ( D G h u I ϵ ) , w h ϵ ) L 2 ( Ω ) C ϵ h γ ∥ u ϵ ∥ H 3 ( Ω ) ∥ D G h w h ϵ ∥ L 2 ( T h ) + ( det ( D 2 u ϵ ) − det ( D G h u I ϵ ) , w h ϵ ) L 2 ( Ω ) . By the mean value theorem, we have
( det ( D 2 u ϵ ) − det ( D G h u I ϵ ) , w h ϵ ) L 2 ( Ω ) = ( Ψ ϵ : ( D 2 u ϵ − D G h u I ϵ ) , w h ϵ ) L 2 ( Ω ) , \left(\det(D^2u^\epsilon)-\det(D_Gh u_I^\epsilon),w_h^\epsilon\right)_{L^2(\Omega)}
=\left(\Psi^\epsilon:(D^2u^\epsilon-D_Gh u_I^\epsilon),w_h^\epsilon\right)_{L^2(\Omega)}, ( det ( D 2 u ϵ ) − det ( D G h u I ϵ ) , w h ϵ ) L 2 ( Ω ) = ( Ψ ϵ : ( D 2 u ϵ − D G h u I ϵ ) , w h ϵ ) L 2 ( Ω ) , where Ψ ϵ = cof ( D 2 u ϵ − θ ( D 2 u ϵ − D G h u I ϵ ) ) \Psi^\epsilon=\operatorname{cof}(D^2u^\epsilon-\theta(D^2u^\epsilon-D_Gh u_I^\epsilon)) Ψ ϵ = cof ( D 2 u ϵ − θ ( D 2 u ϵ − D G h u I ϵ )) θ ∈ [ 0 , 1 ] \theta\in[0,1] θ ∈ [ 0 , 1 ]
∥ Ψ ϵ ∥ L ∞ ( Ω ) ≤ C ∥ D 2 u ϵ ∥ L ∞ ( Ω ) . \lVert\Psi^\epsilon\rVert_{L^\infty(\Omega)}\le C\lVert D^2u^\epsilon\rVert_{L^\infty(\Omega)}. ∥ Ψ ϵ ∥ L ∞ ( Ω ) ≤ C ∥ D 2 u ϵ ∥ L ∞ ( Ω ) . Using Lemma A.7, we obtain
A h ϵ ( w h ϵ , w h ϵ ) ≤ C ϵ h γ ∥ u ϵ ∥ H 3 ( Ω ) ∥ D G h w h ϵ ∥ L 2 ( T h ) + C h γ ∥ u ϵ ∥ W 2 , ∞ ( Ω ) ∥ u ϵ ∥ H 3 ( Ω ) ∥ D G h w h ϵ ∥ L 2 ( T h ) . A_h^\epsilon(w_h^\epsilon,w_h^\epsilon)\le C\epsilon h^\gamma\lVert u^\epsilon\rVert_{H^3(\Omega)}\lVert D_Gh w_h^\epsilon\rVert_{L^2(\mathcal{T}_h)}
+Ch^\gamma\lVert u^\epsilon\rVert_{W^{2,\infty}(\Omega)}\lVert u^\epsilon\rVert_{H^3(\Omega)}\lVert D_Gh w_h^\epsilon\rVert_{L^2(\mathcal{T}_h)}. A h ϵ ( w h ϵ , w h ϵ ) ≤ C ϵ h γ ∥ u ϵ ∥ H 3 ( Ω ) ∥ D G h w h ϵ ∥ L 2 ( T h ) + C h γ ∥ u ϵ ∥ W 2 , ∞ ( Ω ) ∥ u ϵ ∥ H 3 ( Ω ) ∥ D G h w h ϵ ∥ L 2 ( T h ) . Then we can complete the proof by the inequality A h ϵ ( w h ϵ , w h ϵ ) ≥ C ϵ ∥ D G h w h ϵ ∥ L 2 ( T h ) 2 . A_h^\epsilon(w_h^\epsilon,w_h^\epsilon)\ge C\epsilon\lVert D_Gh w_h^\epsilon\rVert^2_{L^2(\mathcal{T}_h)}. A h ϵ ( w h ϵ , w h ϵ ) ≥ C ϵ ∥ D G h w h ϵ ∥ L 2 ( T h ) 2 .
Lemma 4.2. For any v h , w h ∈ B h ( ρ ) v_h,w_h\in B_h(\rho) v h , w h ∈ B h ( ρ ) there holds
∥ D G h ( T ( v h ) − T ( w h ) ) ∥ L 2 ( T h ) ≤ C ( h , ρ , ϵ ) ∥ D G h ( v h − w h ) ∥ L 2 ( T h ) , (23) \lVert D_Gh(T(v_h)-T(w_h))\rVert_{L^2(\mathcal{T}_h)}
\le C(h,\rho,\epsilon)\lVert D_Gh(v_h-w_h)\rVert_{L^2(\mathcal{T}_h)}, \tag*{(23)} ∥ D G h ( T ( v h ) − T ( w h )) ∥ L 2 ( T h ) ≤ C ( h , ρ , ϵ ) ∥ D G h ( v h − w h ) ∥ L 2 ( T h ) , (23) where C ( h , ρ , ϵ ) = C 2 ϵ − 1 ( ϵ − 1 h γ + ρ ) C(h,\rho,\epsilon)=C_2\epsilon^{-1}(\epsilon^{-1}h^\gamma+\rho) C ( h , ρ , ϵ ) = C 2 ϵ − 1 ( ϵ − 1 h γ + ρ ) for some constant C 2 > 0 C_2>0 C 2 > 0
Proof. Let z h = v h − w h z_h=v_h-w_h z h = v h − w h T T T η h ∈ V h 0 \eta_h\in V_h^0 η h ∈ V h 0
A h ϵ ( T ( v h ) − T ( w h ) , η h ) = ( det ( D G h v h ) − det ( D G h w h ) , η h ) L 2 ( Ω ) + ( Φ ϵ ∇ z h , ∇ η h ) L 2 ( Ω ) = ( Ψ h : D G h z h , η h ) L 2 ( Ω ) + ( Φ ϵ ∇ z h , ∇ η h ) L 2 ( Ω ) , \begin{aligned}
A_h^\epsilon(T(v_h)-T(w_h),\eta_h)
={}&\left(\det(D_Gh v_h)-\det(D_Gh w_h),\eta_h\right)_{L^2(\Omega)}
+\left(\Phi^\epsilon\nabla z_h,\nabla\eta_h\right)_{L^2(\Omega)}\\
={}&\left(\Psi_h:D_Gh z_h,\eta_h\right)_{L^2(\Omega)}
+\left(\Phi^\epsilon\nabla z_h,\nabla\eta_h\right)_{L^2(\Omega)},
\end{aligned} A h ϵ ( T ( v h ) − T ( w h ) , η h ) = = ( det ( D G h v h ) − det ( D G h w h ) , η h ) L 2 ( Ω ) + ( Φ ϵ ∇ z h , ∇ η h ) L 2 ( Ω ) ( Ψ h : D G h z h , η h ) L 2 ( Ω ) + ( Φ ϵ ∇ z h , ∇ η h ) L 2 ( Ω ) , where Ψ h = cof ( D G h w h + θ ( D G h v h − D G h w h ) ) \Psi_h=\operatorname{cof}(D_Gh w_h+\theta(D_Gh v_h-D_Gh w_h)) Ψ h = cof ( D G h w h + θ ( D G h v h − D G h w h )) θ ∈ [ 0 , 1 ] \theta\in[0,1] θ ∈ [ 0 , 1 ]
A h ϵ ( T ( v h ) − T ( w h ) , η h ) = ( Ψ h : D G h z h , η h ) L 2 ( Ω ) + ( Φ ϵ G h z h , ∇ η h ) L 2 ( Ω ) + ( Φ ϵ ( ∇ z h − G h z h ) , ∇ η h ) L 2 ( Ω ) = ( ( Ψ h − Φ ϵ ) : D G h z h , η h ) L 2 ( Ω ) ⏟ = : R 1 + ∑ e ∈ E h I ( [ [ Φ ϵ G h z h ⋅ n e ] ] , η h ) L 2 ( e ) ⏟ = : R 2 . + ( Φ ε ( ∇ z h − D G h z h ) , ∇ η h ) L 2 ( Ω ) \begin{aligned}
A_h^\epsilon(T(v_h)-T(w_h),\eta_h)
={}&\left(\Psi_h:D_Gh z_h,\eta_h\right)_{L^2(\Omega)}
+\left(\Phi^\epsilon G_hz_h,\nabla\eta_h\right)_{L^2(\Omega)}
+\left(\Phi^\epsilon(\nabla z_h-G_hz_h),\nabla\eta_h\right)_{L^2(\Omega)}\\
={}&\underbrace{\left((\Psi_h-\Phi^\epsilon):D_Gh z_h,\eta_h\right)_{L^2(\Omega)}}_{=:R_1}
+\underbrace{\sum_{e\in\mathcal{E}_h^I}\left([[\,\Phi^\epsilon G_hz_h\cdot n_e\,]],\eta_h\right)_{L^2(e)}}_{=:R_2}. \\
&+ \left(\Phi^\varepsilon(\nabla z_h-DG_hz_h),\nabla\eta_h\right)_{L^2(\Omega)}
\end{aligned} A h ϵ ( T ( v h ) − T ( w h ) , η h ) = = ( Ψ h : D G h z h , η h ) L 2 ( Ω ) + ( Φ ϵ G h z h , ∇ η h ) L 2 ( Ω ) + ( Φ ϵ ( ∇ z h − G h z h ) , ∇ η h ) L 2 ( Ω ) =: R 1 ( ( Ψ h − Φ ϵ ) : D G h z h , η h ) L 2 ( Ω ) + =: R 2 e ∈ E h I ∑ ( [[ Φ ϵ G h z h ⋅ n e ]] , η h ) L 2 ( e ) . + ( Φ ε ( ∇ z h − D G h z h ) , ∇ η h ) L 2 ( Ω ) = : R 3 =:R_3 =: R 3
Then we have
∣ R 2 ∣ ≤ C h ∥ D G h z h ∥ L 2 ( T h ) ∥ Φ ε η h ∥ H 1 ( Ω ) ≤ C h ∥ D G h z h ∥ L 2 ( T h ) ( ∥ Φ ε ∥ H 1 ( Ω ) ∥ η h ∥ L ∞ ( Ω ) + ∥ η h ∥ H 1 ( Ω ) ∥ Φ ε ∥ L ∞ ( Ω ) ) ≤ C ( ∥ u ε ∥ W 2 , ∞ ( Ω ) + ∥ u ε ∥ H 3 ( Ω ) h ) ∥ D G h z h ∥ L 2 ( T h ) ∥ D G h η h ∥ L 2 ( T h ) , \begin{aligned}
|R_2|&\le Ch\|DG_hz_h\|_{L^2(\mathcal{T}_h)}\|\Phi^\varepsilon\eta_h\|_{H^1(\Omega)}\\
&\le Ch\|DG_hz_h\|_{L^2(\mathcal{T}_h)}
\left(\|\Phi^\varepsilon\|_{H^1(\Omega)}\|\eta_h\|_{L^\infty(\Omega)}
+\|\eta_h\|_{H^1(\Omega)}\|\Phi^\varepsilon\|_{L^\infty(\Omega)}\right)\\
&\le C\left(\|u^\varepsilon\|_{W^{2,\infty}(\Omega)}+\|u^\varepsilon\|_{H^3(\Omega)}h\right)
\|DG_hz_h\|_{L^2(\mathcal{T}_h)}\|DG_h\eta_h\|_{L^2(\mathcal{T}_h)},
\end{aligned} ∣ R 2 ∣ ≤ C h ∥ D G h z h ∥ L 2 ( T h ) ∥ Φ ε η h ∥ H 1 ( Ω ) ≤ C h ∥ D G h z h ∥ L 2 ( T h ) ( ∥ Φ ε ∥ H 1 ( Ω ) ∥ η h ∥ L ∞ ( Ω ) + ∥ η h ∥ H 1 ( Ω ) ∥ Φ ε ∥ L ∞ ( Ω ) ) ≤ C ( ∥ u ε ∥ W 2 , ∞ ( Ω ) + ∥ u ε ∥ H 3 ( Ω ) h ) ∥ D G h z h ∥ L 2 ( T h ) ∥ D G h η h ∥ L 2 ( T h ) , and
∣ R 3 ∣ ≤ C ∥ u ε ∥ W 2 , ∞ ( Ω ) h ∥ D G h z h ∥ L 2 ( T h ) ∥ D G h η h ∥ L 2 ( T h ) . |R_3|\le C\|u^\varepsilon\|_{W^{2,\infty}(\Omega)}h\|DG_hz_h\|_{L^2(\mathcal{T}_h)}\|DG_h\eta_h\|_{L^2(\mathcal{T}_h)}. ∣ R 3 ∣ ≤ C ∥ u ε ∥ W 2 , ∞ ( Ω ) h ∥ D G h z h ∥ L 2 ( T h ) ∥ D G h η h ∥ L 2 ( T h ) . Moreover, we have
∥ Ψ h − Φ ε ∥ L 2 ( Ω ) ≤ ∥ D 2 u ε − D G h u I ε ∥ L 2 ( Ω ) + ∥ D G h u I ε − D G h v h ∥ L 2 ( Ω ) + ∥ D G h z h ∥ L 2 ( Ω ) ≤ C ( h γ ∥ u ε ∥ H 3 ( Ω ) + ρ ) . \begin{aligned}
\|\Psi_h-\Phi^\varepsilon\|_{L^2(\Omega)}
&\le\|D^2u^\varepsilon-DG_hu_I^\varepsilon\|_{L^2(\Omega)}
+\|DG_hu_I^\varepsilon-DG_hv_h\|_{L^2(\Omega)}
+\|DG_hz_h\|_{L^2(\Omega)}\\
&\le C(h^\gamma\|u^\varepsilon\|_{H^3(\Omega)}+\rho).
\end{aligned} ∥ Ψ h − Φ ε ∥ L 2 ( Ω ) ≤ ∥ D 2 u ε − D G h u I ε ∥ L 2 ( Ω ) + ∥ D G h u I ε − D G h v h ∥ L 2 ( Ω ) + ∥ D G h z h ∥ L 2 ( Ω ) ≤ C ( h γ ∥ u ε ∥ H 3 ( Ω ) + ρ ) . Hence, we have
∣ R 1 ∣ ≤ C ( h γ ∥ u ε ∥ H 3 ( Ω ) + ρ ) ∥ D G h z h ∥ L 2 ( T h ) ∥ η h ∥ L ∞ ( Ω ) ≤ C ( h γ ∥ u ε ∥ H 3 ( Ω ) + ρ ) ∥ D G h z h ∥ L 2 ( T h ) ∥ D G h η h ∥ L 2 ( T h ) . \begin{aligned}
|R_1|&\le C(h^\gamma\|u^\varepsilon\|_{H^3(\Omega)}+\rho)
\|DG_hz_h\|_{L^2(\mathcal{T}_h)}\|\eta_h\|_{L^\infty(\Omega)}\\
&\le C(h^\gamma\|u^\varepsilon\|_{H^3(\Omega)}+\rho)
\|DG_hz_h\|_{L^2(\mathcal{T}_h)}\|DG_h\eta_h\|_{L^2(\mathcal{T}_h)}.
\end{aligned} ∣ R 1 ∣ ≤ C ( h γ ∥ u ε ∥ H 3 ( Ω ) + ρ ) ∥ D G h z h ∥ L 2 ( T h ) ∥ η h ∥ L ∞ ( Ω ) ≤ C ( h γ ∥ u ε ∥ H 3 ( Ω ) + ρ ) ∥ D G h z h ∥ L 2 ( T h ) ∥ D G h η h ∥ L 2 ( T h ) . Then we complete the proof by the above estimates.
Theorem 4.3. For h ≤ h 0 h\le h_0 h ≤ h 0 (21) has a unique solution u h ε ∈ B h ( ρ ) u_h^\varepsilon\in B_h(\rho) u h ε ∈ B h ( ρ ) H 2 H^2 H 2
∥ D 2 u ε − D G h u h ε ∥ L 2 ( T h ) ≤ C h γ ( 1 + ε − 1 ∥ u ε ∥ W 2 , ∞ ( Ω ) ) ∥ u ε ∥ H 3 ( Ω ) \|D^2u^\varepsilon-DG_hu_h^\varepsilon\|_{L^2(\mathcal{T}_h)}
\le Ch^\gamma\left(1+\varepsilon^{-1}\|u^\varepsilon\|_{W^{2,\infty}(\Omega)}\right)\|u^\varepsilon\|_{H^3(\Omega)} ∥ D 2 u ε − D G h u h ε ∥ L 2 ( T h ) ≤ C h γ ( 1 + ε − 1 ∥ u ε ∥ W 2 , ∞ ( Ω ) ) ∥ u ε ∥ H 3 ( Ω ) ≤ C ( ε − 1 + ε − 3 ) h γ . (24) \le C\left(\varepsilon^{-1}+\varepsilon^{-3}\right)h^\gamma. \tag*{(24)} ≤ C ( ε − 1 + ε − 3 ) h γ . (24) Proof. Set ρ 0 = 2 C h γ ( 1 + ε − 1 ∥ u ε ∥ W 2 , ∞ ( Ω ) ) ∥ u ε ∥ H 3 ( Ω ) \rho_0=2Ch^\gamma\left(1+\varepsilon^{-1}\|u^\varepsilon\|_{W^{2,\infty}(\Omega)}\right)\|u^\varepsilon\|_{H^3(\Omega)} ρ 0 = 2 C h γ ( 1 + ε − 1 ∥ u ε ∥ W 2 , ∞ ( Ω ) ) ∥ u ε ∥ H 3 ( Ω ) h 0 h_0 h 0 C ( h , ρ 0 , ε ) ≤ 1 / 2 C(h,\rho_0,\varepsilon)\le1/2 C ( h , ρ 0 , ε ) ≤ 1/2 h ≤ h 0 h\le h_0 h ≤ h 0
∥ D G h ( T ( v h ) − T ( w h ) ) ∥ L 2 ( T h ) ≤ 1 2 ∥ D G h ( v h − w h ) ∥ L 2 ( T h ) ∀ v h , w h ∈ B h ( ρ 0 ) , \|DG_h(T(v_h)-T(w_h))\|_{L^2(\mathcal{T}_h)}
\le\frac{1}{2}\|DG_h(v_h-w_h)\|_{L^2(\mathcal{T}_h)}
\quad\forall v_h,w_h\in B_h(\rho_0), ∥ D G h ( T ( v h ) − T ( w h )) ∥ L 2 ( T h ) ≤ 2 1 ∥ D G h ( v h − w h ) ∥ L 2 ( T h ) ∀ v h , w h ∈ B h ( ρ 0 ) , and for any z h ∈ B h ( ρ 0 ) z_h\in B_h(\rho_0) z h ∈ B h ( ρ 0 )
∥ D G h T ( z h ) − D G h u I ε ∥ L 2 ( T h ) ≤ ∥ D G h T ( z h ) − D G h T ( u I ε ) ∥ L 2 ( T h ) + ∥ D G h T ( u I ε ) − D G h u I ε ∥ L 2 ( T h ) ≤ 1 2 ρ 0 + 1 2 ρ 0 = ρ 0 . \begin{aligned}
\|DG_hT(z_h)-DG_hu_I^\varepsilon\|_{L^2(\mathcal{T}_h)}
&\le\|DG_hT(z_h)-DG_hT(u_I^\varepsilon)\|_{L^2(\mathcal{T}_h)}
+\|DG_hT(u_I^\varepsilon)-DG_hu_I^\varepsilon\|_{L^2(\mathcal{T}_h)}\\
&\le\frac{1}{2}\rho_0+\frac{1}{2}\rho_0=\rho_0.
\end{aligned} ∥ D G h T ( z h ) − D G h u I ε ∥ L 2 ( T h ) ≤ ∥ D G h T ( z h ) − D G h T ( u I ε ) ∥ L 2 ( T h ) + ∥ D G h T ( u I ε ) − D G h u I ε ∥ L 2 ( T h ) ≤ 2 1 ρ 0 + 2 1 ρ 0 = ρ 0 . Hence, by the Banach fixed-point theorem, there exists a unique u h ε ∈ B h ( ρ 0 ) u_h^\varepsilon\in B_h(\rho_0) u h ε ∈ B h ( ρ 0 ) T ( u h ε ) = u h ε T(u_h^\varepsilon)=u_h^\varepsilon T ( u h ε ) = u h ε u h ε u_h^\varepsilon u h ε
∥ D 2 u ε − D G h u h ε ∥ L 2 ( T h ) ≤ ∥ D 2 u ε − D G h u I ε ∥ L 2 ( T h ) + ∥ D G h u I ε − D G h u h ε ∥ L 2 ( T h ) ≤ C h γ ∥ u ε ∥ H 3 ( Ω ) + ρ 0 ≤ C h γ ( 1 + ε − 1 ∥ u ε ∥ W 2 , ∞ ( Ω ) ) ∥ u ε ∥ H 3 ( Ω ) , \begin{aligned}
\|D^2u^\varepsilon-DG_hu_h^\varepsilon\|_{L^2(\mathcal{T}_h)}
&\le\|D^2u^\varepsilon-DG_hu_I^\varepsilon\|_{L^2(\mathcal{T}_h)}
+\|DG_hu_I^\varepsilon-DG_hu_h^\varepsilon\|_{L^2(\mathcal{T}_h)}\\
&\le Ch^\gamma\|u^\varepsilon\|_{H^3(\Omega)}+\rho_0\\
&\le Ch^\gamma\left(1+\varepsilon^{-1}\|u^\varepsilon\|_{W^{2,\infty}(\Omega)}\right)\|u^\varepsilon\|_{H^3(\Omega)},
\end{aligned} ∥ D 2 u ε − D G h u h ε ∥ L 2 ( T h ) ≤ ∥ D 2 u ε − D G h u I ε ∥ L 2 ( T h ) + ∥ D G h u I ε − D G h u h ε ∥ L 2 ( T h ) ≤ C h γ ∥ u ε ∥ H 3 ( Ω ) + ρ 0 ≤ C h γ ( 1 + ε − 1 ∥ u ε ∥ W 2 , ∞ ( Ω ) ) ∥ u ε ∥ H 3 ( Ω ) , which completes the proof.
H1 error estimate
In this subsection, we give an H 1 H^1 H 1
L u ϵ ( U ) = φ in Ω , (25) L_{u^\epsilon}(\mathcal{U})=\varphi\quad\text{in }\Omega, \tag*{(25)} L u ϵ ( U ) = φ in Ω , (25) U = 0 on ∂ Ω , (26) \mathcal{U}=0 \quad\text{on }\partial\Omega, \tag*{(26)} U = 0 on ∂ Ω , (26) Δ U = 0 on ∂ Ω , (27) \Delta\mathcal{U}=0 \quad\text{on }\partial\Omega, \tag*{(27)} Δ U = 0 on ∂ Ω , (27) where φ ∈ H − 1 ( Ω ) ⊂ V 0 ∗ \varphi\in H^{-1}(\Omega)\subset V_0^* φ ∈ H − 1 ( Ω ) ⊂ V 0 ∗
⟨ φ , w ⟩ = ( ∇ ( u h ϵ − u I ϵ ) , ∇ w ) L 2 ( Ω ) ∀ w ∈ H 0 1 ( Ω ) . \langle\varphi,w\rangle=\left(\nabla(u_h^\epsilon-u_I^\epsilon),\nabla w\right)_{L^2(\Omega)}\quad\forall w\in H_0^1(\Omega). ⟨ φ , w ⟩ = ( ∇ ( u h ϵ − u I ϵ ) , ∇ w ) L 2 ( Ω ) ∀ w ∈ H 0 1 ( Ω ) . As in [32 ] , we can make the following assumption about the auxiliary problem.
Assumption 4.4. We assume that the solution of (4.6) satisfies the following regularity property:
∥ U ∥ H 3 ( Ω ) ≤ C s ( ϵ ) ∥ φ ∥ H − 1 ( Ω ) ≤ C s ( ϵ ) ∥ ∇ ( u h ϵ − u I ϵ ) ∥ L 2 ( Ω ) . \|\mathcal{U}\|_{H^3(\Omega)}\le C_s(\epsilon)\|\varphi\|_{H^{-1}(\Omega)}\le C_s(\epsilon)\|\nabla(u_h^\epsilon-u_I^\epsilon)\|_{L^2(\Omega)}. ∥ U ∥ H 3 ( Ω ) ≤ C s ( ϵ ) ∥ φ ∥ H − 1 ( Ω ) ≤ C s ( ϵ ) ∥∇ ( u h ϵ − u I ϵ ) ∥ L 2 ( Ω ) . The weak form of (4.6) reads as follows: find U ∈ V 0 \mathcal{U}\in V_0 U ∈ V 0
a ( w , U ) = ( ∇ ( u h ϵ − u I ϵ ) , ∇ w ) L 2 ( Ω ) ∀ w ∈ V 0 . a(w,\mathcal{U})=\left(\nabla(u_h^\epsilon-u_I^\epsilon),\nabla w\right)_{L^2(\Omega)}\quad\forall w\in V_0. a ( w , U ) = ( ∇ ( u h ϵ − u I ϵ ) , ∇ w ) L 2 ( Ω ) ∀ w ∈ V 0 . The discrete auxiliary problem is to find U h ∈ V h 0 \mathcal{U}_h\in V_h^0 U h ∈ V h 0
A h ϵ ( w h , U h ) = ( ∇ ( u h ϵ − u I ϵ ) , ∇ w h ) L 2 ( Ω ) ∀ w h ∈ V h 0 . A_h^\epsilon(w_h,\mathcal{U}_h)=\left(\nabla(u_h^\epsilon-u_I^\epsilon),\nabla w_h\right)_{L^2(\Omega)}\quad\forall w_h\in V_h^0. A h ϵ ( w h , U h ) = ( ∇ ( u h ϵ − u I ϵ ) , ∇ w h ) L 2 ( Ω ) ∀ w h ∈ V h 0 . Thus, we have
∥ U ∥ H 2 ( Ω ) ≤ C ϵ − 1 ∥ ∇ ( u h ϵ − u I ϵ ) ∥ L 2 ( Ω ) , \|\mathcal{U}\|_{H^2(\Omega)}\le C\epsilon^{-1}\|\nabla(u_h^\epsilon-u_I^\epsilon)\|_{L^2(\Omega)}, ∥ U ∥ H 2 ( Ω ) ≤ C ϵ − 1 ∥∇ ( u h ϵ − u I ϵ ) ∥ L 2 ( Ω ) , ∥ D G h U h ∥ L 2 ( T h ) ≤ C ϵ − 1 ∥ ∇ ( u h ϵ − u I ϵ ) ∥ L 2 ( Ω ) , \|DG_h\mathcal{U}_h\|_{L^2(\mathcal{T}_h)}\le C\epsilon^{-1}\|\nabla(u_h^\epsilon-u_I^\epsilon)\|_{L^2(\Omega)}, ∥ D G h U h ∥ L 2 ( T h ) ≤ C ϵ − 1 ∥∇ ( u h ϵ − u I ϵ ) ∥ L 2 ( Ω ) , and by Theorem 3.3
∥ D 2 U − D G h U h ∥ L 2 ( T h ) ≤ C ( C s ( ϵ ) h γ + ϵ − 3 h ) ∥ ∇ ( u h ϵ − u I ϵ ) ∥ L 2 ( Ω ) . \|D^2\mathcal{U}-DG_h\mathcal{U}_h\|_{L^2(\mathcal{T}_h)}\le C\left(C_s(\epsilon)h^\gamma+\epsilon^{-3}h\right)\|\nabla(u_h^\epsilon-u_I^\epsilon)\|_{L^2(\Omega)}. ∥ D 2 U − D G h U h ∥ L 2 ( T h ) ≤ C ( C s ( ϵ ) h γ + ϵ − 3 h ) ∥∇ ( u h ϵ − u I ϵ ) ∥ L 2 ( Ω ) . Theorem 4.5. The solution u h ϵ u_h^\epsilon u h ϵ
∥ ∇ ( u h ϵ − u I ϵ ) ∥ L 2 ( Ω ) ≤ C ( ϵ ) ( h 2 γ + h 1 + γ + h 2 ) + C ϵ − 1 h , (28) \|\nabla(u_h^\epsilon-u_I^\epsilon)\|_{L^2(\Omega)}\le C(\epsilon)(h^{2\gamma}+h^{1+\gamma}+h^2)+C\epsilon^{-1}h, \tag*{(28)} ∥∇ ( u h ϵ − u I ϵ ) ∥ L 2 ( Ω ) ≤ C ( ϵ ) ( h 2 γ + h 1 + γ + h 2 ) + C ϵ − 1 h , (28) and this implies the following H 1 H^1 H 1
∥ ∇ ( u ϵ − u h ϵ ) ∥ L 2 ( Ω ) ≤ C ( ϵ ) ( h 2 γ + h 1 + γ + h 2 ) + C ( ϵ − 1 2 + ϵ − 1 ) h . (29) \|\nabla(u^\epsilon-u_h^\epsilon)\|_{L^2(\Omega)}\le C(\epsilon)(h^{2\gamma}+h^{1+\gamma}+h^2)+C\left(\epsilon^{-\frac12}+\epsilon^{-1}\right)h. \tag*{(29)} ∥∇ ( u ϵ − u h ϵ ) ∥ L 2 ( Ω ) ≤ C ( ϵ ) ( h 2 γ + h 1 + γ + h 2 ) + C ( ϵ − 2 1 + ϵ − 1 ) h . (29) Proof. We have the decomposition
∥ ∇ ( u h ϵ − u I ϵ ) ∥ L 2 ( Ω ) 2 = A h ϵ ( u h ϵ − u I ϵ , U h ) = A h ϵ ( u h ϵ , U h ) − A h ϵ ( u I ϵ , U h ) = : ∑ i = 1 6 G i , \|\nabla(u_h^\epsilon-u_I^\epsilon)\|_{L^2(\Omega)}^2=A_h^\epsilon(u_h^\epsilon-u_I^\epsilon,\mathcal{U}_h)=A_h^\epsilon(u_h^\epsilon,\mathcal{U}_h)-A_h^\epsilon(u_I^\epsilon,\mathcal{U}_h)=:\sum_{i=1}^{6}\mathcal{G}_i, ∥∇ ( u h ϵ − u I ϵ ) ∥ L 2 ( Ω ) 2 = A h ϵ ( u h ϵ − u I ϵ , U h ) = A h ϵ ( u h ϵ , U h ) − A h ϵ ( u I ϵ , U h ) =: i = 1 ∑ 6 G i , where
G 1 = ϵ ( Δ u ϵ − div ( G h u I ϵ ) , div ( G h U h ) ) L 2 ( Ω ) \mathcal{G}_1=\epsilon\left(\Delta u^\epsilon-\operatorname{div}(G_hu_I^\epsilon),\operatorname{div}(G_h\mathcal{U}_h)\right)_{L^2(\Omega)} G 1 = ϵ ( Δ u ϵ − div ( G h u I ϵ ) , div ( G h U h ) ) L 2 ( Ω ) G 2 = − ϵ ( ∇ Δ u ϵ , ∇ U h − G h U h ) L 2 ( Ω ) , G 3 = − ϵ ∑ e ∈ E h I ( Δ u ϵ , [ [ G h U h ⋅ n e ] ] ) L 2 ( e ) , G 4 = ( Φ ϵ ∇ ( u ϵ − u I ϵ ) , ∇ U h ) L 2 ( Ω ) , G 5 = − ( det ( D 2 u ϵ ) − det ( D G h u h ϵ ) , U h ) L 2 ( Ω ) + ( Φ ϵ ∇ ( u h ϵ − u ϵ ) , ∇ U h ) L 2 ( Ω ) , G 6 = − ϵ b h ( u I ϵ , U h ) . \begin{aligned}
G_2&=-\epsilon(\nabla\Delta u^\epsilon,\nabla U_h-G_hU_h)_{L^2(\Omega)},\\
G_3&=-\epsilon\sum_{e\in\mathcal{E}_h^I}(\Delta u^\epsilon,[[G_hU_h\cdot n_e]])_{L^2(e)},\\
G_4&=(\Phi^\epsilon\nabla(u^\epsilon-u_I^\epsilon),\nabla U_h)_{L^2(\Omega)},\\
G_5&=-\left(\det(D^2u^\epsilon)-\det(DG_hu_h^\epsilon),U_h\right)_{L^2(\Omega)}
+(\Phi^\epsilon\nabla(u_h^\epsilon-u^\epsilon),\nabla U_h)_{L^2(\Omega)},\\
G_6&=-\epsilon b_h(u_I^\epsilon,U_h).
\end{aligned} G 2 G 3 G 4 G 5 G 6 = − ϵ ( ∇Δ u ϵ , ∇ U h − G h U h ) L 2 ( Ω ) , = − ϵ e ∈ E h I ∑ ( Δ u ϵ , [[ G h U h ⋅ n e ]] ) L 2 ( e ) , = ( Φ ϵ ∇ ( u ϵ − u I ϵ ) , ∇ U h ) L 2 ( Ω ) , = − ( det ( D 2 u ϵ ) − det ( D G h u h ϵ ) , U h ) L 2 ( Ω ) + ( Φ ϵ ∇ ( u h ϵ − u ϵ ) , ∇ U h ) L 2 ( Ω ) , = − ϵ b h ( u I ϵ , U h ) . Then we can obtain the following error bounds
∣ G 2 ∣ ≤ C h ∥ u ϵ ∥ H 3 ( Ω ) ∥ ∇ ( u h ϵ − u I ϵ ) ∥ L 2 ( Ω ) ≤ C ϵ − 1 h ∥ ∇ ( u h ϵ − u I ϵ ) ∥ L 2 ( Ω ) , |G_2|\leq Ch\|u^\epsilon\|_{H^3(\Omega)}\|\nabla(u_h^\epsilon-u_I^\epsilon)\|_{L^2(\Omega)}
\leq C\epsilon^{-1}h\|\nabla(u_h^\epsilon-u_I^\epsilon)\|_{L^2(\Omega)}, ∣ G 2 ∣ ≤ C h ∥ u ϵ ∥ H 3 ( Ω ) ∥∇ ( u h ϵ − u I ϵ ) ∥ L 2 ( Ω ) ≤ C ϵ − 1 h ∥∇ ( u h ϵ − u I ϵ ) ∥ L 2 ( Ω ) , and
∣ G 3 ∣ ≤ C ϵ h 1 + γ ∥ u ϵ ∥ H 3 ( Ω ) ∥ U ∥ H 3 ( Ω ) ≤ C C s ( ϵ ) h 1 + γ ∥ ∇ ( u h ϵ − u I ϵ ) ∥ L 2 ( Ω ) , |G_3|\leq C\epsilon h^{1+\gamma}\|u^\epsilon\|_{H^3(\Omega)}\|U\|_{H^3(\Omega)}
\leq CC_s(\epsilon)h^{1+\gamma}\|\nabla(u_h^\epsilon-u_I^\epsilon)\|_{L^2(\Omega)}, ∣ G 3 ∣ ≤ C ϵ h 1 + γ ∥ u ϵ ∥ H 3 ( Ω ) ∥ U ∥ H 3 ( Ω ) ≤ C C s ( ϵ ) h 1 + γ ∥∇ ( u h ϵ − u I ϵ ) ∥ L 2 ( Ω ) , and
∣ G 6 ∣ ≤ ϵ ∣ b h ( u I ϵ , U I ) ∣ + ϵ ∣ b h ( u I ϵ , U h − U I ) ∣ ≤ C ϵ h 2 γ ∥ u ϵ ∥ H 3 ( Ω ) ∥ U ∥ H 3 ( Ω ) + C ϵ h γ ∥ u ϵ ∥ H 3 ( Ω ) ∥ D G h ( U h − U I ) ∥ L 2 ( T h ) ≤ C C s ( ϵ ) h 2 γ ∥ ∇ ( u h ϵ − u I ϵ ) ∥ L 2 ( Ω ) + C h γ ( ∥ D 2 U − D G h U h ∥ L 2 ( T h ) + ∥ D 2 U − D G h U I ∥ L 2 ( T h ) ) ≤ C ( C s ( ϵ ) h 2 γ + ϵ − 3 h 1 + γ ) ∥ ∇ ( u h ϵ − u I ϵ ) ∥ L 2 ( Ω ) . \begin{aligned}
|G_6|&\leq\epsilon|b_h(u_I^\epsilon,U_I)|+\epsilon|b_h(u_I^\epsilon,U_h-U_I)|\\
&\leq C\epsilon h^{2\gamma}\|u^\epsilon\|_{H^3(\Omega)}\|U\|_{H^3(\Omega)}
+C\epsilon h^\gamma\|u^\epsilon\|_{H^3(\Omega)}
\|DG_h(U_h-U_I)\|_{L^2(\mathcal{T}_h)}\\
&\leq CC_s(\epsilon)h^{2\gamma}\|\nabla(u_h^\epsilon-u_I^\epsilon)\|_{L^2(\Omega)}\\
&\quad+Ch^\gamma\left(\|D^2U-DG_hU_h\|_{L^2(\mathcal{T}_h)}
+\|D^2U-DG_hU_I\|_{L^2(\mathcal{T}_h)}\right)\\
&\leq C\left(C_s(\epsilon)h^{2\gamma}+\epsilon^{-3}h^{1+\gamma}\right)
\|\nabla(u_h^\epsilon-u_I^\epsilon)\|_{L^2(\Omega)}.
\end{aligned} ∣ G 6 ∣ ≤ ϵ ∣ b h ( u I ϵ , U I ) ∣ + ϵ ∣ b h ( u I ϵ , U h − U I ) ∣ ≤ C ϵ h 2 γ ∥ u ϵ ∥ H 3 ( Ω ) ∥ U ∥ H 3 ( Ω ) + C ϵ h γ ∥ u ϵ ∥ H 3 ( Ω ) ∥ D G h ( U h − U I ) ∥ L 2 ( T h ) ≤ C C s ( ϵ ) h 2 γ ∥∇ ( u h ϵ − u I ϵ ) ∥ L 2 ( Ω ) + C h γ ( ∥ D 2 U − D G h U h ∥ L 2 ( T h ) + ∥ D 2 U − D G h U I ∥ L 2 ( T h ) ) ≤ C ( C s ( ϵ ) h 2 γ + ϵ − 3 h 1 + γ ) ∥∇ ( u h ϵ − u I ϵ ) ∥ L 2 ( Ω ) . For G 1 G_1 G 1
G 1 = ϵ ( Δ u ϵ − div ( G h u I ϵ ) , div ( G h U h ) − Δ U ) L 2 ( Ω ) + ϵ ( Δ u ϵ − div ( G h u I ϵ ) , Δ U ) L 2 ( Ω ) = ϵ ( Δ u ϵ − div ( G h u I ϵ ) , div ( G h U h ) − Δ U ) L 2 ( Ω ) − ϵ ( ∇ u I ϵ − G h u I ϵ , ∇ Δ U ) L 2 ( Ω ) + ϵ ∑ e ∈ E h I ( [ [ ( ∇ u ϵ − G h u I ϵ ) ⋅ n e ] ] , Δ U ) L 2 ( e ) . \begin{aligned}
G_1={}&\epsilon(\Delta u^\epsilon-\operatorname{div}(G_hu_I^\epsilon),
\operatorname{div}(G_hU_h)-\Delta U)_{L^2(\Omega)}
+\epsilon(\Delta u^\epsilon-\operatorname{div}(G_hu_I^\epsilon),\Delta U)_{L^2(\Omega)}\\
={}&\epsilon(\Delta u^\epsilon-\operatorname{div}(G_hu_I^\epsilon),
\operatorname{div}(G_hU_h)-\Delta U)_{L^2(\Omega)}
-\epsilon(\nabla u_I^\epsilon-G_hu_I^\epsilon,\nabla\Delta U)_{L^2(\Omega)}\\
&+\epsilon\sum_{e\in\mathcal{E}_h^I}
([[(\nabla u^\epsilon-G_hu_I^\epsilon)\cdot n_e]],\Delta U)_{L^2(e)}.
\end{aligned} G 1 = = ϵ ( Δ u ϵ − div ( G h u I ϵ ) , div ( G h U h ) − Δ U ) L 2 ( Ω ) + ϵ ( Δ u ϵ − div ( G h u I ϵ ) , Δ U ) L 2 ( Ω ) ϵ ( Δ u ϵ − div ( G h u I ϵ ) , div ( G h U h ) − Δ U ) L 2 ( Ω ) − ϵ ( ∇ u I ϵ − G h u I ϵ , ∇Δ U ) L 2 ( Ω ) + ϵ e ∈ E h I ∑ ([[( ∇ u ϵ − G h u I ϵ ) ⋅ n e ]] , Δ U ) L 2 ( e ) . Hence, we have
∣ G 1 ∣ ≤ C ϵ h 2 γ ∥ u ϵ ∥ H 3 ( Ω ) ∥ U ∥ H 3 ( Ω ) + C ϵ h 1 + γ ∥ u ϵ ∥ H 3 ( Ω ) ∥ U ∥ H 3 ( Ω ) ≤ C C s ( ϵ ) h 2 γ ∥ ∇ ( u h ϵ − u I ϵ ) ∥ L 2 ( Ω ) . \begin{aligned}
|G_1|&\leq C\epsilon h^{2\gamma}\|u^\epsilon\|_{H^3(\Omega)}\|U\|_{H^3(\Omega)}
+C\epsilon h^{1+\gamma}\|u^\epsilon\|_{H^3(\Omega)}\|U\|_{H^3(\Omega)}\\
&\leq CC_s(\epsilon)h^{2\gamma}\|\nabla(u_h^\epsilon-u_I^\epsilon)\|_{L^2(\Omega)}.
\end{aligned} ∣ G 1 ∣ ≤ C ϵ h 2 γ ∥ u ϵ ∥ H 3 ( Ω ) ∥ U ∥ H 3 ( Ω ) + C ϵ h 1 + γ ∥ u ϵ ∥ H 3 ( Ω ) ∥ U ∥ H 3 ( Ω ) ≤ C C s ( ϵ ) h 2 γ ∥∇ ( u h ϵ − u I ϵ ) ∥ L 2 ( Ω ) . For G 4 G_4 G 4
G 4 = ( Φ ϵ ∇ ( u ϵ − u I ϵ ) , ∇ U h ) L 2 ( Ω ) = ( ∇ ( u ϵ − u I ϵ ) , Φ ϵ ∇ U h ) L 2 ( Ω ) = ( ∇ ( u ϵ − u I ϵ ) , Φ ϵ G h U h ) L 2 ( Ω ) + ( ∇ ( u ϵ − u I ϵ ) , Φ ϵ ∇ U h − Φ ϵ G h U h ) L 2 ( Ω ) = ( ∇ ( u ϵ − u I ϵ ) , Φ ϵ ∇ U h − Φ ϵ G h U h ) L 2 ( Ω ) − ( u ϵ − u I ϵ , Φ ϵ : D G h U h ) L 2 ( Ω ) + ∑ e ∈ E h I ( [ [ Φ ϵ G h U h ⋅ n e ] ] , u ϵ − u I ϵ ) L 2 ( e ) + ∑ e ∈ E h B ( g − g I , Φ ϵ G h U h ⋅ n e ) L 2 ( e ) . \begin{aligned}
G_4={}&(\Phi^\epsilon\nabla(u^\epsilon-u_I^\epsilon),\nabla U_h)_{L^2(\Omega)}
=(\nabla(u^\epsilon-u_I^\epsilon),\Phi^\epsilon\nabla U_h)_{L^2(\Omega)}\\
={}&(\nabla(u^\epsilon-u_I^\epsilon),\Phi^\epsilon G_hU_h)_{L^2(\Omega)}
+(\nabla(u^\epsilon-u_I^\epsilon),\Phi^\epsilon\nabla U_h-\Phi^\epsilon G_hU_h)_{L^2(\Omega)}\\
={}&(\nabla(u^\epsilon-u_I^\epsilon),\Phi^\epsilon\nabla U_h-\Phi^\epsilon G_hU_h)_{L^2(\Omega)}
-(u^\epsilon-u_I^\epsilon,\Phi^\epsilon:D G_hU_h)_{L^2(\Omega)}\\
&\quad+\sum_{e\in\mathcal{E}_h^I}([[\Phi^\epsilon G_hU_h\cdot n_e]],u^\epsilon-u_I^\epsilon)_{L^2(e)}
+\sum_{e\in\mathcal{E}_h^B}(g-g_I,\Phi^\epsilon G_hU_h\cdot n_e)_{L^2(e)}.
\end{aligned} G 4 = = = ( Φ ϵ ∇ ( u ϵ − u I ϵ ) , ∇ U h ) L 2 ( Ω ) = ( ∇ ( u ϵ − u I ϵ ) , Φ ϵ ∇ U h ) L 2 ( Ω ) ( ∇ ( u ϵ − u I ϵ ) , Φ ϵ G h U h ) L 2 ( Ω ) + ( ∇ ( u ϵ − u I ϵ ) , Φ ϵ ∇ U h − Φ ϵ G h U h ) L 2 ( Ω ) ( ∇ ( u ϵ − u I ϵ ) , Φ ϵ ∇ U h − Φ ϵ G h U h ) L 2 ( Ω ) − ( u ϵ − u I ϵ , Φ ϵ : D G h U h ) L 2 ( Ω ) + e ∈ E h I ∑ ([[ Φ ϵ G h U h ⋅ n e ]] , u ϵ − u I ϵ ) L 2 ( e ) + e ∈ E h B ∑ ( g − g I , Φ ϵ G h U h ⋅ n e ) L 2 ( e ) . We can estimate G 41 \mathcal{G}_{41} G 41
∣ G 41 ∣ ≤ C ∥ u ε ∥ W 2 , ∞ ( Ω ) h 2 ∥ u ε ∥ H 2 ( Ω ) ∥ D G h U h ∥ L 2 ( T h ) ≤ C ε − 5 2 h 2 ∥ ∇ ( u h ε − u I ε ) ∥ L 2 ( Ω ) . \left|\mathcal{G}_{41}\right| \le C\left\|u^{\varepsilon}\right\|_{W^{2,\infty}(\Omega)}h^{2}\left\|u^{\varepsilon}\right\|_{H^{2}(\Omega)}\left\|DG_{h}U_{h}\right\|_{L^{2}(\mathcal{T}_{h})}
\le C\varepsilon^{-\frac{5}{2}}h^{2}\left\|\nabla\left(u_{h}^{\varepsilon}-u_{I}^{\varepsilon}\right)\right\|_{L^{2}(\Omega)}. ∣ G 41 ∣ ≤ C ∥ u ε ∥ W 2 , ∞ ( Ω ) h 2 ∥ u ε ∥ H 2 ( Ω ) ∥ D G h U h ∥ L 2 ( T h ) ≤ C ε − 2 5 h 2 ∥ ∇ ( u h ε − u I ε ) ∥ L 2 ( Ω ) . In addition, we have
∣ G 42 ∣ ≤ C h ∥ D G h U h ∥ L 2 ( T h ) ∣ ( u ε − u I ε ) Φ ε ∣ H 1 ( Ω ) ≤ C h ∥ D G h U h ∥ L 2 ( T h ) ( ∥ u ε − u I ε ∥ L ∞ ( Ω ) ∣ Φ ε ∣ H 1 ( Ω ) + ∥ Φ ε ∥ L ∞ ( Ω ) ∣ u ε − u I ε ∣ H 1 ( Ω ) ) ≤ C ( ε − 2 + ε − 5 2 ) h 2 ∥ ∇ ( u h ε − u I ε ) ∥ L 2 ( Ω ) , \begin{aligned}
\left|\mathcal{G}_{42}\right| \le Ch\left\|DG_{h}U_{h}\right\|_{L^{2}(\mathcal{T}_{h})}\left|\left(u^{\varepsilon}-u_{I}^{\varepsilon}\right)\Phi^{\varepsilon}\right|_{H^{1}(\Omega)} \\
&\le Ch\left\|DG_{h}U_{h}\right\|_{L^{2}(\mathcal{T}_{h})}\left(\left\|u^{\varepsilon}-u_{I}^{\varepsilon}\right\|_{L^{\infty}(\Omega)}\left|\Phi^{\varepsilon}\right|_{H^{1}(\Omega)}+\left\|\Phi^{\varepsilon}\right\|_{L^{\infty}(\Omega)}\left|u^{\varepsilon}-u_{I}^{\varepsilon}\right|_{H^{1}(\Omega)}\right) \\
&\le C\left(\varepsilon^{-2}+\varepsilon^{-\frac{5}{2}}\right)h^{2}\left\|\nabla\left(u_{h}^{\varepsilon}-u_{I}^{\varepsilon}\right)\right\|_{L^{2}(\Omega)},
\end{aligned} ∣ G 42 ∣ ≤ C h ∥ D G h U h ∥ L 2 ( T h ) ∣ ( u ε − u I ε ) Φ ε ∣ H 1 ( Ω ) ≤ C h ∥ D G h U h ∥ L 2 ( T h ) ( ∥ u ε − u I ε ∥ L ∞ ( Ω ) ∣ Φ ε ∣ H 1 ( Ω ) + ∥ Φ ε ∥ L ∞ ( Ω ) ∣ u ε − u I ε ∣ H 1 ( Ω ) ) ≤ C ( ε − 2 + ε − 2 5 ) h 2 ∥ ∇ ( u h ε − u I ε ) ∥ L 2 ( Ω ) , and
∣ G 43 ∣ ≤ C ∥ Φ ε ∥ L ∞ ( Ω ) ∥ g − g I ∥ L 2 ( ∂ Ω ) ∥ D G h U h ∥ L 2 ( T h ) ≤ C ε − 3 h 2 ∥ ∇ ( u h ε − u I ε ) ∥ L 2 ( Ω ) . \left|\mathcal{G}_{43}\right| \le C\left\|\Phi^{\varepsilon}\right\|_{L^{\infty}(\Omega)}\left\|g-g_{I}\right\|_{L^{2}(\partial\Omega)}\left\|DG_{h}U_{h}\right\|_{L^{2}(\mathcal{T}_{h})}
\le C\varepsilon^{-3}h^{2}\left\|\nabla\left(u_{h}^{\varepsilon}-u_{I}^{\varepsilon}\right)\right\|_{L^{2}(\Omega)}. ∣ G 43 ∣ ≤ C ∥ Φ ε ∥ L ∞ ( Ω ) ∥ g − g I ∥ L 2 ( ∂ Ω ) ∥ D G h U h ∥ L 2 ( T h ) ≤ C ε − 3 h 2 ∥ ∇ ( u h ε − u I ε ) ∥ L 2 ( Ω ) . These estimates imply
∣ G 4 ∣ ≤ C ( ε − 2 + ε − 5 2 + ε − 3 ) h 2 ∥ ∇ ( u h ε − u I ε ) ∥ L 2 ( Ω ) . \left|\mathcal{G}_{4}\right| \le C\left(\varepsilon^{-2}+\varepsilon^{-\frac{5}{2}}+\varepsilon^{-3}\right)h^{2}\left\|\nabla\left(u_{h}^{\varepsilon}-u_{I}^{\varepsilon}\right)\right\|_{L^{2}(\Omega)}. ∣ G 4 ∣ ≤ C ( ε − 2 + ε − 2 5 + ε − 3 ) h 2 ∥ ∇ ( u h ε − u I ε ) ∥ L 2 ( Ω ) . For G 5 \mathcal{G}_{5} G 5
G 5 = − ( ( det ( D 2 u ε ) − det ( D G h u h ε ) , U h ) L 2 ( Ω ) ) + ( Φ ε ( ∇ u h ε − G h u h ε ) , ∇ U h ) L 2 ( Ω ) + ( Φ ε ( G h u h ε − ∇ u ε ) , ∇ U h ) L 2 ( Ω ) = − ( ( det ( D 2 u ε ) − det ( D G h u h ε ) , U h ) L 2 ( Ω ) ) + ( Φ ε ( ∇ u h ε − G h u h ε ) , ∇ U h ) L 2 ( Ω ) − ( Φ ε : ( D G h u h ε − D 2 u ε ) , U h ) L 2 ( Ω ) − ∑ e ∈ E h I ( [ [ Φ ε ( ∇ u h ε − G h u h ε ) ⋅ n e ] ] , U h ) L 2 ( e ) = ( ( Ψ h − Φ ε ) : ( D G h u h ε − D 2 u ε ) , U h ) L 2 ( Ω ) + ( Φ ε ( ∇ u h ε − G h u h ε ) , ∇ U h ) L 2 ( Ω ) − ∑ e ∈ E h I ( [ [ Φ ε ( ∇ u h ε − G h u h ε ) ⋅ n e ] ] , U h ) L 2 ( e ) = ( ( Ψ h − Φ ε ) : ( D G h u h ε − D 2 u ε ) , U h ) L 2 ( Ω ) ⏟ = : G 51 − ∑ e ∈ E h I ( [ [ Φ ε ( ∇ u h ε − G h u h ε ) ⋅ n e ] ] , U h ) L 2 ( e ) ⏟ = : G 52 + ( Φ ε ( ∇ u h ε − G h u h ε ) , ∇ U h − ∇ U ) L 2 ( Ω ) ⏟ = : G 53 + ( Φ ε ( ∇ u h ε − G h u h ε ) , ∇ U ) L 2 ( Ω ) ⏟ = : G 54 . \begin{aligned}
\mathcal{G}_{5}
={}&-\left(\left(\det(D^{2}u^{\varepsilon})-\det(DG_{h}u_{h}^{\varepsilon}),U_{h}\right)_{L^{2}(\Omega)}\right)
+\left(\Phi^{\varepsilon}\left(\nabla u_{h}^{\varepsilon}-G_{h}u_{h}^{\varepsilon}\right),\nabla U_{h}\right)_{L^{2}(\Omega)}\\
&+\left(\Phi^{\varepsilon}\left(G_{h}u_{h}^{\varepsilon}-\nabla u^{\varepsilon}\right),\nabla U_{h}\right)_{L^{2}(\Omega)}\\
={}&-\left(\left(\det(D^{2}u^{\varepsilon})-\det(DG_{h}u_{h}^{\varepsilon}),U_{h}\right)_{L^{2}(\Omega)}\right)
+\left(\Phi^{\varepsilon}\left(\nabla u_{h}^{\varepsilon}-G_{h}u_{h}^{\varepsilon}\right),\nabla U_{h}\right)_{L^{2}(\Omega)}\\
&-\left(\Phi^{\varepsilon}:\left(DG_{h}u_{h}^{\varepsilon}-D^{2}u^{\varepsilon}\right),U_{h}\right)_{L^{2}(\Omega)}
-\sum_{e\in\mathcal{E}_{h}^{I}}\left(\left[\left[\Phi^{\varepsilon}\left(\nabla u_{h}^{\varepsilon}-G_{h}u_{h}^{\varepsilon}\right)\cdot n_{e}\right]\right],U_{h}\right)_{L^{2}(e)}\\
={}&\left((\Psi_{h}-\Phi^{\varepsilon}):\left(DG_{h}u_{h}^{\varepsilon}-D^{2}u^{\varepsilon}\right),U_{h}\right)_{L^{2}(\Omega)}
+\left(\Phi^{\varepsilon}\left(\nabla u_{h}^{\varepsilon}-G_{h}u_{h}^{\varepsilon}\right),\nabla U_{h}\right)_{L^{2}(\Omega)}\\
&-\sum_{e\in\mathcal{E}_{h}^{I}}\left(\left[\left[\Phi^{\varepsilon}\left(\nabla u_{h}^{\varepsilon}-G_{h}u_{h}^{\varepsilon}\right)\cdot n_{e}\right]\right],U_{h}\right)_{L^{2}(e)}\\
={}&\underbrace{\left((\Psi_{h}-\Phi^{\varepsilon}):\left(DG_{h}u_{h}^{\varepsilon}-D^{2}u^{\varepsilon}\right),U_{h}\right)_{L^{2}(\Omega)}}_{=:\mathcal{G}_{51}}\\
&-\underbrace{\sum_{e\in\mathcal{E}_{h}^{I}}\left(\left[\left[\Phi^{\varepsilon}\left(\nabla u_{h}^{\varepsilon}-G_{h}u_{h}^{\varepsilon}\right)\cdot n_{e}\right]\right],U_{h}\right)_{L^{2}(e)}}_{=:\mathcal{G}_{52}}\\
&+\underbrace{\left(\Phi^{\varepsilon}\left(\nabla u_{h}^{\varepsilon}-G_{h}u_{h}^{\varepsilon}\right),\nabla U_{h}-\nabla U\right)_{L^{2}(\Omega)}}_{=:\mathcal{G}_{53}}\\
&+\underbrace{\left(\Phi^{\varepsilon}\left(\nabla u_{h}^{\varepsilon}-G_{h}u_{h}^{\varepsilon}\right),\nabla U\right)_{L^{2}(\Omega)}}_{=:\mathcal{G}_{54}}.
\end{aligned} G 5 = = = = − ( ( det ( D 2 u ε ) − det ( D G h u h ε ) , U h ) L 2 ( Ω ) ) + ( Φ ε ( ∇ u h ε − G h u h ε ) , ∇ U h ) L 2 ( Ω ) + ( Φ ε ( G h u h ε − ∇ u ε ) , ∇ U h ) L 2 ( Ω ) − ( ( det ( D 2 u ε ) − det ( D G h u h ε ) , U h ) L 2 ( Ω ) ) + ( Φ ε ( ∇ u h ε − G h u h ε ) , ∇ U h ) L 2 ( Ω ) − ( Φ ε : ( D G h u h ε − D 2 u ε ) , U h ) L 2 ( Ω ) − e ∈ E h I ∑ ( [ [ Φ ε ( ∇ u h ε − G h u h ε ) ⋅ n e ] ] , U h ) L 2 ( e ) ( ( Ψ h − Φ ε ) : ( D G h u h ε − D 2 u ε ) , U h ) L 2 ( Ω ) + ( Φ ε ( ∇ u h ε − G h u h ε ) , ∇ U h ) L 2 ( Ω ) − e ∈ E h I ∑ ( [ [ Φ ε ( ∇ u h ε − G h u h ε ) ⋅ n e ] ] , U h ) L 2 ( e ) =: G 51 ( ( Ψ h − Φ ε ) : ( D G h u h ε − D 2 u ε ) , U h ) L 2 ( Ω ) − =: G 52 e ∈ E h I ∑ ( [ [ Φ ε ( ∇ u h ε − G h u h ε ) ⋅ n e ] ] , U h ) L 2 ( e ) + =: G 53 ( Φ ε ( ∇ u h ε − G h u h ε ) , ∇ U h − ∇ U ) L 2 ( Ω ) + =: G 54 ( Φ ε ( ∇ u h ε − G h u h ε ) , ∇ U ) L 2 ( Ω ) . where Ψ h = cof ( D 2 u ε + θ ( D G h u h ε − D 2 u ε ) ) \Psi_{h}=\operatorname{cof}\left(D^{2}u^{\varepsilon}+\theta\left(DG_{h}u_{h}^{\varepsilon}-D^{2}u^{\varepsilon}\right)\right) Ψ h = cof ( D 2 u ε + θ ( D G h u h ε − D 2 u ε ) ) θ \theta θ [ 0 , 1 ] [0,1] [ 0 , 1 ]
∣ G 51 ∣ ≤ C ∥ D 2 u ε − D G h u h ε ∥ L 2 ( Ω ) 2 ∥ U h ∥ L ∞ ( Ω ) ≤ C ∥ D 2 u ε − D G h u h ε ∥ L 2 ( Ω ) 2 ∥ D G h U h ∥ L 2 ( T h ) ≤ C ( ε − 3 + ε − 5 + ε − 7 ) h 2 γ ∥ ∇ ( u h ε − u I ε ) ∥ L 2 ( Ω ) , \begin{aligned}
\left|\mathcal{G}_{51}\right| \le C\left\|D^{2}u^{\varepsilon}-DG_{h}u_{h}^{\varepsilon}\right\|_{L^{2}(\Omega)}^{2}\left\|U_{h}\right\|_{L^{\infty}(\Omega)} \\
&\le C\left\|D^{2}u^{\varepsilon}-DG_{h}u_{h}^{\varepsilon}\right\|_{L^{2}(\Omega)}^{2}\left\|DG_{h}U_{h}\right\|_{L^{2}(\mathcal{T}_{h})} \\
&\le C\left(\varepsilon^{-3}+\varepsilon^{-5}+\varepsilon^{-7}\right)h^{2\gamma}\left\|\nabla\left(u_{h}^{\varepsilon}-u_{I}^{\varepsilon}\right)\right\|_{L^{2}(\Omega)},
\end{aligned} ∣ G 51 ∣ ≤ C D 2 u ε − D G h u h ε L 2 ( Ω ) 2 ∥ U h ∥ L ∞ ( Ω ) ≤ C D 2 u ε − D G h u h ε L 2 ( Ω ) 2 ∥ D G h U h ∥ L 2 ( T h ) ≤ C ( ε − 3 + ε − 5 + ε − 7 ) h 2 γ ∥ ∇ ( u h ε − u I ε ) ∥ L 2 ( Ω ) , and
∣ G 52 ∣ ≤ C h ∥ D G h ( u h ε − u I ε ) ∥ L 2 ( T h ) ∣ Φ ε U h ∣ H 1 ( Ω ) + C h 1 + γ ∥ u ε ∥ H 3 ( Ω ) ∣ Φ ε U h ∣ H 1 ( Ω ) \left|\mathcal{G}_{52}\right| \le Ch\left\|DG_{h}\left(u_{h}^{\varepsilon}-u_{I}^{\varepsilon}\right)\right\|_{L^{2}(\mathcal{T}_{h})}\left|\Phi^{\varepsilon}U_{h}\right|_{H^{1}(\Omega)}
+Ch^{1+\gamma}\left\|u^{\varepsilon}\right\|_{H^{3}(\Omega)}\left|\Phi^{\varepsilon}U_{h}\right|_{H^{1}(\Omega)} ∣ G 52 ∣ ≤ C h ∥ D G h ( u h ε − u I ε ) ∥ L 2 ( T h ) ∣ Φ ε U h ∣ H 1 ( Ω ) + C h 1 + γ ∥ u ε ∥ H 3 ( Ω ) ∣ Φ ε U h ∣ H 1 ( Ω ) ≤ C ( h 1 + γ ∥ u ε ∥ H 3 ( Ω ) + h ∥ D 2 u ε − D G h u h ε ∥ L 2 ( Ω ) ) × ( ∥ U h ∥ L ∞ ( Ω ) ∣ Φ ε ∣ H 1 ( Ω ) + ∥ Φ ε ∥ L ∞ ( Ω ) ∣ U h ∣ H 1 ( Ω ) ) ≤ C ε − 2 ( ε − 1 h 1 + γ + h 1 + γ ( ε − 1 + ε − 3 ) ) ∥ ∇ ( u h ε − u I ε ) ∥ L 2 ( Ω ) ≤ C ( ε − 3 + ε − 5 ) h 1 + γ ∥ ∇ ( u h ε − u I ε ) ∥ L 2 ( Ω ) . \begin{aligned}
&\le C\left(h^{1+\gamma}\lVert u^\varepsilon\rVert_{H^3(\Omega)}+h\lVert D^2u^\varepsilon-DG_hu_h^\varepsilon\rVert_{L^2(\Omega)}\right)\\
&\qquad\times\left(\lVert U_h\rVert_{L^\infty(\Omega)}\lvert\Phi^\varepsilon\rvert_{H^1(\Omega)}+\lVert\Phi^\varepsilon\rVert_{L^\infty(\Omega)}\lvert U_h\rvert_{H^1(\Omega)}\right)\\
&\le C\varepsilon^{-2}\left(\varepsilon^{-1}h^{1+\gamma}+h^{1+\gamma}(\varepsilon^{-1}+\varepsilon^{-3})\right)\lVert\nabla(u_h^\varepsilon-u_I^\varepsilon)\rVert_{L^2(\Omega)}\\
&\le C\left(\varepsilon^{-3}+\varepsilon^{-5}\right)h^{1+\gamma}\lVert\nabla(u_h^\varepsilon-u_I^\varepsilon)\rVert_{L^2(\Omega)}.
\end{aligned} ≤ C ( h 1 + γ ∥ u ε ∥ H 3 ( Ω ) + h ∥ D 2 u ε − D G h u h ε ∥ L 2 ( Ω ) ) × ( ∥ U h ∥ L ∞ ( Ω ) ∣ Φ ε ∣ H 1 ( Ω ) + ∥ Φ ε ∥ L ∞ ( Ω ) ∣ U h ∣ H 1 ( Ω ) ) ≤ C ε − 2 ( ε − 1 h 1 + γ + h 1 + γ ( ε − 1 + ε − 3 ) ) ∥ ∇ ( u h ε − u I ε ) ∥ L 2 ( Ω ) ≤ C ( ε − 3 + ε − 5 ) h 1 + γ ∥ ∇ ( u h ε − u I ε ) ∥ L 2 ( Ω ) . Then, we estimate G 53 G_{53} G 53
∣ G 53 ∣ ≤ C ∥ u ε ∥ W 2 , ∞ ( Ω ) h ∥ D G h u h ε ∥ L 2 ( T h ) ∥ D G h ( U h − U I ) ∥ L 2 ( T h ) + C ∥ u ε ∥ W 2 , ∞ ( Ω ) h 2 ∥ D G h u h ε ∥ L 2 ( T h ) ∣ U ∣ H 2 ( Ω ) ≤ C ε − 1 h ∥ D G h u h ε ∥ L 2 ( T h ) ( ∥ D 2 U − D G h U h ∥ L 2 ( T h ) + ∥ D 2 U − D G h U I ∥ L 2 ( T h ) ) + C ε − 2 h 2 ∥ D G h u h ε ∥ L 2 ( T h ) ∥ ∇ ( u h ε − u I ε ) ∥ L 2 ( Ω ) ≤ C ε − 1 h ( C s ( ε ) h γ + ε − 3 h ) ∥ D G h u h ε ∥ L 2 ( T h ) ∥ ∇ ( u h ε − u I ε ) ∥ L 2 ( Ω ) + C ε − 2 h 2 ∥ D G h u h ε ∥ L 2 ( T h ) ∥ ∇ ( u h ε − u I ε ) ∥ L 2 ( Ω ) ≤ C ( ε − 1 C s ( ε ) + ε − 2 + ε − 4 ) h 1 + γ ∥ D G h u h ε ∥ L 2 ( T h ) ∥ ∇ ( u h ε − u I ε ) ∥ L 2 ( Ω ) . \begin{aligned}
\lvert G_{53}\rvert
&\le C\lVert u^\varepsilon\rVert_{W^{2,\infty}(\Omega)}h\lVert DG_hu_h^\varepsilon\rVert_{L^2(\mathcal{T}_h)}\lVert DG_h(U_h-U_I)\rVert_{L^2(\mathcal{T}_h)}\\
&\qquad+C\lVert u^\varepsilon\rVert_{W^{2,\infty}(\Omega)}h^2\lVert DG_hu_h^\varepsilon\rVert_{L^2(\mathcal{T}_h)}\lvert U\rvert_{H^2(\Omega)}\\
&\le C\varepsilon^{-1}h\lVert DG_hu_h^\varepsilon\rVert_{L^2(\mathcal{T}_h)}
\left(\lVert D^2U-DG_hU_h\rVert_{L^2(\mathcal{T}_h)}+\lVert D^2U-DG_hU_I\rVert_{L^2(\mathcal{T}_h)}\right)\\
&\qquad+C\varepsilon^{-2}h^2\lVert DG_hu_h^\varepsilon\rVert_{L^2(\mathcal{T}_h)}
\lVert\nabla(u_h^\varepsilon-u_I^\varepsilon)\rVert_{L^2(\Omega)}\\
&\le C\varepsilon^{-1}h\left(C_s(\varepsilon)h^\gamma+\varepsilon^{-3}h\right)\lVert DG_hu_h^\varepsilon\rVert_{L^2(\mathcal{T}_h)}
\lVert\nabla(u_h^\varepsilon-u_I^\varepsilon)\rVert_{L^2(\Omega)}\\
&\qquad+C\varepsilon^{-2}h^2\lVert DG_hu_h^\varepsilon\rVert_{L^2(\mathcal{T}_h)}
\lVert\nabla(u_h^\varepsilon-u_I^\varepsilon)\rVert_{L^2(\Omega)}\\
&\le C\left(\varepsilon^{-1}C_s(\varepsilon)+\varepsilon^{-2}+\varepsilon^{-4}\right)h^{1+\gamma}
\lVert DG_hu_h^\varepsilon\rVert_{L^2(\mathcal{T}_h)}
\lVert\nabla(u_h^\varepsilon-u_I^\varepsilon)\rVert_{L^2(\Omega)}.
\end{aligned} ∣ G 53 ∣ ≤ C ∥ u ε ∥ W 2 , ∞ ( Ω ) h ∥ D G h u h ε ∥ L 2 ( T h ) ∥ D G h ( U h − U I ) ∥ L 2 ( T h ) + C ∥ u ε ∥ W 2 , ∞ ( Ω ) h 2 ∥ D G h u h ε ∥ L 2 ( T h ) ∣ U ∣ H 2 ( Ω ) ≤ C ε − 1 h ∥ D G h u h ε ∥ L 2 ( T h ) ( ∥ D 2 U − D G h U h ∥ L 2 ( T h ) + ∥ D 2 U − D G h U I ∥ L 2 ( T h ) ) + C ε − 2 h 2 ∥ D G h u h ε ∥ L 2 ( T h ) ∥ ∇ ( u h ε − u I ε ) ∥ L 2 ( Ω ) ≤ C ε − 1 h ( C s ( ε ) h γ + ε − 3 h ) ∥ D G h u h ε ∥ L 2 ( T h ) ∥ ∇ ( u h ε − u I ε ) ∥ L 2 ( Ω ) + C ε − 2 h 2 ∥ D G h u h ε ∥ L 2 ( T h ) ∥ ∇ ( u h ε − u I ε ) ∥ L 2 ( Ω ) ≤ C ( ε − 1 C s ( ε ) + ε − 2 + ε − 4 ) h 1 + γ ∥ D G h u h ε ∥ L 2 ( T h ) ∥ ∇ ( u h ε − u I ε ) ∥ L 2 ( Ω ) . Recalling the H 2 H^2 H 2
∥ D 2 u ε − D G h u h ε ∥ L 2 ( T h ) ≤ C ( ε − 1 + ε − 3 ) h γ , \lVert D^2u^\varepsilon-DG_hu_h^\varepsilon\rVert_{L^2(\mathcal{T}_h)}
\le C\left(\varepsilon^{-1}+\varepsilon^{-3}\right)h^\gamma, ∥ D 2 u ε − D G h u h ε ∥ L 2 ( T h ) ≤ C ( ε − 1 + ε − 3 ) h γ , we obtain, for h h h
∥ D G h u h ε ∥ L 2 ( T h ) ≤ C ε − 1 2 , \lVert DG_hu_h^\varepsilon\rVert_{L^2(\mathcal{T}_h)}
\le C\varepsilon^{-\frac{1}{2}}, ∥ D G h u h ε ∥ L 2 ( T h ) ≤ C ε − 2 1 , which implies that
∣ G 53 ∣ ≤ C ( ε − 3 2 C s ( ε ) + ε − 5 2 + ε − 9 2 ) h 1 + γ ∥ ∇ ( u h ε − u I ε ) ∥ L 2 ( Ω ) . \lvert G_{53}\rvert\le C\left(\varepsilon^{-\frac{3}{2}}C_s(\varepsilon)+\varepsilon^{-\frac{5}{2}}+\varepsilon^{-\frac{9}{2}}\right)h^{1+\gamma}
\lVert\nabla(u_h^\varepsilon-u_I^\varepsilon)\rVert_{L^2(\Omega)}. ∣ G 53 ∣ ≤ C ( ε − 2 3 C s ( ε ) + ε − 2 5 + ε − 2 9 ) h 1 + γ ∥ ∇ ( u h ε − u I ε ) ∥ L 2 ( Ω ) . Moreover,
∣ G 54 ∣ ≤ C h 2 ∥ D G h u h ε ∥ L 2 ( T h ) ∣ Φ ε ∇ U ∣ H 1 ( Ω ) ≤ C ε − 1 2 h 2 ( ∥ Φ ε ∥ L ∞ ( Ω ) ∣ U ∣ H 2 ( Ω ) + ∣ Φ ε ∣ H 1 ( Ω ) ∥ ∇ U ∥ L ∞ ( Ω ) ) ≤ C ( ε − 5 2 + ε − 3 2 C s ( ε ) ) h 2 ∥ ∇ ( u h ε − u I ε ) ∥ L 2 ( Ω ) . \begin{aligned}
\lvert G_{54}\rvert
&\le Ch^2\lVert DG_hu_h^\varepsilon\rVert_{L^2(\mathcal{T}_h)}\lvert\Phi^\varepsilon\nabla U\rvert_{H^1(\Omega)}\\
&\le C\varepsilon^{-\frac{1}{2}}h^2\left(\lVert\Phi^\varepsilon\rVert_{L^\infty(\Omega)}\lvert U\rvert_{H^2(\Omega)}+\lvert\Phi^\varepsilon\rvert_{H^1(\Omega)}\lVert\nabla U\rVert_{L^\infty(\Omega)}\right)\\
&\le C\left(\varepsilon^{-\frac{5}{2}}+\varepsilon^{-\frac{3}{2}}C_s(\varepsilon)\right)h^2
\lVert\nabla(u_h^\varepsilon-u_I^\varepsilon)\rVert_{L^2(\Omega)}.
\end{aligned} ∣ G 54 ∣ ≤ C h 2 ∥ D G h u h ε ∥ L 2 ( T h ) ∣ Φ ε ∇ U ∣ H 1 ( Ω ) ≤ C ε − 2 1 h 2 ( ∥ Φ ε ∥ L ∞ ( Ω ) ∣ U ∣ H 2 ( Ω ) + ∣ Φ ε ∣ H 1 ( Ω ) ∥ ∇ U ∥ L ∞ ( Ω ) ) ≤ C ( ε − 2 5 + ε − 2 3 C s ( ε ) ) h 2 ∥ ∇ ( u h ε − u I ε ) ∥ L 2 ( Ω ) . The above estimates imply that
∣ G 5 ∣ ≤ C ( ε − 3 + ε − 5 + ε − 7 ) h 2 γ ∥ ∇ ( u h ε − u I ε ) ∥ L 2 ( Ω ) + C ( ε − 3 + ε − 5 + ε − 3 2 C s ( ε ) + ε − 5 2 + ε − 9 2 ) h 1 + γ ∥ ∇ ( u h ε − u I ε ) ∥ L 2 ( Ω ) + C ( ε − 5 2 + ε − 3 2 C s ( ε ) ) h 2 ∥ ∇ ( u h ε − u I ε ) ∥ L 2 ( Ω ) . \begin{aligned}
\lvert G_5\rvert
&\le C\left(\varepsilon^{-3}+\varepsilon^{-5}+\varepsilon^{-7}\right)h^{2\gamma}
\lVert\nabla(u_h^\varepsilon-u_I^\varepsilon)\rVert_{L^2(\Omega)}\\
&\qquad+C\left(\varepsilon^{-3}+\varepsilon^{-5}+\varepsilon^{-\frac{3}{2}}C_s(\varepsilon)+\varepsilon^{-\frac{5}{2}}+\varepsilon^{-\frac{9}{2}}\right)h^{1+\gamma}
\lVert\nabla(u_h^\varepsilon-u_I^\varepsilon)\rVert_{L^2(\Omega)}\\
&\qquad+C\left(\varepsilon^{-\frac{5}{2}}+\varepsilon^{-\frac{3}{2}}C_s(\varepsilon)\right)h^2
\lVert\nabla(u_h^\varepsilon-u_I^\varepsilon)\rVert_{L^2(\Omega)}.
\end{aligned} ∣ G 5 ∣ ≤ C ( ε − 3 + ε − 5 + ε − 7 ) h 2 γ ∥ ∇ ( u h ε − u I ε ) ∥ L 2 ( Ω ) + C ( ε − 3 + ε − 5 + ε − 2 3 C s ( ε ) + ε − 2 5 + ε − 2 9 ) h 1 + γ ∥ ∇ ( u h ε − u I ε ) ∥ L 2 ( Ω ) + C ( ε − 2 5 + ε − 2 3 C s ( ε ) ) h 2 ∥ ∇ ( u h ε − u I ε ) ∥ L 2 ( Ω ) . Consequently, we have
∥ ∇ ( u h ε − u I ε ) ∥ L 2 ( Ω ) ≤ C ( ε ) ( h 2 γ + h 1 + γ + h 2 ) + C ε − 1 h , \lVert\nabla(u_h^\varepsilon-u_I^\varepsilon)\rVert_{L^2(\Omega)}
\le C(\varepsilon)\left(h^{2\gamma}+h^{1+\gamma}+h^2\right)+C\varepsilon^{-1}h, ∥ ∇ ( u h ε − u I ε ) ∥ L 2 ( Ω ) ≤ C ( ε ) ( h 2 γ + h 1 + γ + h 2 ) + C ε − 1 h , and the proof is completed by applying the triangle inequality.
Numerical experiments
In this section, we provide several experiments to gauge the efficiency of the finite element method developed in the previous sections. Unless otherwise stated, all computations are performed on the domain Ω = ( 0 , 1 ) 2 \Omega=(0,1)^2 Ω = ( 0 , 1 ) 2
e = ∥ u h − u ∥ L 2 ( Ω ) , D e = ∥ ∇ u h − ∇ u ∥ L 2 ( T h ) , D 2 e = ∥ D G h u h − D 2 u ∥ L 2 ( T h ) . e=\|u_h-u\|_{L^2(\Omega)},\quad De=\|\nabla u_h-\nabla u\|_{L^2(\mathcal{T}_h)},\quad D^2e=\|DG_hu_h-D^2u\|_{L^2(\mathcal{T}_h)}. e = ∥ u h − u ∥ L 2 ( Ω ) , D e = ∥∇ u h − ∇ u ∥ L 2 ( T h ) , D 2 e = ∥ D G h u h − D 2 u ∥ L 2 ( T h ) . Numerical Test 1
In this test, we choose the data f ( x , y ) = 24 x 2 f(x,y)=24x^2 f ( x , y ) = 24 x 2 g ( x , y ) = x 4 + y 2 g(x,y)=x^4+y^2 g ( x , y ) = x 4 + y 2 u ( x , y ) = x 4 + y 2 u(x,y)=x^4+y^2 u ( x , y ) = x 4 + y 2 x 1.2 + y 1.2 x^{1.2}+y^{1.2} x 1.2 + y 1.2 u 0 u_0 u 0
We first focus on the numerical investigation of the behavior of the error u − u h ϵ u-u_h^\epsilon u − u h ϵ h = 0.01 h=0.01 h = 0.01 ϵ \epsilon ϵ h h h u h ϵ u_h^\epsilon u h ϵ u ϵ u^\epsilon u ϵ Table 5.1 . From the table, we clearly observe that the error u − u h ϵ u-u_h^\epsilon u − u h ϵ O ( ϵ ) O(\epsilon) O ( ϵ ) L 2 L^2 L 2 O ( ϵ 3 / 4 ) O(\epsilon^{3/4}) O ( ϵ 3/4 ) H 1 H^1 H 1 O ( ϵ 1 / 4 ) O(\epsilon^{1/4}) O ( ϵ 1/4 ) H 2 H^2 H 2 u − u ϵ u-u^\epsilon u − u ϵ Table 5.2 . Similar decay rates with respect to ϵ \epsilon ϵ
Table 5.1. Numerical error of numerical test 1 on the uniform mesh
Table 5.2. Numerical error of numerical test 1 on the unstructured mesh
Then, we turn to explore a relation between ϵ \epsilon ϵ h h h h h h ϵ \epsilon ϵ ∣ u − u h ϵ ∣ |u-u_h^\epsilon| ∣ u − u h ϵ ∣ ∣ u − u ϵ ∣ |u-u^\epsilon| ∣ u − u ϵ ∣ h = ϵ h=\sqrt{\epsilon} h = ϵ Figure 5.1 . Based on the numerical data, we fit the constant β \beta β y = β ϵ α y=\beta\epsilon^\alpha y = β ϵ α α \alpha α α = 0.25 \alpha=0.25 α = 0.25 H 2 H^2 H 2 α = 0.75 \alpha=0.75 α = 0.75 H 1 H^1 H 1 α = 1 \alpha=1 α = 1 L 2 L^2 L 2 h = ϵ h=\sqrt{\epsilon} h = ϵ
Figure 5.1. Numerical errors with respect to ϵ \epsilon ϵ y = β ϵ α y = \beta\epsilon^\alpha y = β ϵ α
Numerical Test 2
The purpose of this test is to calculate the rate of convergence of ∣ u ϵ − u h ϵ ∣ |u^\epsilon-u_h^\epsilon| ∣ u ϵ − u h ϵ ∣ ϵ = 0.001 \epsilon=0.001 ϵ = 0.001 u ϵ u^\epsilon u ϵ f ϵ ( x , y ) = x 2 y 2 − 4 ϵ f^\epsilon(x,y)=x^2y^2-4\epsilon f ϵ ( x , y ) = x 2 y 2 − 4 ϵ g ϵ ( x , y ) = 1 12 ( x 4 + y 4 ) g^\epsilon(x,y)=\frac{1}{12}(x^4+y^4) g ϵ ( x , y ) = 12 1 ( x 4 + y 4 ) ψ ϵ ( x , y ) = x 2 + y 2 \psi^\epsilon(x,y)=x^2+y^2 ψ ϵ ( x , y ) = x 2 + y 2 u ϵ = 1 12 ( x 4 + y 4 ) u^\epsilon=\frac{1}{12}(x^4+y^4) u ϵ = 12 1 ( x 4 + y 4 )
The numerical results on uniform meshes are summarized in Table 5.3 . It can be seen from the data that the discrete H 2 H^2 H 2 O ( h 0.5 ) O(h^{0.5}) O ( h 0.5 ) H 1 H^1 H 1 O ( h ) O(h) O ( h ) L 2 L^2 L 2 O ( h 2 ) O(h^2) O ( h 2 ) [12 ] , our method achieves improved convergence rates in terms of both L 2 L^2 L 2 H 1 H^1 H 1
Table 5.3. Numerical error of numerical test 2 on uniform meshes
The numerical results on unstructured meshes are reported in Table 5.4 , where similar convergence rates can be observed for the H 1 H^1 H 1 H 2 H^2 H 2
Table 5.4. Numerical error of numerical test 2 on unstructured meshes
5.3. Numerical Test 3. In this test, we choose f = 1 f = 1 f = 1 g = 0 g = 0 g = 0 ψ ϵ ≡ ϵ \psi^\epsilon\equiv\epsilon ψ ϵ ≡ ϵ Figure 5.2 . From the figure, it is clear that the computed solution successfully captures the convexity of the nonsmooth solution.
Figure 5.2. Numerical solution of Test 3 computed by the proposed linear element method with ϵ = 0.001 \epsilon= 0.001 ϵ = 0.001 h = 1 / 64 h = 1/64 h = 1/64
Numerical Test 4
In this experiment, we consider the computational domain Ω = ( − 1 , 1 ) 2 \Omega= (-1, 1)^2 Ω = ( − 1 , 1 ) 2 u ( x , y ) = ∣ x ∣ ∈ H 1 ( Ω ) u(x,y) = |x| \in H^1(\Omega) u ( x , y ) = ∣ x ∣ ∈ H 1 ( Ω ) f = 0 f = 0 f = 0 u u u ϵ = 0.001 \epsilon= 0.001 ϵ = 0.001 Table 5.5 . The numerical results shows the error is decreasing.
Table 5.5. Numerical results for the Monge–Ampère equation with solution u = ∣ x ∣ u = |x| u = ∣ x ∣
Conclusion
In this paper, we propose a recovery-based linear finite element method for solving the regularized Monge–Ampère equation, inspired by the discrete MT estimate of the gradient recovery operator. We establish convergence in both the discrete H 2 H^2 H 2 H 1 H^1 H 1
Acknowledgments. This work was supported in part by the Dyason Fellowship and the Faculty Science Researcher Development Grant of the University of Melbourne, and the NSFC grant 12131005.
Properties of the recovery operator
In this appendix, we list some properties of G h G_h G h [14 ] .
Lemma A.1. For any v h ∈ V h v_h \in V_h v h ∈ V h
∥ ∇ v h − G h v h ∥ L 2 ( Ω ) ≤ C h ∥ D G h v h ∥ L 2 ( T h ) . \lVert\nabla v_h-G_hv_h\rVert_{L^2(\Omega)} \leq Ch\lVert DG_hv_h\rVert_{L^2(\mathcal{T}_h)}. ∥ ∇ v h − G h v h ∥ L 2 ( Ω ) ≤ C h ∥ D G h v h ∥ L 2 ( T h ) . Consequently, we obtain the following bound:
∥ G h v h ∥ L 2 ( Ω ) ≤ C ∥ ∇ v h ∥ L 2 ( Ω ) ≤ C ∥ G h v h ∥ L 2 ( Ω ) . \lVert G_hv_h\rVert_{L^2(\Omega)} \leq C\lVert\nabla v_h\rVert_{L^2(\Omega)} \leq C\lVert G_hv_h\rVert_{L^2(\Omega)}. ∥ G h v h ∥ L 2 ( Ω ) ≤ C ∥ ∇ v h ∥ L 2 ( Ω ) ≤ C ∥ G h v h ∥ L 2 ( Ω ) . Lemma A.2. For any v h ∈ V h 0 v_h \in V_h^0 v h ∈ V h 0
∥ ∇ v h ∥ L 2 ( Ω ) ≤ C ∥ D G h v h ∥ L 2 ( T h ) . \lVert\nabla v_h\rVert_{L^2(\Omega)} \leq C\lVert DG_hv_h\rVert_{L^2(\mathcal{T}_h)}. ∥ ∇ v h ∥ L 2 ( Ω ) ≤ C ∥ D G h v h ∥ L 2 ( T h ) . Theorem A.3. Assuming that v ∈ W 2 , p ( Ω ) v \in W^{2,p}(\Omega) v ∈ W 2 , p ( Ω ) 1 ≤ p ≤ ∞ 1 \leq p \leq\infty 1 ≤ p ≤ ∞
∥ ∇ v − G h v I ∥ L p ( T ) ≤ C h T ∣ v ∣ W 2 , p ( K T ) , ∀ T ∈ T h . \lVert\nabla v-G_hv_I\rVert_{L^p(T)} \leq C h_T \lvert v\rvert_{W^{2,p}(K_T)}, \qquad\forall T \in\mathcal{T}_h. ∥ ∇ v − G h v I ∥ L p ( T ) ≤ C h T ∣ v ∣ W 2 , p ( K T ) , ∀ T ∈ T h . This yields a global approximation property:
∥ ∇ v − G h v I ∥ L p ( Ω ) ≤ C h ∣ v ∣ W 2 , p ( Ω ) . \lVert\nabla v-G_hv_I\rVert_{L^p(\Omega)} \leq Ch\lvert v\rvert_{W^{2,p}(\Omega)}. ∥ ∇ v − G h v I ∥ L p ( Ω ) ≤ C h ∣ v ∣ W 2 , p ( Ω ) . Theorem A.4. Let T h \mathcal{T}_h T h ( ν , μ ) (\nu,\mu) ( ν , μ ) [14 ] ). Let ρ = min { 1 , ν , μ / 2 } \rho= \min\{1,\nu,\mu/2\} ρ = min { 1 , ν , μ /2 } v ∈ H 3 ( Ω ) v \in H^3(\Omega) v ∈ H 3 ( Ω ) h ≤ e − 2 h \leq e^{-2} h ≤ e − 2 β ∈ [ 0 , 2 ] \beta\in[0,2] β ∈ [ 0 , 2 ]
( ∑ T ∈ T h h T − β ∥ ∇ v − G h v I ∥ L 2 ( T ) 2 ) 1 / 2 ≤ C h ( 1 + ρ ) − β / 2 ∣ log h ∣ 1 / 2 ∥ v ∥ H 3 ( Ω ) . \left(\sum_{T\in\mathcal{T}_h}h_T^{-\beta}\lVert\nabla v-G_hv_I\rVert_{L^2(T)}^2\right)^{1/2}
\leq Ch^{(1+\rho)-\beta/2}\lvert\log h\rvert^{1/2}\lVert v\rVert_{H^3(\Omega)}. ( T ∈ T h ∑ h T − β ∥ ∇ v − G h v I ∥ L 2 ( T ) 2 ) 1/2 ≤ C h ( 1 + ρ ) − β /2 ∣ log h ∣ 1/2 ∥ v ∥ H 3 ( Ω ) . Next, we will prove a superconvergence result of the operator G h G_h G h
Theorem A.5. Suppose v ∈ ( H 1 ( Ω ) ) 2 v \in\left(H^1(\Omega)\right)^2 v ∈ ( H 1 ( Ω ) ) 2 v h ∈ V h v_h \in V_h v h ∈ V h
∣ ∫ Ω v ⋅ ( ∇ v h − G h v h ) d x ∣ ≤ C h 2 ∣ v ∣ H 1 ( Ω ) ∥ D G h v h ∥ L 2 ( T h ) . \left\lvert\int_{\Omega}v\cdot(\nabla v_h-G_hv_h)\,dx\right\rvert\leq Ch^2\lvert v\rvert_{H^1(\Omega)}\lVert DG_hv_h\rVert_{L^2(\mathcal{T}_h)}. ∫ Ω v ⋅ ( ∇ v h − G h v h ) d x ≤ C h 2 ∣ v ∣ H 1 ( Ω ) ∥ D G h v h ∥ L 2 ( T h ) . Proof. Let { ϕ m } m ∈ M h \{\phi_m\}_{m\in M_h} { ϕ m } m ∈ M h V h C R V_h^{CR} V h C R T m T_m T m m m m w h ∈ V h C R × V h C R w_h\in V_h^{CR}\times V_h^{CR} w h ∈ V h C R × V h C R
∫ Ω ∑ m ∈ M h w h ( m ) ⋅ G h v h ( m ) ϕ m d x = ∑ m ∈ M h w h ( m ) ⋅ G h v h ( m ) ∣ T m ∣ 3 = ∑ m ∈ M h w h ( m ) 3 ⋅ ∫ T m ∇ v h d x = ∑ T ∈ T h ∫ T ( ∑ m ∈ M h ∩ T w h ( m ) 3 ) ⋅ ∇ v h d x = ∑ T ∈ T h ∫ T ∑ m ∈ M h ∩ T w h ( m ) ϕ m ⋅ ∇ v h d x = ∑ T ∈ T h ∫ T w h ⋅ ∇ v h = ∫ Ω w h ⋅ ∇ v h d x . \begin{aligned}
\int_{\Omega}\sum_{m\in M_h}w_h(m)\cdot G_hv_h(m)\phi_m\,\mathrm{d}x
&=\sum_{m\in M_h}w_h(m)\cdot G_hv_h(m)\frac{|T_m|}{3}
=\sum_{m\in M_h}\frac{w_h(m)}{3}\cdot\int_{T_m}\nabla v_h\,\mathrm{d}x\\
&=\sum_{T\in\mathcal{T}_h}\int_T\left(\sum_{m\in M_h\cap T}\frac{w_h(m)}{3}\right)\cdot\nabla v_h\,\mathrm{d}x
=\sum_{T\in\mathcal{T}_h}\int_T\sum_{m\in M_h\cap T}w_h(m)\phi_m\cdot\nabla v_h\,\mathrm{d}x\\
&=\sum_{T\in\mathcal{T}_h}\int_Tw_h\cdot\nabla v_h
=\int_{\Omega}w_h\cdot\nabla v_h\,\mathrm{d}x.
\end{aligned} ∫ Ω m ∈ M h ∑ w h ( m ) ⋅ G h v h ( m ) ϕ m d x = m ∈ M h ∑ w h ( m ) ⋅ G h v h ( m ) 3 ∣ T m ∣ = m ∈ M h ∑ 3 w h ( m ) ⋅ ∫ T m ∇ v h d x = T ∈ T h ∑ ∫ T ( m ∈ M h ∩ T ∑ 3 w h ( m ) ) ⋅ ∇ v h d x = T ∈ T h ∑ ∫ T m ∈ M h ∩ T ∑ w h ( m ) ϕ m ⋅ ∇ v h d x = T ∈ T h ∑ ∫ T w h ⋅ ∇ v h = ∫ Ω w h ⋅ ∇ v h d x . where we have used the fact that ∇ v h \nabla v_h ∇ v h T T T w h = ( v ) I C R w_h=(v)_{ICR} w h = ( v ) I C R V h C R V_h^{CR} V h C R
∥ v − w h ∥ L 2 ( Ω ) ≤ C h ∣ v ∣ H 1 ( Ω ) , ∥ D w h ∥ L 2 ( T h ) ≤ C ∣ v ∣ H 1 ( Ω ) . \|v-w_h\|_{L^2(\Omega)}\leq Ch|v|_{H^1(\Omega)},
\qquad
\|Dw_h\|_{L^2(\mathcal{T}_h)}\leq C|v|_{H^1(\Omega)}. ∥ v − w h ∥ L 2 ( Ω ) ≤ C h ∣ v ∣ H 1 ( Ω ) , ∥ D w h ∥ L 2 ( T h ) ≤ C ∣ v ∣ H 1 ( Ω ) . We have the decomposition
∫ Ω v ⋅ ( ∇ v h − G h v h ) d x = ∫ Ω ( v − w h ) ⋅ ( ∇ v h − G h v h ) d x ⏟ = : I 1 + ∫ Ω w h ⋅ ( ∇ v h − G h v h ) d x ⏟ = : I 2 . \begin{aligned}
\int_{\Omega}v\cdot(\nabla v_h-G_hv_h)\,\mathrm{d}x
&=\underbrace{\int_{\Omega}(v-w_h)\cdot(\nabla v_h-G_hv_h)\,\mathrm{d}x}_{=:I_1}
+\underbrace{\int_{\Omega}w_h\cdot(\nabla v_h-G_hv_h)\,\mathrm{d}x}_{=:I_2}.
\end{aligned} ∫ Ω v ⋅ ( ∇ v h − G h v h ) d x = =: I 1 ∫ Ω ( v − w h ) ⋅ ( ∇ v h − G h v h ) d x + =: I 2 ∫ Ω w h ⋅ ( ∇ v h − G h v h ) d x . For I 1 I_1 I 1
∣ I 1 ∣ ≤ ∥ v − w h ∥ L 2 ( Ω ) ∥ ∇ v h − G h v h ∥ L 2 ( Ω ) ≤ C h 2 ∣ v ∣ H 1 ( Ω ) ∥ D G h v h ∥ L 2 ( T h ) , |I_1|\leq\|v-w_h\|_{L^2(\Omega)}\|\nabla v_h-G_hv_h\|_{L^2(\Omega)}
\leq Ch^2|v|_{H^1(\Omega)}\|DG_hv_h\|_{L^2(\mathcal{T}_h)}, ∣ I 1 ∣ ≤ ∥ v − w h ∥ L 2 ( Ω ) ∥∇ v h − G h v h ∥ L 2 ( Ω ) ≤ C h 2 ∣ v ∣ H 1 ( Ω ) ∥ D G h v h ∥ L 2 ( T h ) , and for I 2 I_2 I 2
I 2 = ∫ Ω w h ⋅ ( ∇ v h − G h v h ) d x = ∫ Ω w h ⋅ ∇ v h d x − ∫ Ω w h ⋅ G h v h d x = ∫ Ω ∑ m ∈ M h w h ( m ) ⋅ G h v h ( m ) ϕ m d x − ∫ Ω w h ⋅ G h v h d x = ∑ T ∈ T h ∫ T ( ( w h ⋅ G h v h ) I C R − w h ⋅ G h v h ) d x , \begin{aligned}
I_2
&=\int_{\Omega}w_h\cdot(\nabla v_h-G_hv_h)\,\mathrm{d}x\\
&=\int_{\Omega}w_h\cdot\nabla v_h\,\mathrm{d}x-\int_{\Omega}w_h\cdot G_hv_h\,\mathrm{d}x\\
&=\int_{\Omega}\sum_{m\in M_h}w_h(m)\cdot G_hv_h(m)\phi_m\,\mathrm{d}x-\int_{\Omega}w_h\cdot G_hv_h\,\mathrm{d}x\\
&=\sum_{T\in\mathcal{T}_h}\int_T\left((w_h\cdot G_hv_h)_{ICR}-w_h\cdot G_hv_h\right)\,\mathrm{d}x,
\end{aligned} I 2 = ∫ Ω w h ⋅ ( ∇ v h − G h v h ) d x = ∫ Ω w h ⋅ ∇ v h d x − ∫ Ω w h ⋅ G h v h d x = ∫ Ω m ∈ M h ∑ w h ( m ) ⋅ G h v h ( m ) ϕ m d x − ∫ Ω w h ⋅ G h v h d x = T ∈ T h ∑ ∫ T ( ( w h ⋅ G h v h ) I C R − w h ⋅ G h v h ) d x , then we have
∣ I 2 ∣ ≤ ∑ T ∈ T h ∥ ( w h ⋅ G h v h ) I C R − w h ⋅ G h v h ∥ L 1 ( T ) ≤ C h 2 ∑ T ∈ T h ∥ D 2 ( w h ⋅ G h v h ) ∥ L 1 ( T ) ≤ C h 2 ∑ T ∈ T h ∥ D w h ∥ L 2 ( T ) ∥ D G h v h ∥ L 2 ( T ) ≤ C h 2 ∥ D w h ∥ L 2 ( T h ) ∥ D G h v h ∥ L 2 ( T h ) ≤ C h 2 ∣ v ∣ H 1 ( Ω ) ∥ D G h v h ∥ L 2 ( T h ) . \begin{aligned}
|I_2|
&\leq\sum_{T\in\mathcal{T}_h}\|(w_h\cdot G_hv_h)_{ICR}-w_h\cdot G_hv_h\|_{L^1(T)}
\leq Ch^2\sum_{T\in\mathcal{T}_h}\|D^2(w_h\cdot G_hv_h)\|_{L^1(T)}\\
&\leq Ch^2\sum_{T\in\mathcal{T}_h}\|Dw_h\|_{L^2(T)}\|DG_hv_h\|_{L^2(T)}
\leq Ch^2\|Dw_h\|_{L^2(\mathcal{T}_h)}\|DG_hv_h\|_{L^2(\mathcal{T}_h)}\\
&\leq Ch^2|v|_{H^1(\Omega)}\|DG_hv_h\|_{L^2(\mathcal{T}_h)}.
\end{aligned} ∣ I 2 ∣ ≤ T ∈ T h ∑ ∥ ( w h ⋅ G h v h ) I C R − w h ⋅ G h v h ∥ L 1 ( T ) ≤ C h 2 T ∈ T h ∑ ∥ D 2 ( w h ⋅ G h v h ) ∥ L 1 ( T ) ≤ C h 2 T ∈ T h ∑ ∥ D w h ∥ L 2 ( T ) ∥ D G h v h ∥ L 2 ( T ) ≤ C h 2 ∥ D w h ∥ L 2 ( T h ) ∥ D G h v h ∥ L 2 ( T h ) ≤ C h 2 ∣ v ∣ H 1 ( Ω ) ∥ D G h v h ∥ L 2 ( T h ) . Then we can complete the proof by the above estimates.
Lemma A.6. There holds
∥ v h ∥ L ∞ ( Ω ) ≤ C ∥ Δ h v h ∥ L 2 ( Ω ) ∀ v h ∈ V h 0 (30) \|v_h\|_{L^\infty(\Omega)} \le C\|\Delta_h v_h\|_{L^2(\Omega)} \quad\forall v_h \in V_h^0 \tag*{(30)} ∥ v h ∥ L ∞ ( Ω ) ≤ C ∥ Δ h v h ∥ L 2 ( Ω ) ∀ v h ∈ V h 0 (30) where Δ h : V h 0 → V h 0 \Delta_h: V_h^0 \to V_h^0 Δ h : V h 0 → V h 0
( Δ h u h , v h ) L 2 ( Ω ) = − ( ∇ u h , ∇ v h ) L 2 ( Ω ) ∀ v h ∈ V h 0 . (31) (\Delta_h u_h,v_h)_{L^2(\Omega)} = -(\nabla u_h,\nabla v_h)_{L^2(\Omega)} \quad\forall v_h \in V_h^0. \tag*{(31)} ( Δ h u h , v h ) L 2 ( Ω ) = − ( ∇ u h , ∇ v h ) L 2 ( Ω ) ∀ v h ∈ V h 0 . (31) Proof. By the discrete Meyers estimate in [2 ] , there exists ε > 0 \varepsilon> 0 ε > 0 p ∈ ( 2 , 2 + ε ) p \in(2,2+\varepsilon) p ∈ ( 2 , 2 + ε )
∥ v h ∥ W 1 , p ( Ω ) ≤ C p ∥ Δ h v h ∥ W − 1 , p ( Ω ) . \|v_h\|_{W^{1,p}(\Omega)} \le C_p\|\Delta_h v_h\|_{W^{-1,p}(\Omega)}. ∥ v h ∥ W 1 , p ( Ω ) ≤ C p ∥ Δ h v h ∥ W − 1 , p ( Ω ) . Fix such a p p p p ′ = p / ( p − 1 ) p' = p/(p-1) p ′ = p / ( p − 1 ) W 0 1 , p ′ ( Ω ) ↪ L 2 ( Ω ) W_0^{1,p'}(\Omega) \hookrightarrow L^2(\Omega) W 0 1 , p ′ ( Ω ) ↪ L 2 ( Ω ) φ ∈ W 0 1 , p ′ ( Ω ) \varphi\in W_0^{1,p'}(\Omega) φ ∈ W 0 1 , p ′ ( Ω )
∣ ( Δ h v h , φ ) L 2 ( Ω ) ∣ ≤ ∥ Δ h v h ∥ L 2 ( Ω ) ∥ φ ∥ L 2 ( Ω ) ≤ C ∥ Δ h v h ∥ L 2 ( Ω ) ∥ φ ∥ W 0 1 , p ′ ( Ω ) . |(\Delta_h v_h,\varphi)_{L^2(\Omega)}| \le\|\Delta_h v_h\|_{L^2(\Omega)}\|\varphi\|_{L^2(\Omega)} \le C\|\Delta_h v_h\|_{L^2(\Omega)}\|\varphi\|_{W_0^{1,p'}(\Omega)}. ∣ ( Δ h v h , φ ) L 2 ( Ω ) ∣ ≤ ∥ Δ h v h ∥ L 2 ( Ω ) ∥ φ ∥ L 2 ( Ω ) ≤ C ∥ Δ h v h ∥ L 2 ( Ω ) ∥ φ ∥ W 0 1 , p ′ ( Ω ) . This implies
∥ Δ h v h ∥ W − 1 , p ( Ω ) ≤ C ∥ Δ h v h ∥ L 2 ( Ω ) . \|\Delta_h v_h\|_{W^{-1,p}(\Omega)} \le C\|\Delta_h v_h\|_{L^2(\Omega)}. ∥ Δ h v h ∥ W − 1 , p ( Ω ) ≤ C ∥ Δ h v h ∥ L 2 ( Ω ) . Finally, by the embedding W 1 , p ( Ω ) ↪ L ∞ ( Ω ) W^{1,p}(\Omega) \hookrightarrow L^\infty(\Omega) W 1 , p ( Ω ) ↪ L ∞ ( Ω )
∥ v h ∥ L ∞ ( Ω ) ≤ C ∥ v h ∥ W 1 , p ( Ω ) ≤ C ∥ Δ h v h ∥ W − 1 , p ( Ω ) ≤ C ∥ Δ h v h ∥ L 2 ( Ω ) , \|v_h\|_{L^\infty(\Omega)} \le C\|v_h\|_{W^{1,p}(\Omega)} \le C\|\Delta_h v_h\|_{W^{-1,p}(\Omega)} \le C\|\Delta_h v_h\|_{L^2(\Omega)}, ∥ v h ∥ L ∞ ( Ω ) ≤ C ∥ v h ∥ W 1 , p ( Ω ) ≤ C ∥ Δ h v h ∥ W − 1 , p ( Ω ) ≤ C ∥ Δ h v h ∥ L 2 ( Ω ) , which completes the proof.
Lemma A.7. We have the following estimate:
∥ v h ∥ L ∞ ( Ω ) ≤ C ∥ D G h v h ∥ L 2 ( T h ) ∀ v h ∈ V h 0 . (32) \|v_h\|_{L^\infty(\Omega)} \le C\|DG_hv_h\|_{L^2(\mathcal{T}_h)} \quad\forall v_h \in V_h^0. \tag*{(32)} ∥ v h ∥ L ∞ ( Ω ) ≤ C ∥ D G h v h ∥ L 2 ( T h ) ∀ v h ∈ V h 0 . (32) Proof. From Lemma A.6 , we only need to prove that
∥ Δ h v h ∥ L 2 ( Ω ) ≤ C ∥ D G h v h ∥ L 2 ( T h ) . (33) \|\Delta_hv_h\|_{L^2(\Omega)} \le C\|DG_hv_h\|_{L^2(\mathcal{T}_h)}. \tag*{(33)} ∥ Δ h v h ∥ L 2 ( Ω ) ≤ C ∥ D G h v h ∥ L 2 ( T h ) . (33) Note that ∥ Δ h v h ∥ L 2 ( Ω ) = sup 0 ≠ u h ∈ V h ∣ ( Δ h v h , u h ) L 2 ( Ω ) ∣ / ∥ u h ∥ L 2 ( Ω ) \|\Delta_hv_h\|_{L^2(\Omega)} = \sup_{0\ne u_h\in V_h}|(\Delta_hv_h,u_h)_{L^2(\Omega)}|/\|u_h\|_{L^2(\Omega)} ∥ Δ h v h ∥ L 2 ( Ω ) = sup 0 = u h ∈ V h ∣ ( Δ h v h , u h ) L 2 ( Ω ) ∣/∥ u h ∥ L 2 ( Ω ) 0 ≠ u h ∈ V h 0\ne u_h\in V_h 0 = u h ∈ V h
( Δ h v h , u h ) L 2 ( Ω ) = − ( ∇ v h , ∇ u h ) L 2 ( Ω ) = ( G h v h − ∇ v h , ∇ u h ) L 2 ( Ω ) − ( G h v h , ∇ u h ) L 2 ( Ω ) = ( G h v h − ∇ v h , ∇ u h ) L 2 ( Ω ) + ( div ( G h v h ) , u h ) L 2 ( Ω ) − ∑ e ∈ E h I ( [ [ G h v h ] ] ⋅ n e , u h ) L 2 ( e ) . \begin{aligned}
(\Delta_hv_h,u_h)_{L^2(\Omega)}
&=-(\nabla v_h,\nabla u_h)_{L^2(\Omega)}\\
&=(G_hv_h-\nabla v_h,\nabla u_h)_{L^2(\Omega)}-(G_hv_h,\nabla u_h)_{L^2(\Omega)}\\
&=(G_hv_h-\nabla v_h,\nabla u_h)_{L^2(\Omega)}+(\operatorname{div}(G_hv_h),u_h)_{L^2(\Omega)}-\sum_{e\in\mathcal{E}_h^I}([[G_hv_h]]\cdot n_e,u_h)_{L^2(e)}.
\end{aligned} ( Δ h v h , u h ) L 2 ( Ω ) = − ( ∇ v h , ∇ u h ) L 2 ( Ω ) = ( G h v h − ∇ v h , ∇ u h ) L 2 ( Ω ) − ( G h v h , ∇ u h ) L 2 ( Ω ) = ( G h v h − ∇ v h , ∇ u h ) L 2 ( Ω ) + ( div ( G h v h ) , u h ) L 2 ( Ω ) − e ∈ E h I ∑ ([[ G h v h ]] ⋅ n e , u h ) L 2 ( e ) . The above identity implies
∣ ( Δ h v h , u h ) L 2 ( Ω ) ∣ ≤ C h ∥ D G h v h ∥ L 2 ( T h ) ∥ ∇ u h ∥ L 2 ( Ω ) + ∥ D G h v h ∥ L 2 ( T h ) ∥ u h ∥ L 2 ( Ω ) + ( ∑ e ∈ E h I h e − 1 ∥ [ [ G h v h ] ] ∥ L 2 ( e ) 2 ) 1 2 ( ∑ e ∈ E h I h e ∥ u h ∥ L 2 ( e ) 2 ) 1 2 ≤ C ∥ D G h v h ∥ L 2 ( T h ) ∥ u h ∥ L 2 ( Ω ) . \begin{aligned}
|(\Delta_hv_h,u_h)_{L^2(\Omega)}|
&\le Ch\|DG_hv_h\|_{L^2(\mathcal{T}_h)}\|\nabla u_h\|_{L^2(\Omega)}+\|DG_hv_h\|_{L^2(\mathcal{T}_h)}\|u_h\|_{L^2(\Omega)}\\
&\quad+\left(\sum_{e\in\mathcal{E}_h^I}h_e^{-1}\|[[G_hv_h]]\|_{L^2(e)}^2\right)^{\frac12}\left(\sum_{e\in\mathcal{E}_h^I}h_e\|u_h\|_{L^2(e)}^2\right)^{\frac12}\\
&\le C\|DG_hv_h\|_{L^2(\mathcal{T}_h)}\|u_h\|_{L^2(\Omega)}.
\end{aligned} ∣ ( Δ h v h , u h ) L 2 ( Ω ) ∣ ≤ C h ∥ D G h v h ∥ L 2 ( T h ) ∥∇ u h ∥ L 2 ( Ω ) + ∥ D G h v h ∥ L 2 ( T h ) ∥ u h ∥ L 2 ( Ω ) + e ∈ E h I ∑ h e − 1 ∥ [[ G h v h ]] ∥ L 2 ( e ) 2 2 1 e ∈ E h I ∑ h e ∥ u h ∥ L 2 ( e ) 2 2 1 ≤ C ∥ D G h v h ∥ L 2 ( T h ) ∥ u h ∥ L 2 ( Ω ) . Here we used the inverse inequality and trace inequality in the last line. This completes the proof.
