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Convergence analysis of a weak Galerkin finite element method on a Bakhvalov-type mesh for a singularly perturbed convection-diffusion equation in 2D

Shicheng Liu [email protected] Xiangyun Meng [email protected] Qilong Zhai [email protected] School of Mathematics, Jilin University, Changchun 130012, China School of Mathematics and Statistics, Beijing Jiaotong University, Beijing 100044, China
Abstract

In this paper, we propose a weak Galerkin finite element method (WG) for solving singularly perturbed convection-diffusion problems on a Bakhvalov-type mesh in 2D. Our method is flexible and allows the use of discontinuous approximation functions on the meshe. An error estimate is devised in a suitable norm and the optimal convergence order is obtained. Finally, numerical experiments are given to support the theory and to show the efficiency of the proposed method.

keywords:
Weak Galerkin finite element method, convection-diffusion, singularly perturbed, Bakhvalov-type mesh.
MSC:
[2020] 65N15 , 65N30 , 35B25
††journal: Journal of Computational and Applied Mathematics

1 Introduction

Consider the following singularly perturbed convection-diffusion problem

−ε​Δ​u−𝐛⋅∇u+c​u\displaystyle-\varepsilon\Delta u-\mathbf{b}\cdot\nabla u+cu =f,in​Ω,\displaystyle=f,\quad\text{in}~{}\Omega, (1.1)
u\displaystyle u =0,on​∂Ω,\displaystyle=0,\quad\text{on}~{}\partial\Omega, (1.2)

with a positive parameter ε\varepsilon satisfying 0<ε≪10<\varepsilon\ll 1, and 𝐛∈[W1,∞​(Ω)]2\mathbf{b}\in[W^{1,\infty}(\Omega)]^{2}. The functions 𝐛\mathbf{b}, cc and ff are assumed to be smooth on Ω\Omega. For any (x,y)∈Ω¯(x,y)\in\overline{\Omega}, Assume that

b1​(x,y)≥β1≥0,b2​(x,y)≥β2≥0,c​(x,y)+12​∇⋅𝐛​(x,y)≥γ≥0.\displaystyle\begin{aligned} &b_{1}(x,y)\geq\beta_{1}\geq 0,&&b_{2}(x,y)\geq\beta_{2}\geq 0,&&c(x,y)+\frac{1}{2}\nabla\cdot\mathbf{b}(x,y)\geq\gamma\geq 0.\end{aligned} (1.3)

where β1\beta_{1}, β2\beta_{2} and γ\gamma are some positive constants. Assumption (1.3) makes problem (1.1)-(1.2) has a unique solution in H2​(Ω)∪H01​(Ω)H^{2}(\Omega)\cup H_{0}^{1}(\Omega) for all f∈L2​(Ω)f\in L^{2}(\Omega), see details in [17].

Singularly perturbed problems are one of the important topics in scientific computation. It is well known that the solution of the boundary value problem usually has layers, which are thin regions where the solution or its derivatives change rapidly, due to the diffusion coefficient is very small. In order to resolve the difficulty, numerical stabilization techniques have been developed, which can be divided into fitted operator methods and fitted mesh methods. One of the effective methods of solving singularly perturbed problems is to use layer-adapted meshes. Boundary layers can be resolved by designing layer-adapted meshes if we know a prior knowledge of the layers structure. Commonly used layer-adapted meshes for solving singularly perturbed problems include Bakhvalov-type meshes and Shishkin-type meshes. Bakhvalov mesh is proposed for the first time in [2], its application needs a nonlinear equation which cannot be solved explicitly. In order to avoid this difficulty, meshes that arise from an approximation of Bakhvalov’s mesh generating function are called Bakhvalov-type meshes, which are one of the most popular layer-adapted meshes, see details in [12]. Another piecewise equidistant mesh proposed by Shishkin in [19], but a logarithmic factor will be present in the error bounds when one uses a Shishkin-type mesh. Therefore, Bakhvalov-type meshes have better numerical performance than Shishkin-type meshes in general. Even if the layer-adapted meshe is used, the numerical solution of the convection-dominated problem still has some oscillation, as detailed in [4]. Additional stabilization is added to the numerical scheme to solve these oscillatory behaviour, examples for singularly perturbed onvection-diffusion problem, such as the streamline-diffusion finite element method [13, 14], the classical finite difference method of up-winding flavor [1, 6, 11, 20] and the discontinuous Galerkin methods [7, 8, 18, 30, 37, 35].

In this paper, we consider the WG method to solve the singularly perturbed convection-diffusion boundary value problem on Bakhvalov-type mesh. The WG method proves to be an effective numerical technique for the partial differential equations(PDEs). The main idea of this method is that the classical derivative is replaced by weak derivative, which allows the use of discontinuous functions in numerical schemes with parameter independent stabilizers. The initial proposal for its application in solving second-order elliptic problems was made by Junping Wang and Xiu Ye in [25]. The WG method has been applied to all kinds of problems including Stokes equations [26, 27, 29], Maxwell’s equations [16], Brinkman equations [15, 28, 31], fractional time convection-diffusion problems [21] and so on. For singular perturbed value problems, the WG method has also yielded some results, such as [9, 23, 24, 22, 34, 36, 33, 3]. The main purpose of this paper is to present optimal order uniform convergence in the energy norm on Bakhvalov-type mesh for convection-dominated problems in 2D.

This paper is organized as follows. In Section 2, we describe the assumptions and introduce a Bakhvalov-type mesh. In Section 3, we introduce the definition of weak operator, the WG scheme and some properties of projection operator involved. In Section 4, we provide convergence analysis. In Section 5, the numerical results verify the correctness of our theory.

2 Assumption and Partition

In this section, we will introduce the construction of the Bakhvalov-type mesh and the decomposition of the solution uu. As in [10] we shall introduce the following assumptions which describes the structure of uu.

Assumption 2.1.

For analysis, the solution uu of the equation (1.1) can can be decomposed as follows

u=S+E1+E2+E12.\displaystyle u=S+E_{1}+E_{2}+E_{12}.

where SS represents the smooth part, E1E_{1} and E2E_{2} corresponds to boundary layer components, E12E_{12} is corner layer part. Then, for 0≤i+j≤k+10\leq i+j\leq k+1, there exists a constant CC such taht

|∂i+jS∂xi​∂yj|≤C,\displaystyle\left|\frac{\partial^{i+j}S}{\partial x^{i}\partial y^{j}}\right|\leq C, |∂i+jE1∂xi​∂yj|≤C​ε−j​e−β2​yε,\displaystyle\left|\frac{\partial^{i+j}E_{1}}{\partial x^{i}\partial y^{j}}\right|\leq C\varepsilon^{-j}e^{-\frac{\beta_{2}y}{\varepsilon}}, (2.1)
|∂i+jE2∂xi​∂yj|≤C​ε−i​e−β1​xε,\displaystyle\left|\frac{\partial^{i+j}E_{2}}{\partial x^{i}\partial y^{j}}\right|\leq C\varepsilon^{-i}e^{-\frac{\beta_{1}x}{\varepsilon}}, |∂i+jE12∂xi​∂yj|≤C​ε−(i+j)​e−β1​x+β2​yε.\displaystyle\left|\frac{\partial^{i+j}E_{12}}{\partial x^{i}\partial y^{j}}\right|\leq C\varepsilon^{-(i+j)}e^{-\frac{\beta_{1}x+\beta_{2}y}{\varepsilon}}. (2.2)

For the convection-diffusion problem (1.1)-(1.2), we consider the following Bakhvalov-type mesh introduced in [12], which is defined by

xi={−σ​εβ1​ln⁡(1−2​(1−ε)​i/N),for​i=0,⋯,N/21−(1−xN2)​2​(N−i)/N,for​i=N2+1,⋯,Nyj={−σ​εβ1​ln⁡(1−2​(1−ε)​j/N),for​j=0,⋯,N/21−(1−yN2)​2​(N−j)/N,for​j=N2+1,⋯,N\displaystyle\begin{aligned} x_{i}=\begin{cases}-\frac{\sigma\varepsilon}{\beta_{1}}\ln(1-2(1-\varepsilon)i/N),\quad\text{for}~{}i=0,\cdots,N/2\\ 1-(1-x_{\frac{N}{2}})2(N-i)/N,\quad\text{for}~{}i=\frac{N}{2}+1,\cdots,N\end{cases}\\ y_{j}=\begin{cases}-\frac{\sigma\varepsilon}{\beta_{1}}\ln(1-2(1-\varepsilon)j/N),\quad\text{for}~{}j=0,\cdots,N/2\\ 1-(1-y_{\frac{N}{2}})2(N-j)/N,\quad\text{for}~{}j=\frac{N}{2}+1,\cdots,N\end{cases}\end{aligned} (2.3)

where N≥4N\geq 4 is an positive integer and σ≥k+1\sigma\geq k+1, see Figure 1. Transition points are xN/2x_{N/2} and yN/2y_{N/2}, indicating a shift in mesh from coarse to fine. Then, we get a rectangulation mesh denoted as 𝒯N\mathcal{T}_{N}. For the sake of briefness, set

Ω12=:[x0,xN/2−1]×[y0,yN/2−1],\displaystyle\Omega_{12}=:[x_{0},x_{N/2-1}]\times[y_{0},y_{N/2-1}], Ω1=:[xN/2−1,xN]×[y0,yN/2−1],\displaystyle\Omega_{1}=:[x_{N/2-1},x_{N}]\times[y_{0},y_{N/2-1}],
Ω2=:[x0,xN/2−1]×[yN/2−1,yN],\displaystyle\Omega_{2}=:[x_{0},x_{N/2-1}]\times[y_{N/2-1},y_{N}], Ω0=:[xN/2−1,xN]×[yN/2−1,yN].\displaystyle\Omega_{0}=:[x_{N/2-1},x_{N}]\times[y_{N/2-1},y_{N}].
Refer to caption
Figure 1: A Bakhvalov-type mesh with N=8N=8 and dissection of Ω\Omega.

Let hx,i=xi−xi−1h_{x,i}=x_{i}-x_{i-1} and hy,j=yj−yj−1h_{y,j}=y_{j}-y_{j-1}, we omit the subscript xx or yy if there is no confusion. According to [32], we have the following lemma which introduce some important properties of Bakhvalov-type mesh.

Lemma 2.1.

In this paper, suppose that ε≤N−1\varepsilon\leq N^{-1}, then for Bakhvalov-type mesh (2.3), one can obtain the properties

C1​ε​N−1≤h1≤C2​ε​N−1,\displaystyle C_{1}\varepsilon N^{-1}\leq h_{1}\leq C_{2}\varepsilon N^{-1}, (2.4)
h1≤h2≤⋯≤hN/2−1,\displaystyle h_{1}\leq h_{2}\leq\cdots\leq h_{N/2-1}, (2.5)
14​σ​ε≤hN/2−1≤σ​ε,\displaystyle\frac{1}{4}\sigma\varepsilon\leq h_{N/2-1}\leq\sigma\varepsilon, (2.6)
12​σ​ε≤hN/2≤2​σ​N−1,\displaystyle\frac{1}{2}\sigma\varepsilon\leq h_{N/2}\leq 2\sigma N^{-1}, (2.7)
N−1≤hi≤2​N−1,N/2+1≤i≤N,\displaystyle N^{-1}\leq h_{i}\leq 2N^{-1},\quad N/2+1\leq i\leq N, (2.8)
C3​σ​ε​ln⁡N≤xN/2≤C4​σ​ln⁡N,XN/2≥C​σ​ε​|ln⁡ε|,\displaystyle C_{3}\sigma\varepsilon\ln N\leq x_{N/2}\leq C_{4}\sigma\ln N,\quad X_{N/2}\geq C\sigma\varepsilon|\ln\varepsilon|, (2.9)
hiρ​e−β1​x/ε≤C​ερ​N−ρ,1≤i≤N/2−1,0≤ρ≤σ.\displaystyle h_{i}^{\rho}e^{-\beta_{1}x/\varepsilon}\leq C\varepsilon^{\rho}N^{-\rho},\quad 1\leq i\leq N/2-1,0\leq\rho\leq\sigma. (2.10)

3 WG scheme

In this section, we introduce the notions of WG method. Let T∈𝒯NT\in\mathcal{T}_{N} be any element with boundary ∂T\partial T. We introduce a weak function v={v0,vb}v=\{v_{0},v_{b}\} on the element TT, where v0∈L2​(T)v_{0}\in L^{2}(T), and vb∈L2​(∂T)v_{b}\in L^{2}(\partial T). Note that vbv_{b} has a single value on each edge ee. Additionally, component vbv_{b} may not necessarily be the same as the trace of v0v_{0} on ∂T\partial T.

For any integer k≥1k\geq 1, we introduce the space of weak functions

VN={v={v0,vb}:v0∈ℚk​(T),vb∈ℙk​(e),e∈∂T,∀T∈𝒯N},V_{N}=\left\{v=\{v_{0},v_{b}\}:v_{0}\in\mathbb{Q}_{k}(T),v_{b}\in\mathbb{P}_{k}(e),e\in\partial T,\forall T\in\mathcal{T}_{N}\right\},

where ℚk\mathbb{Q}_{k} is the space of polynomials which are of degree not exceeding kk with respect to each one of the variables xx and yy. Let VN0V_{N}^{0} represent the subspace of VhV_{h} defined by

VN0={v∈Vh,vb|e=0,e⊂∂T∩∂Ω}.V_{N}^{0}=\left\{v\in V_{h},v_{b}|_{e}=0,e\subset\partial T\cap\partial\Omega\right\}.
Definition 3.1.

For any v∈VNv\in V_{N}, a discrete weak gradient ∇w\nabla_{w} is defined on TT as a unique polynomial ∇wv∈[ℚk​(T)]2\nabla_{w}v\in[\mathbb{Q}_{k}(T)]^{2} satisfying:

(∇wv,𝐪)T=−(v0,∇⋅𝐪)T+⟨vb,𝐪⋅𝐧⟩∂T,∀𝐪∈[ℚk​(T)]2.\displaystyle\left(\nabla_{w}v,\mathbf{q}\right)_{T}=-\left(v_{0},\nabla\cdot\mathbf{q}\right)_{T}+\left\langle v_{b},\mathbf{q}\cdot\mathbf{n}\right\rangle_{\partial T},\quad\forall\mathbf{q}\in\left[\mathbb{Q}_{k}(T)\right]^{2}. (3.1)
Definition 3.2.

For any v∈VNv\in V_{N}, a discrete weak convection divergence ∇wbv∈ℚk​(T)\nabla_{w}^{b}v\in\mathbb{Q}_{k}(T) related to 𝐛\mathbf{b} is defined on TT as a unique polynomial satisfyings

(∇w,kbv,τ)T=−(v0,∇⋅(𝐛​φ))T+⟨vb,𝐛⋅𝐧​φ⟩∂T,∀φ∈ℚk​(T).\displaystyle\left(\nabla_{w,k}^{b}v,\tau\right)_{T}=-\left(v_{0},\nabla\cdot(\mathbf{b}\varphi)\right)_{T}+\left\langle v_{b},\mathbf{b}\cdot\mathbf{n}\varphi\right\rangle_{\partial T},\quad\forall\varphi\in\mathbb{Q}_{k}(T). (3.2)

The following notations are often used

(φ,ϕ)𝒯N=∑T∈𝒯N(φ,ϕ)T,\displaystyle(\varphi,\phi)_{\mathcal{T}_{N}}=\sum_{T\in\mathcal{T}_{N}}(\varphi,\phi)_{T}, ⟨φ,ϕ⟩∂𝒯N=∑T∈𝒯N⟨φ,ϕ⟩∂T,\displaystyle\langle\varphi,\phi\rangle_{\partial\mathcal{T}_{N}}=\sum_{T\in\mathcal{T}_{N}}\langle\varphi,\phi\rangle_{\partial T}, ‖φ‖𝒯N2=∑T∈𝒯N‖φ‖T2,\displaystyle\|\varphi\|_{\mathcal{T}_{N}}^{2}=\sum_{T\in\mathcal{T}_{N}}\|\varphi\|_{T}^{2},

We define ∂Tx\partial T_{x} and ∂Ty\partial T_{y} as the sets of element edges that are parallel to the xx and yy axes, respectively. Denote ∂+T={(x,y)∈∂T|𝐛​(x,y)⋅𝐧​(x,y)≤0}\partial_{+}T=\{(x,y)\in\partial T|\mathbf{b}(x,y)\cdot\mathbf{n}(x,y)\leq 0\}, and STS_{T} is the penalization parameter given by

ϑT={N,if​T∈Ω0,ε​hy−1,if​T∈Ω\Ω0,on​∂Tx,ε​hx−1,if​T∈Ω\Ω0,on​∂Ty.\displaystyle\vartheta_{T}=\begin{cases}N,&\text{if}~{}T\in\Omega_{0},\\ \varepsilon h_{y}^{-1},&\text{if}~{}T\in\Omega\backslash\Omega_{0},\text{on}~{}\partial T_{x},\\ \varepsilon h_{x}^{-1},&\text{if}~{}T\in\Omega\backslash\Omega_{0},\text{on}~{}\partial T_{y}.\end{cases}

Then, a WG algorithm is proposed in the following.

Find approximate solution uN={u0,ub}∈VN0u_{N}=\{u_{0},u_{b}\}\in V_{N}^{0} satisfying

A​(uN,v)=(f,v0),∀v∈VN0.\displaystyle A(u_{N},v)=(f,v_{0}),\quad\forall v\in V_{N}^{0}. (3.3)

where

A​(uN,v)=\displaystyle A(u_{N},v)= Ad​(uN,v)+Ac​(uN,v),\displaystyle A_{d}(u_{N},v)+A_{c}(u_{N},v),

and

Ad​(uN,v)=\displaystyle A_{d}(u_{N},v)= (∇wuN,∇wv)𝒯N+sd​(uN,v),\displaystyle(\nabla_{w}u_{N},\nabla_{w}v)_{\mathcal{T}_{N}}+s_{d}\left(u_{N},v\right),
Ac​(uN,v)=\displaystyle A_{c}(u_{N},v)= (∇wbuN,v0)𝒯N+(c​u0,v0)𝒯N+sc​(uN,v)\displaystyle(\nabla_{w}^{b}u_{N},v_{0})_{\mathcal{T}_{N}}+(cu_{0},v_{0})_{\mathcal{T}_{N}}+s_{c}\left(u_{N},v\right)
sd​(uN,v)=\displaystyle s_{d}(u_{N},v)= ∑i=x,y∑T∈𝒯NϑT​⟨u0−ub,v0−vb⟩∂Ti,\displaystyle\sum_{i=x,y}\sum_{T\in\mathcal{T}_{N}}\vartheta_{T}\left\langle u_{0}-u_{b},v_{0}-v_{b}\right\rangle_{\partial T_{i}},
sc​(uN,v)=\displaystyle s_{c}(u_{N},v)= ∑T∈𝒯Ne∈∂+T⟨−𝐛⋅𝐧​(u0−ub),v0−vb⟩e,\displaystyle\sum_{\begin{subarray}{c}T\in\mathcal{T}_{N}\\ e\in\partial_{+}T\end{subarray}}\left\langle-\mathbf{b}\cdot\mathbf{n}(u_{0}-u_{b}),v_{0}-v_{b}\right\rangle_{e},
Definition 3.3.

From the bilinear form in (3.3) , for all v∈VNv\in V_{N} we can derive an energy norm defined by

‖|v|‖2=Ad​(v,v)+‖v0‖𝒯h+‖|𝐛⋅𝐧|12​v0−vb‖∂𝒯h,\displaystyle{|||}v{|||}^{2}=A_{d}(v,v)+\|v_{0}\|_{\mathcal{T}_{h}}+\||\mathbf{b}\cdot\mathbf{n}|^{\frac{1}{2}}v_{0}-v_{b}\|_{\partial{\mathcal{T}_{h}}}, (3.4)
Lemma 3.1.

There exists a positive constant α\alpha, independent of ε\varepsilon, such that for v∈VN0v\in V_{N}^{0}

A​(v,v)≥α​‖|v|‖2,A(v,v)\geq\alpha{|||}v{|||}^{2}, (3.5)

where α=min⁡{γ,12}\alpha=\min\{\gamma,\frac{1}{2}\}.

Proof.

For any v∈VN0v\in V_{N}^{0}, it follows from the definition of the weak divergence (3.2) that

(∇wbv,v0)𝒯N=(v0,∇⋅(𝐛​v0))𝒯N+⟨vb,𝐛⋅𝐧​v0⟩∂𝒯N=−12​(v0,∇⋅(𝐛​v0))𝒯N−12​(v0,∇⋅(𝐛​v0))𝒯N+⟨vb,𝐛⋅𝐧​v0⟩∂𝒯N=−12​(v0,∇⋅(𝐛​v0))𝒯N+12​(𝐛⋅∇v0,v0)𝒯N−12​⟨v0,𝐛⋅𝐧​v0⟩∂𝒯N+⟨vb,𝐛⋅𝐧​v0⟩∂𝒯N=−12​(v0,∇⋅𝐛​v0)𝒯N−12​⟨v0−vb,𝐛⋅𝐧​v0⟩∂𝒯N+12​⟨vb,𝐛⋅𝐧​(v0−vb)⟩∂𝒯N=−12​(v0,∇⋅𝐛​v0)𝒯N−12​⟨𝐛⋅𝐧​(v0−vb),v0−vb⟩∂𝒯N\displaystyle\begin{aligned} (\nabla_{w}^{b}v,v_{0})_{\mathcal{T}_{N}}&=(v_{0},\nabla\cdot(\mathbf{b}v_{0}))_{\mathcal{T}_{N}}+\langle v_{b},\mathbf{b}\cdot\mathbf{n}v_{0}\rangle_{\partial\mathcal{T}_{N}}\\ &=-\frac{1}{2}(v_{0},\nabla\cdot(\mathbf{b}v_{0}))_{\mathcal{T}_{N}}-\frac{1}{2}(v_{0},\nabla\cdot(\mathbf{b}v_{0}))_{\mathcal{T}_{N}}+\langle v_{b},\mathbf{b}\cdot\mathbf{n}v_{0}\rangle_{\partial\mathcal{T}_{N}}\\ &=-\frac{1}{2}(v_{0},\nabla\cdot(\mathbf{b}v_{0}))_{\mathcal{T}_{N}}+\frac{1}{2}(\mathbf{b}\cdot\nabla v_{0},v_{0})_{\mathcal{T}_{N}}-\frac{1}{2}\langle v_{0},\mathbf{b}\cdot\mathbf{n}v_{0}\rangle_{\partial\mathcal{T}_{N}}+\langle v_{b},\mathbf{b}\cdot\mathbf{n}v_{0}\rangle_{\partial\mathcal{T}_{N}}\\ &=-\frac{1}{2}(v_{0},\nabla\cdot\mathbf{b}v_{0})_{\mathcal{T}_{N}}-\frac{1}{2}\langle v_{0}-v_{b},\mathbf{b}\cdot\mathbf{n}v_{0}\rangle_{\partial\mathcal{T}_{N}}+\frac{1}{2}\langle v_{b},\mathbf{b}\cdot\mathbf{n}(v_{0}-v_{b})\rangle_{\partial\mathcal{T}_{N}}\\ &=-\frac{1}{2}(v_{0},\nabla\cdot\mathbf{b}v_{0})_{\mathcal{T}_{N}}-\frac{1}{2}\langle\mathbf{b}\cdot\mathbf{n}(v_{0}-v_{b}),v_{0}-v_{b}\rangle_{\partial\mathcal{T}_{N}}\end{aligned} (3.6)

Then, with together (1.3) and (3.6) we have

Ac​(v,v)\displaystyle A_{c}(v,v) =−(∇wbv,v0)𝒯N+(c​v0,v0)𝒯N+⟨−𝐛⋅𝐧​(v0−vb),v0−vb⟩∂+𝒯N\displaystyle=-(\nabla_{w}^{b}v,v_{0})_{\mathcal{T}_{N}}+(cv_{0},v_{0})_{\mathcal{T}_{N}}+\langle-\mathbf{b}\cdot\mathbf{n}(v_{0}-v_{b}),v_{0}-v_{b}\rangle_{\partial_{+}\mathcal{T}_{N}}
=((c+12​∇⋅𝐛)​v0,v0)𝒯N+12​⟨𝐛⋅𝐧​(v0−vb),v0−vb⟩∂𝒯N+⟨−𝐛⋅𝐧​(v0−vb),v0−vb⟩∂+𝒯N\displaystyle=\left((c+\frac{1}{2}\nabla\cdot\mathbf{b})v_{0},v_{0}\right)_{\mathcal{T}_{N}}+\frac{1}{2}\langle\mathbf{b}\cdot\mathbf{n}(v_{0}-v_{b}),v_{0}-v_{b}\rangle_{\partial\mathcal{T}_{N}}+\langle-\mathbf{b}\cdot\mathbf{n}(v_{0}-v_{b}),v_{0}-v_{b}\rangle_{\partial_{+}\mathcal{T}_{N}}
≥(γ​v0,v0)𝒯N+12​⟨−𝐛⋅𝐧​(v0−vb),v0−vb⟩∂+𝒯N+12​⟨𝐛⋅𝐧​(v0−vb),v0−vb⟩∂𝒯N\∂+𝒯N\displaystyle\geq(\gamma v_{0},v_{0})_{\mathcal{T}_{N}}+\frac{1}{2}\langle-\mathbf{b}\cdot\mathbf{n}(v_{0}-v_{b}),v_{0}-v_{b}\rangle_{\partial_{+}\mathcal{T}_{N}}+\frac{1}{2}\langle\mathbf{b}\cdot\mathbf{n}(v_{0}-v_{b}),v_{0}-v_{b}\rangle_{\partial\mathcal{T}_{N}\backslash\partial_{+}\mathcal{T}_{N}}
≥(γ​v0,v0)𝒯N+12​⟨|𝐛⋅𝐧|​(v0−vb),v0−vb⟩∂𝒯N\displaystyle\geq(\gamma v_{0},v_{0})_{\mathcal{T}_{N}}+\frac{1}{2}\langle|\mathbf{b}\cdot\mathbf{n}|(v_{0}-v_{b}),v_{0}-v_{b}\rangle_{\partial\mathcal{T}_{N}}

The above inequality and Ad​(v,v)=a​(v,v)+sd​(v,v)A_{d}(v,v)=a(v,v)+s_{d}(v,v) yield

A​(v,v)≥α​‖|v|‖2,\displaystyle A(v,v)\geq\alpha{|||}v{|||}^{2},

where α=min⁡{γ,12}\alpha=\min\{\gamma,\frac{1}{2}\}. The lemma is proved. ∎

Now, we introduce some L2L^{2} projection operators for each element T∈𝒯NT\in\mathcal{T}_{N} detailed as follows. Let operator 𝒬0\mathcal{Q}_{0} projects onto ℚk​(T)\mathbb{Q}_{k}(T). For each edge e∈∂Te\in\partial T, consider the operator 𝒬b\mathcal{Q}_{b} onto ℙk​(e)\mathbb{P}_{k}(e). Finally, we define a projection operator 𝒬N​u={𝒬0​u,𝒬b​u}\mathcal{Q}_{N}u=\left\{\mathcal{Q}_{0}u,\mathcal{Q}_{b}u\right\} for solution uu onto the space VNV_{N} on each element TT. Additionally, denote by 𝐐N\mathbf{Q}_{N} the projection operator onto [ℚk​(T)]2[\mathbb{Q}_{k}(T)]^{2}. The following lemma is commutative property of weak gradient.

Lemma 3.2.

On each element T∈𝒯NT\in\mathcal{T}_{N}, for any v∈H2​(T)v\in H^{2}(T),

∇w(𝒬N​u)=𝐐N​(∇u).\displaystyle\nabla_{w}(\mathcal{Q}_{N}u)=\mathbf{Q}_{N}(\nabla u). (3.7)
Proof.

For any 𝐪∈[ℚk​(T)]2\mathbf{q}\in[\mathbb{Q}_{k}(T)]^{2}, from the weak gradient definition (3.1), we get

(∇w𝒬N​u,𝐪)T\displaystyle\left(\nabla_{w}\mathcal{Q}_{N}u,\mathbf{q}\right)_{T} =−(𝒬0​u,∇⋅𝐪)T+⟨𝒬b​u,𝐪⋅𝐧⟩∂T\displaystyle=-\left(\mathcal{Q}_{0}u,\nabla\cdot\mathbf{q}\right)_{T}+\left\langle\mathcal{Q}_{b}u,\mathbf{q}\cdot\mathbf{n}\right\rangle_{\partial T}
=−(u,∇⋅𝐪)T+⟨u,𝐪⋅𝐧⟩∂T\displaystyle=-\left(u,\nabla\cdot\mathbf{q}\right)_{T}+\left\langle u,\mathbf{q}\cdot\mathbf{n}\right\rangle_{\partial T}
=(∇u,𝐪)T\displaystyle=\left(\nabla u,\mathbf{q}\right)_{T}
=(𝐐N​∇u,𝐪)T.\displaystyle=\left(\mathbf{Q}_{N}\nabla u,\mathbf{q}\right)_{T}.

This corresponds to equality (3.7). ∎

4 Error estimate

In this section, we will obtain an error estimate for the WG finite element approximation uNu_{N}. We next derive the following error equation that the error satisfies.

Lemma 4.1.

Suppose that uu is the solutions of (1.1)-(1.2) and uNu_{N} is the solutions of (3.3). Denote eN=𝒬N​u−uNe_{N}=\mathcal{Q}_{N}u-u_{N}. Then for any v∈VN0v\in V_{N}^{0}, we have

A​(eN,v)=l1​(u,v)−l2​(u,v)−l3​(u,v)+sd​(𝒬N​u,v)+sc​(𝒬N​u,v),\displaystyle A(e_{N},v)=l_{1}(u,v)-l_{2}(u,v)-l_{3}(u,v)+s_{d}(\mathcal{Q}_{N}u,v)+s_{c}(\mathcal{Q}_{N}u,v), (4.1)

where

l1​(u,v)=(u−𝒬0​u,∇⋅(𝐛​v0))𝒯h,\displaystyle l_{1}(u,v)=\left(u-\mathcal{Q}_{0}u,\nabla\cdot(\mathbf{b}v_{0})\right)_{\mathcal{T}_{h}},
l2​(u,v)=⟨u−𝒬b​u,𝐛⋅𝐧​(v0−vb)⟩∂𝒯h,\displaystyle l_{2}(u,v)=\left\langle u-\mathcal{Q}_{b}u,\mathbf{b}\cdot\mathbf{n}(v_{0}-v_{b})\right\rangle_{\partial\mathcal{T}_{h}},
l3​(u,v)=ε​⟨(∇u−𝐐h​∇u)⋅𝐧,v0−vb⟩∂𝒯h.\displaystyle l_{3}(u,v)=\varepsilon\left\langle\left(\nabla u-\mathbf{Q}_{h}\nabla u\right)\cdot\mathbf{n},v_{0}-v_{b}\right\rangle_{\partial\mathcal{T}_{h}}.
Proof.

Using (3.1), (3.7) and integration by parts, we obtain

(∇w𝒬N​u,∇wv)𝒯N\displaystyle\left(\nabla_{w}\mathcal{Q}_{N}u,\nabla_{w}v\right)_{\mathcal{T}_{N}} =(𝐐N​∇u,∇wv)𝒯N\displaystyle=\left(\mathbf{Q}_{N}\nabla u,\nabla_{w}v\right)_{\mathcal{T}_{N}}
=−(v0,∇⋅(𝐐N​∇u))𝒯N+⟨vb,(𝐐N​∇u)⋅𝐧⟩∂𝒯N\displaystyle=-\left(v_{0},\nabla\cdot(\mathbf{Q}_{N}\nabla u)\right)_{\mathcal{T}_{N}}+\left\langle v_{b},(\mathbf{Q}_{N}\nabla u)\cdot\mathbf{n}\right\rangle_{\partial\mathcal{T}_{N}}
=(∇v0,𝐐N​∇u)𝒯N−⟨v0−vb,(𝐐N​∇u)⋅𝐧⟩∂𝒯N\displaystyle=\left(\nabla v_{0},\mathbf{Q}_{N}\nabla u\right)_{\mathcal{T}_{N}}-\left\langle v_{0}-v_{b},(\mathbf{Q}_{N}\nabla u)\cdot\mathbf{n}\right\rangle_{\partial\mathcal{T}_{N}}
=(∇u,∇v0)𝒯N−⟨v0−vb,(𝐐N​∇u)⋅𝐧⟩∂𝒯N\displaystyle=\left(\nabla u,\nabla v_{0}\right)_{\mathcal{T}_{N}}-\left\langle v_{0}-v_{b},(\mathbf{Q}_{N}\nabla u)\cdot\mathbf{n}\right\rangle_{\partial\mathcal{T}_{N}}
=(−Δ​u,v0)𝒯N+⟨∇u⋅𝐧,v0⟩∂𝒯N−⟨v0−vb,(𝐐N​∇u)⋅𝐧⟩∂𝒯N−⟨∇u⋅𝐧,vb⟩∂𝒯N,\displaystyle=(-\Delta u,v_{0})_{\mathcal{T}_{N}}+\langle\nabla u\cdot\mathbf{n},v_{0}\rangle_{\partial\mathcal{T}_{N}}-\left\langle v_{0}-v_{b},(\mathbf{Q}_{N}\nabla u)\cdot\mathbf{n}\right\rangle_{\partial\mathcal{T}_{N}}-\langle\nabla u\cdot\mathbf{n},v_{b}\rangle_{\partial\mathcal{T}_{N}},
=(−Δ​u,v0)𝒯N−⟨(𝐐N−∇u)⋅𝐧,v0−vb⟩∂𝒯N\displaystyle=(-\Delta u,v_{0})_{\mathcal{T}_{N}}-\left\langle(\mathbf{Q}_{N}-\nabla u)\cdot\mathbf{n},v_{0}-v_{b}\right\rangle_{\partial\mathcal{T}_{N}}

where we have used the fact that ⟨∇u⋅𝐧,vb⟩∂𝒯N=0\langle\nabla u\cdot\mathbf{n},v_{b}\rangle_{\partial\mathcal{T}_{N}}=0. Similarly, from (3.2) and integration by parts one has

−(∇wb𝒬N​u,v0)𝒯N\displaystyle-(\nabla_{w}^{b}\mathcal{Q}_{N}u,v_{0})_{\mathcal{T}_{N}} =(𝒬0​u,∇⋅(𝐛​v0))𝒯N−⟨𝒬b​u,𝐛⋅𝐧​v0⟩∂𝒯N\displaystyle=(\mathcal{Q}_{0}u,\nabla\cdot(\mathbf{b}v_{0}))_{\mathcal{T}_{N}}-\langle\mathcal{Q}_{b}u,\mathbf{b}\cdot\mathbf{n}v_{0}\rangle_{\partial\mathcal{T}_{N}}
=(𝒬0​u−u,∇⋅(𝐛​v0))𝒯N+(u,∇⋅(𝐛​v0))𝒯N−⟨𝒬b​u,𝐛⋅𝐧​v0⟩∂𝒯N\displaystyle=(\mathcal{Q}_{0}u-u,\nabla\cdot(\mathbf{b}v_{0}))_{\mathcal{T}_{N}}+(u,\nabla\cdot(\mathbf{b}v_{0}))_{\mathcal{T}_{N}}-\langle\mathcal{Q}_{b}u,\mathbf{b}\cdot\mathbf{n}v_{0}\rangle_{\partial\mathcal{T}_{N}}
=(𝒬0​u−u,∇⋅(𝐛​v0))𝒯N−(𝐛⋅∇u,v0)𝒯N−⟨𝒬b​u−u,𝐛⋅𝐧​v0⟩∂𝒯N\displaystyle=(\mathcal{Q}_{0}u-u,\nabla\cdot(\mathbf{b}v_{0}))_{\mathcal{T}_{N}}-(\mathbf{b}\cdot\nabla u,v_{0})_{\mathcal{T}_{N}}-\langle\mathcal{Q}_{b}u-u,\mathbf{b}\cdot\mathbf{n}v_{0}\rangle_{\partial\mathcal{T}_{N}}
=−(𝐛⋅∇u,v0)𝒯N+(𝒬0​u−u,∇⋅(𝐛​v0))𝒯N−⟨𝒬b​u−u,𝐛⋅𝐧​(v0−vb)⟩∂𝒯N\displaystyle=-(\mathbf{b}\cdot\nabla u,v_{0})_{\mathcal{T}_{N}}+(\mathcal{Q}_{0}u-u,\nabla\cdot(\mathbf{b}v_{0}))_{\mathcal{T}_{N}}-\langle\mathcal{Q}_{b}u-u,\mathbf{b}\cdot\mathbf{n}(v_{0}-v_{b})\rangle_{\partial\mathcal{T}_{N}}

where we have used ⟨𝒬b​u−u,𝐛⋅𝐧​vb⟩∂𝒯N=0\langle\mathcal{Q}_{b}u-u,\mathbf{b}\cdot\mathbf{n}v_{b}\rangle_{\partial\mathcal{T}_{N}}=0. Also we have

(c​𝒬N​u,v0)𝒯N=(c​𝒬0​u,v0)𝒯N=(c​u,v0)𝒯N.\displaystyle(c\mathcal{Q}_{N}u,v_{0})_{\mathcal{T}_{N}}=(c\mathcal{Q}_{0}u,v_{0})_{\mathcal{T}_{N}}=(cu,v_{0})_{\mathcal{T}_{N}}.

Notice that

ε​(∇wuN,∇wv)−(∇wbuN,v0)+(c​uN,v0)+sd​(uN,v)+sc​(uN,v)\displaystyle\varepsilon(\nabla_{w}u_{N},\nabla_{w}v)-(\nabla_{w}^{b}u_{N},v_{0})+(cu_{N},v_{0})+s_{d}(u_{N},v)+s_{c}(u_{N},v) =(f,v0),\displaystyle=(f,v_{0}),

and

−ε​(Δ​u,v0)−(𝐛⋅∇u,v0)+(c​u,v0)\displaystyle-\varepsilon(\Delta u,v_{0})-(\mathbf{b}\cdot\nabla u,v_{0})+(cu,v_{0}) =(f,v0).\displaystyle=(f,v_{0}).

Combined with the above equality we yield

ε​(∇wb𝒬N​u,∇wv)−(∇wb𝒬N​u,v0)+(c​𝒬N​u,v0)=(f,v0)+l1​(u,v)−l2​(u,v)−l3​(u,v).\displaystyle\varepsilon(\nabla_{w}^{b}\mathcal{Q}_{N}u,\nabla_{w}v)-(\nabla_{w}^{b}\mathcal{Q}_{N}u,v_{0})+(c\mathcal{Q}_{N}u,v_{0})=(f,v_{0})+l_{1}(u,v)-l_{2}(u,v)-l_{3}(u,v).

By adding sd​(𝒬N​u,v)s_{d}(\mathcal{Q}_{N}u,v) and sc​(𝒬N​u,v)s_{c}(\mathcal{Q}_{N}u,v) to both sides of the above equation, we get

A​(𝒬N​u−uN,v)=l1​(u,v)−l2​(u,v)−l3​(u,v)+sd​(𝒬N​u,v)+sc​(𝒬N​u,v).\displaystyle A(\mathcal{Q}_{N}u-u_{N},v)=l_{1}(u,v)-l_{2}(u,v)-l_{3}(u,v)+s_{d}(\mathcal{Q}_{N}u,v)+s_{c}(\mathcal{Q}_{N}u,v).

The lemma is proved. ∎

Theorem 4.1.

Let σ≥k+1\sigma\geq k+1. Suppose that u∈Hk+1​(Ω)u\in H^{k+1}(\Omega) and uNu_{N} be the solutions of (1.1)-(1.2) and (3.3)(\ref{3.3}), respectively. Then one has

‖|𝒬N​u−uN|‖≤C​N−k.{|||}\mathcal{Q}_{N}u-u_{N}{|||}\leq CN^{-k}. (4.2)
Proof.

Let v=eNv=e_{N} in the error equation (4.1), we derive the following equation

A​(eN,eN)=l1​(u,eN)−l2​(u,eN)−l3​(u,eN)+sd​(𝒬N​u,eN)+sc​(𝒬N​u,eN).\displaystyle A(e_{N},e_{N})=l_{1}(u,e_{N})-l_{2}(u,e_{N})-l_{3}(u,e_{N})+s_{d}(\mathcal{Q}_{N}u,e_{N})+s_{c}(\mathcal{Q}_{N}u,e_{N}).

Using the Cauchy-Schwarz inequality and (A.5), we obtain

l1​(u,eN)=(u−𝒬0​u,∇⋅(𝐛​e0))𝒯h=(𝒬0​u−u,∇⋅𝐛​e0)𝒯h+(𝒬0​u−u,𝐛⋅∇e0)𝒯h≤C​‖𝒬0​u−u‖𝒯h​‖e0‖𝒯h+(𝒬0​u−u,b1​∂xe0+b2​∂ye0)𝒯h≤C​‖𝒬0​u−u‖𝒯h​‖e0‖𝒯h+(𝒬0​u−u,(b1−b1¯)​∂xe0+(b2−b2¯)​∂ye0)𝒯h≤C​‖𝒬0​u−u‖𝒯h​‖e0‖𝒯h≤C​N−(k+1)​‖|eN|‖.\displaystyle\begin{aligned} l_{1}(u,e_{N})&=\left(u-\mathcal{Q}_{0}u,\nabla\cdot(\mathbf{b}e_{0})\right)_{\mathcal{T}_{h}}\\ &=\left(\mathcal{Q}_{0}u-u,\nabla\cdot\mathbf{b}e_{0}\right)_{\mathcal{T}_{h}}+\left(\mathcal{Q}_{0}u-u,\mathbf{b}\cdot\nabla e_{0}\right)_{\mathcal{T}_{h}}\\ &\leq C\left\|\mathcal{Q}_{0}u-u\right\|_{\mathcal{T}_{h}}\left\|e_{0}\right\|_{\mathcal{T}_{h}}+\left(\mathcal{Q}_{0}u-u,b_{1}\partial_{x}e_{0}+b_{2}\partial_{y}e_{0}\right)_{\mathcal{T}_{h}}\\ &\leq C\left\|\mathcal{Q}_{0}u-u\right\|_{\mathcal{T}_{h}}\left\|e_{0}\right\|_{\mathcal{T}_{h}}+\left(\mathcal{Q}_{0}u-u,\left(b_{1}-\overline{b_{1}}\right)\partial_{x}e_{0}+\left(b_{2}-\overline{b_{2}}\right)\partial_{y}e_{0}\right)_{\mathcal{T}_{h}}\\ &\leq C\left\|\mathcal{Q}_{0}u-u\right\|_{\mathcal{T}_{h}}\left\|e_{0}\right\|_{\mathcal{T}_{h}}\\ &\leq CN^{-(k+1)}{|||}e_{N}{|||}.\end{aligned}

where b1¯\overline{b_{1}} and b2¯\overline{b_{2}} are piecewise constant functions whose value are hx−1​∫eb1​𝑑xh_{x}^{-1}\int_{e}b_{1}\,dx and hy−1​∫eb2​𝑑yh_{y}^{-1}\int_{e}b_{2}\,dy, respectively. Similarly, it follows from the Cauchy-Schwarz inequality and (A.6) that

l2​(u,eN)=⟨u−𝒬b​u,𝐛⋅𝐧​(e0−eb)⟩∂𝒯h≤C​∑i=x,y∑T∈𝒯N‖u−𝒬b​u‖∂Ti​‖|𝐛⋅𝐧|12​(e0−eb)‖∂Ti≤C​∑i=x,y∑T∈𝒯N‖u−𝒬0​u‖∂Ti​‖|𝐛⋅𝐧|12​(e0−eb)‖∂Ti≤C​∑i=x,y‖u−𝒬0​u‖∂𝒯N​i​‖|𝐛⋅𝐧|12​(e0−eb)‖𝒯N​i≤C​N−(k+12)​‖|eN|‖.\displaystyle\begin{aligned} l_{2}(u,e_{N})&=\left\langle u-\mathcal{Q}_{b}u,\mathbf{b}\cdot\mathbf{n}(e_{0}-e_{b})\right\rangle_{\partial\mathcal{T}_{h}}\\ &\leq C\sum_{i=x,y}\sum_{T\in\mathcal{T}_{N}}\left\|u-\mathcal{Q}_{b}u\right\|_{\partial T_{i}}\left\|\left|\mathbf{b}\cdot\mathbf{n}\right|^{\frac{1}{2}}(e_{0}-e_{b})\right\|_{\partial T_{i}}\\ &\leq C\sum_{i=x,y}\sum_{T\in\mathcal{T}_{N}}\left\|u-\mathcal{Q}_{0}u\right\|_{\partial T_{i}}\left\|\left|\mathbf{b}\cdot\mathbf{n}\right|^{\frac{1}{2}}(e_{0}-e_{b})\right\|_{\partial T_{i}}\\ &\leq C\sum_{i=x,y}\left\|u-\mathcal{Q}_{0}u\right\|_{\partial\mathcal{T}_{Ni}}\left\|\left|\mathbf{b}\cdot\mathbf{n}\right|^{\frac{1}{2}}(e_{0}-e_{b})\right\|_{\mathcal{T}_{Ni}}\\ &\leq CN^{-(k+\frac{1}{2})}{|||}e_{N}{|||}.\end{aligned}

From the Cauchy-Schwarz inequality and (A.7), we yield

l3​(u,eN)=ε​⟨(∇u−𝐐h​∇u)⋅𝐧,e0−eb⟩∂𝒯h≤C​ε​∑i=x,y∑T∈𝒯N‖∇u−𝐐h​∇u‖∂Ti​‖e0−eb‖∂Ti≤C​∑i=x,y(∑T∈𝒯NθT​‖∇u−𝐐h​∇u‖∂Ti2)12​(∑T∈𝒯NϑT​‖e0−eb‖∂Ti2)12≤C​N−k​‖|eN|‖.\displaystyle\begin{aligned} l_{3}(u,e_{N})&=\varepsilon\left\langle\left(\nabla u-\mathbf{Q}_{h}\nabla u\right)\cdot\mathbf{n},e_{0}-e_{b}\right\rangle_{\partial\mathcal{T}_{h}}\\ &\leq C\varepsilon\sum_{i=x,y}\sum_{T\in\mathcal{T}_{N}}\left\|\nabla u-\mathbf{Q}_{h}\nabla u\right\|_{\partial T_{i}}\left\|e_{0}-e_{b}\right\|_{\partial T_{i}}\\ &\leq C\sum_{i=x,y}\left(\sum_{T\in\mathcal{T}_{N}}\theta_{T}\left\|\nabla u-\mathbf{Q}_{h}\nabla u\right\|_{\partial T_{i}}^{2}\right)^{\frac{1}{2}}\left(\sum_{T\in\mathcal{T}_{N}}\vartheta_{T}\left\|e_{0}-e_{b}\right\|_{\partial T_{i}}^{2}\right)^{\frac{1}{2}}\\ &\leq CN^{-k}{|||}e_{N}{|||}.\end{aligned}

On the other hand, it follows from the definition of stabilization, Cauchy-Schwarz inequality (A.8) and (A.6) that

sd​(𝒬N,eN)\displaystyle s_{d}(\mathcal{Q}_{N},e_{N}) =∑i=x,y∑T∈𝒯NϑT​⟨𝒬0​u−𝒬b​u,e0−eb⟩∂Ti\displaystyle=\sum_{i=x,y}\sum_{T\in\mathcal{T}_{N}}\vartheta_{T}\left\langle\mathcal{Q}_{0}u-\mathcal{Q}_{b}u,e_{0}-e_{b}\right\rangle_{\partial T_{i}}
≤C​∑i=x,y(∑T∈𝒯NϑT​‖𝒬0​u−u‖∂Ti2)12​(∑T∈𝒯NϑT​‖e0−eb‖∂Ti2)12\displaystyle\leq C\sum_{i=x,y}\left(\sum_{T\in\mathcal{T}_{N}}\vartheta_{T}\left\|\mathcal{Q}_{0}u-u\right\|_{\partial T_{i}}^{2}\right)^{\frac{1}{2}}\left(\sum_{T\in\mathcal{T}_{N}}\vartheta_{T}\left\|e_{0}-e_{b}\right\|_{\partial T_{i}}^{2}\right)^{\frac{1}{2}}
≤C​N−k​‖|eN|‖.\displaystyle\leq CN^{-k}{|||}e_{N}{|||}.

and

sc​(𝒬N,eN)\displaystyle s_{c}(\mathcal{Q}_{N},e_{N}) =∑i=x,y∑T∈𝒯N⟨−𝐛⋅𝐧​(𝒬0​u−𝒬b​u),v0−vb⟩∂+Ti\displaystyle=\sum_{i=x,y}\sum_{T\in\mathcal{T}_{N}}\left\langle-\mathbf{b}\cdot\mathbf{n}(\mathcal{Q}_{0}u-\mathcal{Q}_{b}u),v_{0}-v_{b}\right\rangle_{\partial_{+}T_{i}}
≤C​∑i=x,y(∑T∈𝒯N‖𝒬0​u−u‖∂Ti2)12​(∑T∈𝒯N|𝐛⋅𝐧|12​‖e0−eb‖∂Ti2)12\displaystyle\leq C\sum_{i=x,y}\left(\sum_{T\in\mathcal{T}_{N}}\left\|\mathcal{Q}_{0}u-u\right\|_{\partial T_{i}}^{2}\right)^{\frac{1}{2}}\left(\sum_{T\in\mathcal{T}_{N}}\left|\mathbf{b}\cdot\mathbf{n}\right|^{\frac{1}{2}}\left\|e_{0}-e_{b}\right\|_{\partial T_{i}}^{2}\right)^{\frac{1}{2}}
≤C​N−(k+12)​‖|eN|‖.\displaystyle\leq CN^{-(k+\frac{1}{2})}{|||}e_{N}{|||}.

Combining all the estimates above with together (3.5) that

α​‖|eN|‖2\displaystyle\alpha{|||}e_{N}{|||}^{2} ≤A​(eN,eN)\displaystyle\leq A(e_{N},e_{N})
≤|l1​(u,eN)|+|l2​(u,eN)|+|l3​(u,eN)|+|sd​(𝒬N,eN)|+|sc​(𝒬N,eN)|\displaystyle\leq|l_{1}(u,e_{N})|+|l_{2}(u,e_{N})|+|l_{3}(u,e_{N})|+|s_{d}(\mathcal{Q}_{N},e_{N})|+|s_{c}(\mathcal{Q}_{N},e_{N})|
≤C​N−k​‖|eN|‖.\displaystyle\leq CN^{-k}{|||}e_{N}{|||}.

which implies (4.2). This completes the proof of the theorem. ∎

5 Numerical Experiments

In this section we present numerical experiments that support our theoretical results, which solution exhibits typical exponential layer behavior. And we select σ=2​k\sigma=2k for creating the Bakhvalov-type mesh.

Example 5.1.
−ε​Δ​u−(2+2​x−y)​∂xu−(3−x+2​y)​∂yu+u\displaystyle-\varepsilon\Delta u-(2+2x-y)\partial_{x}u-(3-x+2y)\partial_{y}u+u =f,in​Ω=(0,1)2,\displaystyle=f,\quad\text{in}~{}\Omega=(0,1)^{2},
u\displaystyle u =0,on​∂Ω,\displaystyle=0,\quad\text{on}~{}\partial\Omega,

where f​(x,y)f(x,y) is chosen such that the exact solution

u​(x,y)=2​sin⁡(π​x)​(1−e−2​xε)​(1−y)2​(1−e−yε).u(x,y)=2\sin(\pi x)\left(1-e^{-\frac{2x}{\varepsilon}}\right)\left(1-y\right)^{2}\left(1-e^{-\frac{y}{\varepsilon}}\right).

The error ‖|eN|‖{|||}e_{N}{|||} and convergence rates for several different ε\varepsilon are shown in Table 1-2.

Table 1: Example 5.1, k=1
N ε\varepsilon = 1e-6 ε\varepsilon = 1e-7 ε\varepsilon = 1e-8 ε\varepsilon = 1e-9 ε\varepsilon = 1e-10      \bigstrut
8 3.66E-01 0.00 3.84E-01 0.00 3.98E-01 0.00 4.07E-01 0.00 4.14E-01 0.00 \bigstrut
16 1.73E-01 1.08 1.83E-01 1.07 1.91E-01 1.06 1.96E-01 1.05 2.00E-01 1.05 \bigstrut
32 8.24E-02 1.07 8.81E-02 1.05 9.23E-02 1.05 9.54E-02 1.04 9.76E-02 1.03 \bigstrut
64 3.95E-02 1.06 4.27E-02 1.04 4.51E-02 1.03 4.68E-02 1.03 4.80E-02 1.02 \bigstrut
128 1.90E-02 1.06 2.08E-02 1.04 2.21E-02 1.03 2.30E-02 1.02 2.37E-02 1.02 \bigstrut
256 9.14E-03 1.06 1.01E-02 1.04 1.08E-02 1.03 1.14E-02 1.02 1.17E-02 1.01 \bigstrut
Table 2: Example 5.1, k=2
N ε\varepsilon = 1e-6 ε\varepsilon = 1e-7 ε\varepsilon = 1e-8 ε\varepsilon = 1e-9 ε\varepsilon = 1e-10      \bigstrut
8 7.89E-02 0.00 8.27E-02 0.00 8.52E-02 0.00 8.70E-02 0.00 8.85E-02 0.00 \bigstrut
16 1.93E-02 2.03 2.04E-02 2.02 2.11E-02 2.02 2.16E-02 2.01 2.20E-02 2.01 \bigstrut
32 4.70E-03 2.04 5.01E-03 2.02 5.21E-03 2.02 5.35E-03 2.01 5.45E-03 2.01 \bigstrut
64 1.14E-03 2.04 1.23E-03 2.02 1.29E-03 2.02 1.33E-03 2.01 1.35E-03 2.01 \bigstrut
128 2.77E-04 2.05 3.02E-04 2.03 3.18E-04 2.02 3.29E-04 2.01 3.36E-04 2.01 \bigstrut
Example 5.2.
−ε​Δ​u−2​∂xu−3​∂yu+u\displaystyle-\varepsilon\Delta u-2\partial_{x}u-3\partial_{y}u+u =f,in​Ω=(0,1)2,\displaystyle=f,\quad\text{in}~{}\Omega=(0,1)^{2},
u\displaystyle u =0,on​∂Ω,\displaystyle=0,\quad\text{on}~{}\partial\Omega,

where f​(x,y)f(x,y) is chosen such that the exact solution

u​(x,y)=2​sin⁡(1−x)​(1−e−2​xε)​(1−y)2​(1−e−3​yε).u(x,y)=2\sin(1-x)\left(1-e^{-\frac{2x}{\varepsilon}}\right)\left(1-y\right)^{2}\left(1-e^{-\frac{3y}{\varepsilon}}\right).

The error ‖|eN|‖{|||}e_{N}{|||} and convergence rates for several different ε\varepsilon are shown in Table 3-4.

Table 3: Example 5.2, k=1
N ε\varepsilon = 1e-6 ε\varepsilon = 1e-7 ε\varepsilon = 1e-8 ε\varepsilon = 1e-9 ε\varepsilon = 1e-10      \bigstrut
8 8.67E-02 0.00 8.68E-02 0.00 8.69E-02 0.00 8.70E-02 0.00 8.71E-02 0.00 \bigstrut
16 3.53E-02 1.30 3.53E-02 1.30 3.54E-02 1.30 3.54E-02 1.30 3.54E-02 1.30 \bigstrut
32 1.50E-02 1.24 1.50E-02 1.24 1.50E-02 1.24 1.50E-02 1.24 1.50E-02 1.24 \bigstrut
64 6.69E-03 1.16 6.69E-03 1.16 6.69E-03 1.16 6.69E-03 1.16 6.69E-03 1.16 \bigstrut
128 3.14E-03 1.09 3.14E-03 1.09 3.14E-03 1.09 3.14E-03 1.09 3.14E-03 1.09 \bigstrut
256 1.51E-03 1.05 1.51E-03 1.05 1.51E-03 1.05 1.51E-03 1.05 1.51E-03 1.05 \bigstrut
Table 4: Example 5.2, k=2
N ε\varepsilon = 1e-6 ε\varepsilon = 1e-7 ε\varepsilon = 1e-8 ε\varepsilon = 1e-9 ε\varepsilon = 1e-10      \bigstrut
8 1.58E-02 0.00 1.58E-02 0.00 1.58E-02 0.00 1.58E-02 0.00 1.58E-02 0.00 \bigstrut
16 2.91E-03 2.44 2.91E-03 2.44 2.91E-03 2.44 2.91E-03 2.44 2.91E-03 2.44 \bigstrut
32 5.83E-04 2.32 5.83E-04 2.32 5.83E-04 2.32 5.83E-04 2.32 5.83E-04 2.32 \bigstrut
64 1.29E-04 2.17 1.29E-04 2.17 1.29E-04 2.17 1.29E-04 2.17 1.29E-04 2.17 \bigstrut
128 3.05E-05 2.08 3.05E-05 2.08 3.05E-05 2.08 3.05E-05 2.08 3.05E-05 2.08 \bigstrut

Tables 1–4 show our numerical results including the errors in the energy norm and convergence rates for Examples 5.1 and 5.2. These data show uniform convergence of the singular perturbation parameter ε\varepsilon.

Statements and Declarations

Data Availability. The code used in this work will be made available upon request to the authors.

Appendix A  

The goal of this Appendix is to establish some fundamental estimates useful in the error estimate. The following lemmas are employed in the convergence analysis, and readers are directed to [5] for a detailed proof process.

Lemma A.1.

Consider ϕ∈Hk+1​(T)\phi\in H^{k+1}(T) with k≥1k\geq 1. Let 𝒬0​ϕ\mathcal{Q}_{0}\phi denote the L2L^{2}-projection of ϕ\phi onto ℚk​(T)\mathbb{Q}_{k}(T). Then the following inequality estimate holds,

‖ϕ−𝒬0​ϕ‖T≤C​(hxk+1​‖∂xk+1ϕ‖T+hyk+1​‖∂yk+1ϕ‖T)\displaystyle\|\phi-\mathcal{Q}_{0}\phi\|_{T}\leq C\left(h_{x}^{k+1}\left\|\partial_{x}^{k+1}\phi\right\|_{T}+h_{y}^{k+1}\left\|\partial_{y}^{k+1}\phi\right\|_{T}\right) (A.1)
‖ϕ−𝒬0​ϕ‖∂Ti≤C​(hjk+12​‖∂jk+1ϕ‖T+hik+1​hj−12​‖∂ik+1ϕ‖T+hj12​hik​‖∂ik∂jϕ‖T),\displaystyle\|\phi-\mathcal{Q}_{0}\phi\|_{\partial T_{i}}\leq C\left(h_{j}^{k+\frac{1}{2}}\|\partial_{j}^{k+1}\phi\|_{T}+h_{i}^{k+1}h_{j}^{-\frac{1}{2}}\|\partial_{i}^{k+1}\phi\|_{T}+h_{j}^{\frac{1}{2}}h_{i}^{k}\|\partial_{i}^{k}\partial_{j}\phi\|_{T}\right), (A.2)

where i,j∈{x,y}i,j\in\{x,y\}, and i≠ji\neq j.

Lemma A.2.

Let v∈ℚk​(T)v\in\mathbb{Q}_{k}(T) with k≥1k\geq 1 such that the following inequalities holds,

‖v‖∂Ti≤\displaystyle\|v\|_{\partial T_{i}}\leq C​hj−12​‖v‖T,\displaystyle Ch_{j}^{-\frac{1}{2}}\|v\|_{T}, (A.3)
‖∂iv‖T≤\displaystyle\|\partial_{i}v\|_{T}\leq C​hi−1​‖v‖T,\displaystyle Ch_{i}^{-1}\|v\|_{T}, (A.4)

where CC is a constant only depends on kk and i,j∈{x,y}i,j\in\{x,y\}, i≠ji\neq j.

We would like to establish the following estimates which are useful in the convergence analysis for the WG scheme (3.3).

Lemma A.3.

Let k≥1k\geq 1 and u∈Hk+1​(Ω)u\in H^{k+1}(\Omega). There exists a constant CC such that the following estimates hold true,

(∑T∈𝒯N‖u−𝒬0​u‖T2)12≤C​N−(k+1),\displaystyle\left(\sum_{T\in\mathcal{T}_{N}}\left\|u-\mathcal{Q}_{0}u\right\|_{T}^{2}\right)^{\frac{1}{2}}\leq CN^{-(k+1)}, (A.5)
∑i=x,y(∑T∈𝒯N‖u−𝒬0​u‖∂Ti2)12≤C​N−(k+12),\displaystyle\sum_{i=x,y}\left(\sum_{T\in\mathcal{T}_{N}}\left\|u-\mathcal{Q}_{0}u\right\|_{\partial T_{i}}^{2}\right)^{\frac{1}{2}}\leq CN^{-(k+\frac{1}{2})}, (A.6)
∑i=x,y(∑T∈𝒯NθT​‖∇u−𝐐N​∇u‖∂Ti2)12≤C​N−k.\displaystyle\sum_{i=x,y}\left(\sum_{T\in\mathcal{T}_{N}}\theta_{T}\left\|\nabla u-\mathbf{Q}_{N}\nabla u\right\|_{\partial T_{i}}^{2}\right)^{\frac{1}{2}}\leq CN^{-k}. (A.7)
∑i=x,y(∑T∈𝒯NϑT​‖u−𝒬0​u‖∂Ti2)12≤C​N−k,\displaystyle\sum_{i=x,y}\left(\sum_{T\in\mathcal{T}_{N}}\vartheta_{T}\left\|u-\mathcal{Q}_{0}u\right\|_{\partial T_{i}}^{2}\right)^{\frac{1}{2}}\leq CN^{-k}, (A.8)

where

θT={ε2​N−1,if​T∈Ω0,ε​hy,if​T∈Ω\Ω0,on​∂Tx,ε​hx,if​T∈Ω\Ω0,on​∂Ty.\displaystyle\theta_{T}=\begin{cases}\varepsilon^{2}N^{-1},&\text{if}~{}T\in\Omega_{0},\\ \varepsilon h_{y},&\text{if}~{}T\in\Omega\backslash\Omega_{0},\text{on}~{}\partial T_{x},\\ \varepsilon h_{x},&\text{if}~{}T\in\Omega\backslash\Omega_{0},\text{on}~{}\partial T_{y}.\end{cases}
Proof.

Each term in the Assumption2.1 will be considered individually. To derive inequality (A.5), we use (A.1) and Lemma2.1 to obtain

‖S−𝒬0​S‖Ωl\displaystyle\|S-\mathcal{Q}_{0}S\|_{\Omega_{l}} ≤C​(∑T∈Ωl(hx2​(k+1)+hy2​(k+1))⋅hx​hy)12\displaystyle\leq C\left(\sum_{T\in\Omega_{l}}\left(h_{x}^{2(k+1)}+h_{y}^{2(k+1)}\right)\cdot h_{x}h_{y}\right)^{\frac{1}{2}}
≤C​N​(hN2​(k+2))12\displaystyle\leq CN\left(h_{N}^{2(k+2)}\right)^{\frac{1}{2}}
≤C​N−(k+1),\displaystyle\leq CN^{-(k+1)},

Next, considering the boundary layer E1E_{1} in region Ω12∪Ω1\Omega_{12}\cup\Omega_{1}, using (A.1) and Lemma2.1 we have

‖E1−𝒬0​E1‖Ω12∪Ω1\displaystyle\|E_{1}-\mathcal{Q}_{0}E_{1}\|_{\Omega_{12}\cup\Omega_{1}} ≤C​(∑T∈Ω12∪Ω1(hx2​(k+1)​‖∂xk+1E1‖T2+hy2​(k+1)​‖∂yk+1E1‖T2))12\displaystyle\leq C\left(\sum_{T\in\Omega_{12}\cup\Omega_{1}}\left(h_{x}^{2(k+1)}\left\|\partial_{x}^{k+1}E_{1}\right\|_{T}^{2}+h_{y}^{2(k+1)}\left\|\partial_{y}^{k+1}E_{1}\right\|_{T}^{2}\right)\right)^{\frac{1}{2}}
≤C​(∑T∈Ω12∪Ω1(hx2​k+3​hy+ε−2​(k+1)​hx​hy2​k+3)​‖e−β2​yε‖L∞​(T)2)12\displaystyle\leq C\left(\sum_{T\in\Omega_{12}\cup\Omega_{1}}\left(h_{x}^{2k+3}h_{y}+\varepsilon^{-2(k+1)}h_{x}h_{y}^{2k+3}\right)\left\|e^{-\frac{\beta_{2}y}{\varepsilon}}\right\|_{L^{\infty}(T)}^{2}\right)^{\frac{1}{2}}
≤C​N​(hx,N2​(k+3)​ε​N−1+hx,N​hy,N2−1​N−2​(k+1))12\displaystyle\leq CN\left(h_{x,N}^{2(k+3)}\varepsilon N^{-1}+h_{x,N}h_{y,\frac{N}{2}-1}N^{-2(k+1)}\right)^{\frac{1}{2}}
≤C​N​(ε​N−2​(k+2)+ε​N−(2​k+3))12\displaystyle\leq CN\left(\varepsilon N^{-2(k+2)}+\varepsilon N^{-(2k+3)}\right)^{\frac{1}{2}}
≤C​N−(k+1),\displaystyle\leq CN^{-(k+1)},

As for the region Ω2∪Ω0\Omega_{2}\cup\Omega_{0}, we have the following estimate

‖E1−𝒬0​E1‖Ω2∪Ω0\displaystyle\|E_{1}-\mathcal{Q}_{0}E_{1}\|_{\Omega_{2}\cup\Omega_{0}} ≤C​(∑T∈Ω2∪Ω0(‖E1‖T2+‖𝒬0​E1‖T2))12\displaystyle\leq C\left(\sum_{T\in\Omega_{2}\cup\Omega_{0}}\left(\left\|E_{1}\right\|_{T}^{2}+\left\|\mathcal{Q}_{0}E_{1}\right\|_{T}^{2}\right)\right)^{\frac{1}{2}}
≤C​‖E1‖Ω2∪Ω0\displaystyle\leq C\left\|E_{1}\right\|_{\Omega_{2}\cup\Omega_{0}}
≤C​N​(hx,N​hy,N​‖E1‖L∞​(Ω2∪Ω0)2)12\displaystyle\leq CN\left(h_{x,N}h_{y,N}\left\|E_{1}\right\|_{L^{\infty}(\Omega_{2}\cup\Omega_{0})}^{2}\right)^{\frac{1}{2}}
≤C​N−σ\displaystyle\leq CN^{-\sigma}

A similar bound can be readily obtained for E2E_{2}. For the concer layer E12E_{12}, by applying inequalities (A.1) and Lemma2.1, we arrive at

‖E12−𝒬0​E12‖Ω12\displaystyle\|E_{12}-\mathcal{Q}_{0}E_{12}\|_{\Omega_{12}} ≤C​(∑T∈Ω12(hx2​k+3​hy​‖∂xk+1E12‖L∞​(T)2+hx​hy2​k+3​‖∂yk+1E12‖L∞​(T)2))12\displaystyle\leq C\left(\sum_{T\in\Omega_{12}}\left(h_{x}^{2k+3}h_{y}\left\|\partial_{x}^{k+1}E_{12}\right\|_{L^{\infty}(T)}^{2}+h_{x}h_{y}^{2k+3}\left\|\partial_{y}^{k+1}E_{12}\right\|_{L^{\infty}(T)}^{2}\right)\right)^{\frac{1}{2}}
≤C​(∑T∈Ω12ε−2​(k+1)​(hx2​k+3​hy+hx​hy2​k+3)​‖e−β1​x+β2​yε‖L∞​(T)2)12\displaystyle\leq C\left(\sum_{T\in\Omega_{12}}\varepsilon^{-2(k+1)}\left(h_{x}^{2k+3}h_{y}+h_{x}h_{y}^{2k+3}\right)\left\|e^{-\frac{\beta_{1}x+\beta_{2}y}{\varepsilon}}\right\|_{L^{\infty}(T)}^{2}\right)^{\frac{1}{2}}
≤C​N​(ε​N−1​N−2​(k+1)​(hx,N2−1+hy,N2−1))12\displaystyle\leq CN\left(\varepsilon N^{-1}N^{-2(k+1)}\left(h_{x,\frac{N}{2}-1}+h_{y,\frac{N}{2}-1}\right)\right)^{\frac{1}{2}}
≤C​N​(ε2​N−(2​k+3))12\displaystyle\leq CN\left(\varepsilon^{2}N^{-(2k+3)}\right)^{\frac{1}{2}}
≤C​N−(k+32),\displaystyle\leq CN^{-(k+\frac{3}{2})},

Also, we have the estimate in regions Ω1∪Ω0\Omega_{1}\cup\Omega_{0} and Ω2∪Ω0\Omega_{2}\cup\Omega_{0}, as follows

‖E12−𝒬0​E12‖Ω1∪Ω0\displaystyle\|E_{12}-\mathcal{Q}_{0}E_{12}\|_{\Omega_{1}\cup\Omega_{0}} ≤C​(∑T∈Ω1∪Ω0(‖E12‖T2+‖𝒬0​E12‖T2))12\displaystyle\leq C\left(\sum_{T\in\Omega_{1}\cup\Omega_{0}}\left(\left\|E_{12}\right\|_{T}^{2}+\left\|\mathcal{Q}_{0}E_{12}\right\|_{T}^{2}\right)\right)^{\frac{1}{2}}
≤C​‖E12‖Ω1∪Ω0\displaystyle\leq C\left\|E_{12}\right\|_{\Omega_{1}\cup\Omega_{0}}
≤C​N​(hx,N​hy,N​‖E12‖L∞​(Ω1∪Ω0)2)12\displaystyle\leq CN\left(h_{x,N}h_{y,N}\left\|E_{12}\right\|_{L^{\infty}(\Omega_{1}\cup\Omega_{0})}^{2}\right)^{\frac{1}{2}}
≤C​N−σ\displaystyle\leq CN^{-\sigma}

and

‖E12−𝒬0​E12‖Ω2∪Ω0\displaystyle\|E_{12}-\mathcal{Q}_{0}E_{12}\|_{\Omega_{2}\cup\Omega_{0}} ≤C​(∑T∈Ω2∪Ω0(‖E12‖T2+‖𝒬0​E12‖T2))12\displaystyle\leq C\left(\sum_{T\in\Omega_{2}\cup\Omega_{0}}\left(\left\|E_{12}\right\|_{T}^{2}+\left\|\mathcal{Q}_{0}E_{12}\right\|_{T}^{2}\right)\right)^{\frac{1}{2}}
≤C​‖E12‖Ω2∪Ω0\displaystyle\leq C\left\|E_{12}\right\|_{\Omega_{2}\cup\Omega_{0}}
≤C​N​(hx,N​hy,N​‖E12‖L∞​(Ω2∪Ω0)2)12\displaystyle\leq CN\left(h_{x,N}h_{y,N}\left\|E_{12}\right\|_{L^{\infty}(\Omega_{2}\cup\Omega_{0})}^{2}\right)^{\frac{1}{2}}
≤C​N−σ\displaystyle\leq CN^{-\sigma}

Combining the above inequalities, we have completed the proof of (A.5).

Next, we give the proof of (A.6). Let Ωℓ\Omega_{\ell} represents any region in Ω0,Ω1,Ω2\Omega_{0},\Omega_{1},\Omega_{2} and Ω12\Omega_{12}, by using (A.2) and Lemma2.1 we obtain

(∑T∈Ωℓ‖S−𝒬0​S‖∂Ti2)12\displaystyle\left(\sum_{T\in\Omega_{\ell}}\left\|S-\mathcal{Q}_{0}S\right\|_{\partial T_{i}}^{2}\right)^{\frac{1}{2}} ≤C​(∑T∈Ωℓ(hj2​k+1+hi2​(k+1)​hj−1+hi2​k​hj)⋅hi​hj)12\displaystyle\leq C\left(\sum_{T\in\Omega_{\ell}}\left(h_{j}^{2k+1}+h_{i}^{2(k+1)}h_{j}^{-1}+h_{i}^{2k}h_{j}\right)\cdot h_{i}h_{j}\right)^{\frac{1}{2}}
≤C​N​(hN2​k+1)12\displaystyle\leq CN\left(h_{N}^{2k+1}\right)^{\frac{1}{2}}
≤C​N−(k+12),\displaystyle\leq CN^{-(k+\frac{1}{2})},

For the boundary layer E1E_{1} in region Ω12∪Ω1\Omega_{12}\cup\Omega_{1}, using (A.2) and Lemma2.1 we have

(∑T∈Ω12∪Ω1‖E1−𝒬0​E1‖∂Tx2)12\displaystyle\left(\sum_{T\in\Omega_{12}\cup\Omega_{1}}\left\|E_{1}-\mathcal{Q}_{0}E_{1}\right\|_{\partial T_{x}}^{2}\right)^{\frac{1}{2}}
≤C​(∑T∈Ω12∪Ω1(hy2​k+1​‖∂yk+1E1‖T2+hx2​(k+1)​hy−1​‖∂xk+1E1‖T2+hx2​k​hy​‖∂xk∂yE1‖T2))12\displaystyle\leq C\left(\sum_{T\in\Omega_{12}\cup\Omega_{1}}\left(h_{y}^{2k+1}\left\|\partial_{y}^{k+1}E_{1}\right\|_{T}^{2}+h_{x}^{2(k+1)}h_{y}^{-1}\left\|\partial_{x}^{k+1}E_{1}\right\|_{T}^{2}+h_{x}^{2k}h_{y}\left\|\partial_{x}^{k}\partial_{y}E_{1}\right\|_{T}^{2}\right)\right)^{\frac{1}{2}}
≤C​(∑T∈Ω12∪Ω1(ε−2​(k+1)⋅hx​hy2​(k+1)+hx2​k+3+ε−2⋅hx2​k+1​hy2)​‖e−β2​yε‖L∞​(T)2)12\displaystyle\leq C\left(\sum_{T\in\Omega_{12}\cup\Omega_{1}}\left(\varepsilon^{-2(k+1)}\cdot h_{x}h_{y}^{2(k+1)}+h_{x}^{2k+3}+\varepsilon^{-2}\cdot h_{x}^{2k+1}h_{y}^{2}\right)\left\|e^{-\frac{\beta_{2}y}{\varepsilon}}\right\|_{L^{\infty}(T)}^{2}\right)^{\frac{1}{2}}
≤C​N​(hx,N​N−2​(k+1)+hx,N2​k+3+hx,N2​k+1​N−2)12\displaystyle\leq CN\left(h_{x,N}N^{-2(k+1)}+h_{x,N}^{2k+3}+h_{x,N}^{2k+1}N^{-2}\right)^{\frac{1}{2}}
≤C​N−(k+12),\displaystyle\leq CN^{-(k+\frac{1}{2})},

By using (A.3), (A.4) and Lemma2.1 in the rest of the region we get

(∑T∈Ω2∪Ω0‖E1−𝒬0​E1‖∂Tx2)12\displaystyle\left(\sum_{T\in\Omega_{2}\cup\Omega_{0}}\left\|E_{1}-\mathcal{Q}_{0}E_{1}\right\|_{\partial T_{x}}^{2}\right)^{\frac{1}{2}} ≤C​(∑T∈Ω2∪Ω0(‖E1‖∂T12+‖𝒬0​E1‖∂T12))12\displaystyle\leq C\left(\sum_{T\in\Omega_{2}\cup\Omega_{0}}\left(\left\|E_{1}\right\|_{\partial T_{1}}^{2}+\left\|\mathcal{Q}_{0}E_{1}\right\|_{\partial T_{1}}^{2}\right)\right)^{\frac{1}{2}}
≤C​(∑T∈Ω2∪Ω0(‖E1‖∂T12+hy−1​‖E1‖T2))12\displaystyle\leq C\left(\sum_{T\in\Omega_{2}\cup\Omega_{0}}\left(\left\|E_{1}\right\|_{\partial T_{1}}^{2}+h_{y}^{-1}\left\|E_{1}\right\|_{T}^{2}\right)\right)^{\frac{1}{2}}
≤C​(N​∫01e−2​β1​yε​𝑑x+∑T∈Ω2∪Ω0hx​‖e−β1​yε‖L∞​(T)2)12\displaystyle\leq C\left(N\int_{0}^{1}e^{-\frac{2\beta_{1}y}{\varepsilon}}\,dx+\sum_{T\in\Omega_{2}\cup\Omega_{0}}h_{x}\left\|e^{-\frac{\beta_{1}y}{\varepsilon}}\right\|_{L^{\infty}(T)}^{2}\right)^{\frac{1}{2}}
≤C​(N⋅N−2​σ+hx,N⋅N2⋅N−2​σ)12\displaystyle\leq C\left(N\cdot N^{-2\sigma}+h_{x,N}\cdot N^{2}\cdot N^{-2\sigma}\right)^{\frac{1}{2}}
≤C​N12−σ,\displaystyle\leq CN^{\frac{1}{2}-\sigma},

Similar result are easily obtained for ∂Ty\partial T_{y}. The same goes for the boundary layer E2E_{2}. Now discuss the concer layer E12E_{12}, we have

(∑T∈Ω12‖E12−𝒬0​E12‖∂Tx2)12\displaystyle\left(\sum_{T\in\Omega_{12}}\left\|E_{12}-\mathcal{Q}_{0}E_{12}\right\|_{\partial T_{x}}^{2}\right)^{\frac{1}{2}}
≤C​(∑T∈Ω12(hy2​k+1​‖∂yk+1E12‖T2+hx2​(k+1)​hy−1​‖∂xk+1E12‖T2+hx2​k​hy​‖∂xk∂yE12‖T2))12\displaystyle\leq C\left(\sum_{T\in\Omega_{12}}\left(h_{y}^{2k+1}\left\|\partial_{y}^{k+1}E_{12}\right\|_{T}^{2}+h_{x}^{2(k+1)}h_{y}^{-1}\left\|\partial_{x}^{k+1}E_{12}\right\|_{T}^{2}+h_{x}^{2k}h_{y}\left\|\partial_{x}^{k}\partial_{y}E_{12}\right\|_{T}^{2}\right)\right)^{\frac{1}{2}}
≤C​(∑T∈Ω12ε−2​(k+1)​(hx​hy2​(k+1)+hx2​k+3+hx2​k+1​hy2)​‖e−β1​x+β2​yε‖L∞​(T)2)12\displaystyle\leq C\left(\sum_{T\in\Omega_{12}}\varepsilon^{-2(k+1)}\left(h_{x}h_{y}^{2(k+1)}+h_{x}^{2k+3}+h_{x}^{2k+1}h_{y}^{2}\right)\left\|e^{-\frac{\beta_{1}x+\beta_{2}y}{\varepsilon}}\right\|_{L^{\infty}(T)}^{2}\right)^{\frac{1}{2}}
≤C​N​(ε​N−1​N−2​(k+1)+hx,N2−1​N−2​(k+1)+ε​N−(2​k+1)​N−2)12\displaystyle\leq CN\left(\varepsilon N^{-1}N^{-2(k+1)}+h_{x,\frac{N}{2}-1}N^{-2(k+1)}+\varepsilon N^{-(2k+1)}N^{-2}\right)^{\frac{1}{2}}
≤C​N−(k+12),\displaystyle\leq CN^{-(k+\frac{1}{2})},

where we have used (A.2) and Lemma2.1. And in other regions, using (A.3), (A.4) and Lemma2.1 we derive

(∑T∈Ω1∪Ω0‖E12−𝒬0​E12‖∂Tx2)12\displaystyle\left(\sum_{T\in\Omega_{1}\cup\Omega_{0}}\left\|E_{12}-\mathcal{Q}_{0}E_{12}\right\|_{\partial T_{x}}^{2}\right)^{\frac{1}{2}} ≤C​(∑T∈Ω1∪Ω0(‖E12‖∂Tx2+‖𝒬0​E12‖∂Tx2))12\displaystyle\leq C\left(\sum_{T\in\Omega_{1}\cup\Omega_{0}}\left(\left\|E_{12}\right\|_{\partial T_{x}}^{2}+\left\|\mathcal{Q}_{0}E_{12}\right\|_{\partial T_{x}}^{2}\right)\right)^{\frac{1}{2}}
≤C​(∑T∈Ω1∪Ω0(‖E12‖∂Tx2+hy−1​‖E12‖T2))12\displaystyle\leq C\left(\sum_{T\in\Omega_{1}\cup\Omega_{0}}\left(\left\|E_{12}\right\|_{\partial T_{x}}^{2}+h_{y}^{-1}\left\|E_{12}\right\|_{T}^{2}\right)\right)^{\frac{1}{2}}
≤C​(N​∫xN/2−11e−2​(β1​x+β2​y)ε​𝑑x+∑T∈Ω1∪Ω0hx​‖e−β1​x+β2​yε‖L∞​(T)2)12\displaystyle\leq C\left(N\int_{x_{N/2-1}}^{1}e^{-\frac{2(\beta_{1}x+\beta_{2}y)}{\varepsilon}}\,dx+\sum_{T\in\Omega_{1}\cup\Omega_{0}}h_{x}\left\|e^{-\frac{\beta_{1}x+\beta_{2}y}{\varepsilon}}\right\|_{L^{\infty}(T)}^{2}\right)^{\frac{1}{2}}
≤C​(N⋅ε​N−2​σ+hx,N⋅N2⋅N−2​σ)12\displaystyle\leq C\left(N\cdot\varepsilon N^{-2\sigma}+h_{x,N}\cdot N^{2}\cdot N^{-2\sigma}\right)^{\frac{1}{2}}
≤C​N12−σ,\displaystyle\leq CN^{\frac{1}{2}-\sigma},

and

(∑T∈Ω2∪Ω0‖E12−𝒬0​E12‖∂Tx2)12\displaystyle\left(\sum_{T\in\Omega_{2}\cup\Omega_{0}}\left\|E_{12}-\mathcal{Q}_{0}E_{12}\right\|_{\partial T_{x}}^{2}\right)^{\frac{1}{2}} ≤C​(∑T∈Ω2∪Ω0(‖E12‖∂Tx2+‖𝒬0​E12‖∂Tx2))12\displaystyle\leq C\left(\sum_{T\in\Omega_{2}\cup\Omega_{0}}\left(\left\|E_{12}\right\|_{\partial T_{x}}^{2}+\left\|\mathcal{Q}_{0}E_{12}\right\|_{\partial T_{x}}^{2}\right)\right)^{\frac{1}{2}}
≤C​(∑T∈Ω2∪Ω0(‖E12‖∂Tx2+hy−1​‖E12‖T2))12\displaystyle\leq C\left(\sum_{T\in\Omega_{2}\cup\Omega_{0}}\left(\left\|E_{12}\right\|_{\partial T_{x}}^{2}+h_{y}^{-1}\left\|E_{12}\right\|_{T}^{2}\right)\right)^{\frac{1}{2}}
≤C​(N​∫01e−2​(β1​x+β2​y)ε​𝑑x+∑T∈Ω2∪Ω0hx​‖e−β1​x+β2​yε‖L∞​(T)2)12\displaystyle\leq C\left(N\int_{0}^{1}e^{-\frac{2(\beta_{1}x+\beta_{2}y)}{\varepsilon}}\,dx+\sum_{T\in\Omega_{2}\cup\Omega_{0}}h_{x}\left\|e^{-\frac{\beta_{1}x+\beta_{2}y}{\varepsilon}}\right\|_{L^{\infty}(T)}^{2}\right)^{\frac{1}{2}}
≤C​(N⋅ε​N−2​σ+hx,N⋅N2⋅N−2​σ)12\displaystyle\leq C\left(N\cdot\varepsilon N^{-2\sigma}+h_{x,N}\cdot N^{2}\cdot N^{-2\sigma}\right)^{\frac{1}{2}}
≤C​N12−σ,\displaystyle\leq CN^{\frac{1}{2}-\sigma},

Also it is easy to get similar result for ∂Ty\partial T_{y}. Together with the above estimates, we easily get inequality (A.6).

Let’s prove (A.7) now. The notation Ωτ\Omega_{\tau} represents any region in Ω0,Ω1\Omega_{0},\Omega_{1}, and Ω2\Omega_{2}. By the definition of θT\theta_{T} and (A.2), we obtain

(∑T∈ΩτθT​‖∇S−𝐐N​∇S‖∂Ti2)12\displaystyle\left(\sum_{T\in\Omega_{\tau}}\theta_{T}\left\|\nabla S-\mathbf{Q}_{N}\nabla S\right\|_{\partial T_{i}}^{2}\right)^{\frac{1}{2}} =(∑T∈Ωτε​hj​‖∇S−𝐐N​∇S‖∂Ti2)12\displaystyle=\left(\sum_{T\in\Omega_{\tau}}\varepsilon h_{j}\left\|\nabla S-\mathbf{Q}_{N}\nabla S\right\|_{\partial T_{i}}^{2}\right)^{\frac{1}{2}}
≤C​ε12​(∑T∈Ω12(hj2​k+hi2​k+hi2​(k−1)​hj2)⋅hi​hj)12\displaystyle\leq C\varepsilon^{\frac{1}{2}}\left(\sum_{T\in\Omega_{12}}\left(h_{j}^{2k}+h_{i}^{2k}+h_{i}^{2(k-1)}h_{j}^{2}\right)\cdot h_{i}h_{j}\right)^{\frac{1}{2}}
≤C​ε12​N​(hN2​(k+1))12\displaystyle\leq C\varepsilon^{\frac{1}{2}}N\left(h_{N}^{2(k+1)}\right)^{\frac{1}{2}}
≤C​N−(k+12),\displaystyle\leq CN^{-(k+\frac{1}{2})},

And for the region Ω0\Omega_{0}, we also have

(∑T∈Ω0θT​‖∇S−𝐐N​∇S‖∂Ti2)12\displaystyle\left(\sum_{T\in\Omega_{0}}\theta_{T}\left\|\nabla S-\mathbf{Q}_{N}\nabla S\right\|_{\partial T_{i}}^{2}\right)^{\frac{1}{2}} =(∑T∈Ω0ε2​N−1​‖∇S−𝐐N​∇S‖∂Ti2)12\displaystyle=\left(\sum_{T\in\Omega_{0}}\varepsilon^{2}N^{-1}\left\|\nabla S-\mathbf{Q}_{N}\nabla S\right\|_{\partial T_{i}}^{2}\right)^{\frac{1}{2}}
≤C​ε​(∑T∈Ω12N−1​(hj2​k−1+hi2​k​hj−1+hi2​(k−1)​hj)⋅hi​hj)12\displaystyle\leq C\varepsilon\left(\sum_{T\in\Omega_{12}}N^{-1}\left(h_{j}^{2k-1}+h_{i}^{2k}h_{j}^{-1}+h_{i}^{2(k-1)}h_{j}\right)\cdot h_{i}h_{j}\right)^{\frac{1}{2}}
≤C​ε​N​(N−1⋅hN2​k)12\displaystyle\leq C\varepsilon N\left(N^{-1}\cdot h_{N}^{2k}\right)^{\frac{1}{2}}
≤C​N−(k+12),\displaystyle\leq CN^{-(k+\frac{1}{2})},

where i,j∈{x,y}i,j\in\{x,y\} and i≠ji\neq j. For the boundary layer E1E_{1} in region Ω12∪Ω1\Omega_{12}\cup\Omega_{1}, we have

(∑T∈Ω12∪Ω1θT​‖∇E1−𝐐N​∇E1‖∂Tx2)12=(∑T∈Ω12∪Ω1ε​hy​‖∇E1−𝐐N​∇E1‖∂Tx2)12\displaystyle\left(\sum_{T\in\Omega_{12}\cup\Omega_{1}}\theta_{T}\left\|\nabla E_{1}-\mathbf{Q}_{N}\nabla E_{1}\right\|_{\partial T_{x}}^{2}\right)^{\frac{1}{2}}=\left(\sum_{T\in\Omega_{12}\cup\Omega_{1}}\varepsilon h_{y}\left\|\nabla E_{1}-\mathbf{Q}_{N}\nabla E_{1}\right\|_{\partial T_{x}}^{2}\right)^{\frac{1}{2}}
≤Cε12(∑T∈Ω12∪Ω1(hy2​k(∥∂yk∂xE1∥T2+∥∂yk+1E1∥T2)+hx2​k(∥∂xk+1E1∥T2+∥∂xk∂yE1∥T2)\displaystyle\leq C\varepsilon^{\frac{1}{2}}\Bigg{(}\sum_{T\in\Omega_{12}\cup\Omega_{1}}\Big{(}h_{y}^{2k}\left(\left\|\partial_{y}^{k}\partial_{x}E_{1}\right\|_{T}^{2}+\left\|\partial_{y}^{k+1}E_{1}\right\|_{T}^{2}\right)+h_{x}^{2k}\left(\left\|\partial_{x}^{k+1}E_{1}\right\|_{T}^{2}+\left\|\partial_{x}^{k}\partial_{y}E_{1}\right\|_{T}^{2}\right)
+hx2​(k−1)hy2(∥∂xk∂yE1∥T2+∥∂xk−1∂y2E1∥T2)))12\displaystyle\quad+h_{x}^{2(k-1)}h_{y}^{2}\left(\left\|\partial_{x}^{k}\partial_{y}E_{1}\right\|_{T}^{2}+\left\|\partial_{x}^{k-1}\partial_{y}^{2}E_{1}\right\|_{T}^{2}\right)\Big{)}\Bigg{)}^{\frac{1}{2}}
≤Cε12(∑T∈Ω12∪Ω1(hy2​k+1hx(ε−2​k+ε−2​(k+1))+hx2​k+1hy(1+ε−2)\displaystyle\leq C\varepsilon^{\frac{1}{2}}\Bigg{(}\sum_{T\in\Omega_{12}\cup\Omega_{1}}\Big{(}h_{y}^{2k+1}h_{x}\left(\varepsilon^{-2k}+\varepsilon^{-2(k+1)}\right)+h_{x}^{2k+1}h_{y}\left(1+\varepsilon^{-2}\right)
+hx2​k−1hy3(ε−2+ε−4))∥e−β2​yε∥L∞​(T)2)12\displaystyle\quad+h_{x}^{2k-1}h_{y}^{3}\left(\varepsilon^{-2}+\varepsilon^{-4}\right)\Big{)}\left\|e^{-\frac{\beta_{2}y}{\varepsilon}}\right\|_{L^{\infty}(T)}^{2}\Bigg{)}^{\frac{1}{2}}
≤C​ε12​N​((hx,N​N−(2​k+1)+hx,N2​k+1​N−1+hx,N2​k−1​N−3)⋅(ε+ε−1))12\displaystyle\leq C\varepsilon^{\frac{1}{2}}N\left(\left(h_{x,N}N^{-(2k+1)}+h_{x,N}^{2k+1}N^{-1}+h_{x,N}^{2k-1}N^{-3}\right)\cdot\left(\varepsilon+\varepsilon^{-1}\right)\right)^{\frac{1}{2}}
≤C​N−k,\displaystyle\leq CN^{-k},

In region Ω2\Omega_{2}, using (A.3), (A.4) and Lemma2.1 we get

(∑T∈Ω2θT​‖∇E1−𝐐N​∇E1‖∂Tx2)12=(∑T∈Ω2ε​hy​‖∇E1−𝐐N​∇E1‖∂Tx2)12\displaystyle\left(\sum_{T\in\Omega_{2}}\theta_{T}\left\|\nabla E_{1}-\mathbf{Q}_{N}\nabla E_{1}\right\|_{\partial T_{x}}^{2}\right)^{\frac{1}{2}}=\left(\sum_{T\in\Omega_{2}}\varepsilon h_{y}\left\|\nabla E_{1}-\mathbf{Q}_{N}\nabla E_{1}\right\|_{\partial T_{x}}^{2}\right)^{\frac{1}{2}}
≤C​ε12​(∑T∈Ω2hy​(‖∂xE1‖∂T12+‖∂yE1‖∂T12)+‖∂xE1‖T2+‖∂yE1‖T2)12\displaystyle\leq C\varepsilon^{\frac{1}{2}}\left(\sum_{T\in\Omega_{2}}h_{y}\left(\left\|\partial_{x}E_{1}\right\|_{\partial T_{1}}^{2}+\left\|\partial_{y}E_{1}\right\|_{\partial T_{1}}^{2}\right)+\left\|\partial_{x}E_{1}\right\|_{T}^{2}+\left\|\partial_{y}E_{1}\right\|_{T}^{2}\right)^{\frac{1}{2}}
≤C​ε12​(hy,N⋅N​(1+ε−2)​∫0xN/2−1e−2​β2​yε​𝑑x+∑T∈Ω2(1+ε−2)​hx​hy​‖e−β2​yε‖L∞​(T)2)12\displaystyle\leq C\varepsilon^{\frac{1}{2}}\left(h_{y,N}\cdot N\left(1+\varepsilon^{-2}\right)\int_{0}^{x_{N/2-1}}e^{-\frac{2\beta_{2}y}{\varepsilon}}\,dx+\sum_{T\in\Omega_{2}}\left(1+\varepsilon^{-2}\right)h_{x}h_{y}\left\|e^{-\frac{\beta_{2}y}{\varepsilon}}\right\|_{L^{\infty}(T)}^{2}\right)^{\frac{1}{2}}
≤C​ε12​((1+ε−2)​(ε⋅N1−2​σ+ε​N−1⋅N2−2​σ))12\displaystyle\leq C\varepsilon^{\frac{1}{2}}\left(\left(1+\varepsilon^{-2}\right)\left(\varepsilon\cdot N^{1-2\sigma}+\varepsilon N^{-1}\cdot N^{2-2\sigma}\right)\right)^{\frac{1}{2}}
≤C​N12−σ,\displaystyle\leq CN^{\frac{1}{2}-\sigma},

Similarly, for the region Ω0\Omega_{0} we have

(∑T∈Ω0θT​‖∇E1−𝐐N​∇E1‖∂Tx2)12=(∑T∈Ω0ε2​N−1​‖∇E1−𝐐N​∇E1‖∂Tx2)12\displaystyle\left(\sum_{T\in\Omega_{0}}\theta_{T}\left\|\nabla E_{1}-\mathbf{Q}_{N}\nabla E_{1}\right\|_{\partial T_{x}}^{2}\right)^{\frac{1}{2}}=\left(\sum_{T\in\Omega_{0}}\varepsilon^{2}N^{-1}\left\|\nabla E_{1}-\mathbf{Q}_{N}\nabla E_{1}\right\|_{\partial T_{x}}^{2}\right)^{\frac{1}{2}}
≤C​ε​(∑T∈Ω0N−1​(‖∂xE1‖∂Tx2+‖∂yE1‖∂Tx2)+N−1​hy−1​(‖∂xE1‖T2+‖∂yE1‖T2))12\displaystyle\leq C\varepsilon\left(\sum_{T\in\Omega_{0}}N^{-1}\left(\left\|\partial_{x}E_{1}\right\|_{\partial T_{x}}^{2}+\left\|\partial_{y}E_{1}\right\|_{\partial T_{x}}^{2}\right)+N^{-1}h_{y}^{-1}\left(\left\|\partial_{x}E_{1}\right\|_{T}^{2}+\left\|\partial_{y}E_{1}\right\|_{T}^{2}\right)\right)^{\frac{1}{2}}
≤C​ε​(N−1⋅N​(1+ε−2)​∫xN/2−11e−2​β2​yε​𝑑x+∑T∈Ω2(1+ε−2)​N−1​hx​‖e−β2​yε‖L∞​(T)2)12\displaystyle\leq C\varepsilon\left(N^{-1}\cdot N\left(1+\varepsilon^{-2}\right)\int_{x_{N/2-1}}^{1}e^{-\frac{2\beta_{2}y}{\varepsilon}}\,dx+\sum_{T\in\Omega_{2}}\left(1+\varepsilon^{-2}\right)N^{-1}h_{x}\left\|e^{-\frac{\beta_{2}y}{\varepsilon}}\right\|_{L^{\infty}(T)}^{2}\right)^{\frac{1}{2}}
≤C​ε​((1+ε−2)​N−2​σ)12\displaystyle\leq C\varepsilon\left(\left(1+\varepsilon^{-2}\right)N^{-2\sigma}\right)^{\frac{1}{2}}
≤C​N−σ,\displaystyle\leq CN^{-\sigma},

It is easy to get similar result for ∂Ty\partial T_{y} and the boundary layer E2E_{2} in the same way. By arguments similar to the ones used above, for the concer layer E12E_{12}, one has

(∑T∈Ω12θT​‖∇E12−𝐐N​∇E12‖∂Tx2)12=(∑T∈Ω12∪Ω1ε​hy​‖∇E12−𝐐N​∇E12‖∂Tx2)12\displaystyle\left(\sum_{T\in\Omega_{12}}\theta_{T}\left\|\nabla E_{12}-\mathbf{Q}_{N}\nabla E_{12}\right\|_{\partial T_{x}}^{2}\right)^{\frac{1}{2}}=\left(\sum_{T\in\Omega_{12}\cup\Omega_{1}}\varepsilon h_{y}\left\|\nabla E_{12}-\mathbf{Q}_{N}\nabla E_{12}\right\|_{\partial T_{x}}^{2}\right)^{\frac{1}{2}}
≤C​ε12​(∑T∈Ω12(hy2​k​(‖∂yk∂xE12‖T2+‖∂yk+1E12‖T2)+hx2​k​(‖∂xk+1E12‖T2+‖∂xk∂yE12‖T2)+hx2​(k−1)​hy2​(‖∂xk∂yE12‖T2+‖∂xk−1∂y2E12‖T2)))12\displaystyle\leq C\varepsilon^{\frac{1}{2}}\left(\sum_{T\in\Omega_{12}}\left(h_{y}^{2k}\left(\left\|\partial_{y}^{k}\partial_{x}E_{12}\right\|_{T}^{2}+\left\|\partial_{y}^{k+1}E_{12}\right\|_{T}^{2}\right)+h_{x}^{2k}\left(\left\|\partial_{x}^{k+1}E_{12}\right\|_{T}^{2}+\left\|\partial_{x}^{k}\partial_{y}E_{12}\right\|_{T}^{2}\right)+h_{x}^{2(k-1)}h_{y}^{2}\left(\left\|\partial_{x}^{k}\partial_{y}E_{12}\right\|_{T}^{2}+\left\|\partial_{x}^{k-1}\partial_{y}^{2}E_{12}\right\|_{T}^{2}\right)\right)\right)^{\frac{1}{2}}
≤C​ε12​(∑T∈Ω12ε−2​(k+1)​(hy2​k+1​hx+hx2​k+1​hy+hx2​k−1​hy3)​‖e−β1​x+β2​yε‖L∞​(T)2)12\displaystyle\leq C\varepsilon^{\frac{1}{2}}\left(\sum_{T\in\Omega_{12}}\varepsilon^{-2(k+1)}\left(h_{y}^{2k+1}h_{x}+h_{x}^{2k+1}h_{y}+h_{x}^{2k-1}h_{y}^{3}\right)\left\|e^{-\frac{\beta_{1}x+\beta_{2}y}{\varepsilon}}\right\|_{L^{\infty}(T)}^{2}\right)^{\frac{1}{2}}
≤C​ε12​N​(N−(2​k+1)⋅N−1+N−(2​k+1)​N−1+N−(2​k−1)​N−3)12\displaystyle\leq C\varepsilon^{\frac{1}{2}}N\left(N^{-(2k+1)}\cdot N^{-1}+N^{-(2k+1)}N^{-1}+N^{-(2k-1)}N^{-3}\right)^{\frac{1}{2}}
≤C​ε12​N−k,\displaystyle\leq C\varepsilon^{\frac{1}{2}}N^{-k},

Using (A.3), (A.4) and Lemma2.1, we obtain

(∑T∈Ω1θT​‖∇E12−𝐐N​∇E12‖∂Tx2)12=(∑T∈Ω1ε​h2​‖∇E12−𝐐N​∇E12‖∂Tx2)12\displaystyle\left(\sum_{T\in\Omega_{1}}\theta_{T}\left\|\nabla E_{12}-\mathbf{Q}_{N}\nabla E_{12}\right\|_{\partial T_{x}}^{2}\right)^{\frac{1}{2}}=\left(\sum_{T\in\Omega_{1}}\varepsilon h_{2}\left\|\nabla E_{12}-\mathbf{Q}_{N}\nabla E_{12}\right\|_{\partial T_{x}}^{2}\right)^{\frac{1}{2}}
≤C​ε12​(∑T∈Ω1hy​(‖∂xE12‖∂Tx2+‖∂yE12‖∂Tx2)+‖∂xE12‖T2+‖∂yE12‖T2)12\displaystyle\leq C\varepsilon^{\frac{1}{2}}\left(\sum_{T\in\Omega_{1}}h_{y}\left(\left\|\partial_{x}E_{12}\right\|_{\partial T_{x}}^{2}+\left\|\partial_{y}E_{12}\right\|_{\partial T_{x}}^{2}\right)+\left\|\partial_{x}E_{12}\right\|_{T}^{2}+\left\|\partial_{y}E_{12}\right\|_{T}^{2}\right)^{\frac{1}{2}}
≤C​ε12​(hy,N2−1⋅N​ε−2​∫xN/2−11e−2​(β1​x+β2​y)ε​𝑑x+∑T∈Ω1ε−2​hx​hy​‖e−β1​x+β2​yε‖L∞​(T)2)12\displaystyle\leq C\varepsilon^{\frac{1}{2}}\left(h_{y,\frac{N}{2}-1}\cdot N\varepsilon^{-2}\int_{x_{N/2-1}}^{1}e^{-\frac{2(\beta_{1}x+\beta_{2}y)}{\varepsilon}}\,dx+\sum_{T\in\Omega_{1}}\varepsilon^{-2}h_{x}h_{y}\left\|e^{-\frac{\beta_{1}x+\beta_{2}y}{\varepsilon}}\right\|_{L^{\infty}(T)}^{2}\right)^{\frac{1}{2}}
≤C​ε12​(ε−2​(ε2⋅N1−2​σ+ε​N−1⋅N2−2​σ))12\displaystyle\leq C\varepsilon^{\frac{1}{2}}\left(\varepsilon^{-2}\left(\varepsilon^{2}\cdot N^{1-2\sigma}+\varepsilon N^{-1}\cdot N^{2-2\sigma}\right)\right)^{\frac{1}{2}}
≤C​N12−σ,\displaystyle\leq CN^{\frac{1}{2}-\sigma},

In the same way, we also have

(∑T∈Ω2θT​‖∇E12−𝐐N​∇E12‖∂Tx2)12=(∑T∈Ω2ε​h2​‖∇E12−𝐐N​∇E12‖∂Tx2)12\displaystyle\left(\sum_{T\in\Omega_{2}}\theta_{T}\left\|\nabla E_{12}-\mathbf{Q}_{N}\nabla E_{12}\right\|_{\partial T_{x}}^{2}\right)^{\frac{1}{2}}=\left(\sum_{T\in\Omega_{2}}\varepsilon h_{2}\left\|\nabla E_{12}-\mathbf{Q}_{N}\nabla E_{12}\right\|_{\partial T_{x}}^{2}\right)^{\frac{1}{2}}
≤C​ε12​(∑T∈Ω2hy​(‖∂xE12‖∂Tx2+‖∂yE12‖∂Tx2)+‖∂xE12‖T2+‖∂yE12‖T2)12\displaystyle\leq C\varepsilon^{\frac{1}{2}}\left(\sum_{T\in\Omega_{2}}h_{y}\left(\left\|\partial_{x}E_{12}\right\|_{\partial T_{x}}^{2}+\left\|\partial_{y}E_{12}\right\|_{\partial T_{x}}^{2}\right)+\left\|\partial_{x}E_{12}\right\|_{T}^{2}+\left\|\partial_{y}E_{12}\right\|_{T}^{2}\right)^{\frac{1}{2}}
≤C​ε12​(hy,N⋅N​ε−2​∫0xN/2−1e−2​(β1​x+β2​y)ε​𝑑x+∑T∈Ω2ε−2​hx​hy​‖e−β1​x+β2​yε‖L∞​(T)2)12\displaystyle\leq C\varepsilon^{\frac{1}{2}}\left(h_{y,N}\cdot N\varepsilon^{-2}\int_{0}^{x_{N/2-1}}e^{-\frac{2(\beta_{1}x+\beta_{2}y)}{\varepsilon}}\,dx+\sum_{T\in\Omega_{2}}\varepsilon^{-2}h_{x}h_{y}\left\|e^{-\frac{\beta_{1}x+\beta_{2}y}{\varepsilon}}\right\|_{L^{\infty}(T)}^{2}\right)^{\frac{1}{2}}
≤C​ε12​(ε−2​(ε⋅N−2​σ+ε​N−1⋅N2−2​σ))12\displaystyle\leq C\varepsilon^{\frac{1}{2}}\left(\varepsilon^{-2}\left(\varepsilon\cdot N^{-2\sigma}+\varepsilon N^{-1}\cdot N^{2-2\sigma}\right)\right)^{\frac{1}{2}}
≤C​N12−σ,\displaystyle\leq CN^{\frac{1}{2}-\sigma},

and

(∑T∈Ω0θT​‖∇E12−𝐐N​∇E12‖∂Tx2)12=(∑T∈Ω0ε2​N−1​‖∇E12−𝐐N​∇E12‖∂Tx2)12\displaystyle\left(\sum_{T\in\Omega_{0}}\theta_{T}\left\|\nabla E_{12}-\mathbf{Q}_{N}\nabla E_{12}\right\|_{\partial T_{x}}^{2}\right)^{\frac{1}{2}}=\left(\sum_{T\in\Omega_{0}}\varepsilon^{2}N^{-1}\left\|\nabla E_{12}-\mathbf{Q}_{N}\nabla E_{12}\right\|_{\partial T_{x}}^{2}\right)^{\frac{1}{2}}
≤C​ε​(∑T∈Ω0N−1​(‖∂xE12‖∂Tx2+‖∂yE12‖∂Tx2)+N−1​hy−1​(‖∂xE12‖T2+‖∂yE12‖T2))12\displaystyle\leq C\varepsilon\left(\sum_{T\in\Omega_{0}}N^{-1}\left(\left\|\partial_{x}E_{12}\right\|_{\partial T_{x}}^{2}+\left\|\partial_{y}E_{12}\right\|_{\partial T_{x}}^{2}\right)+N^{-1}h_{y}^{-1}\left(\left\|\partial_{x}E_{12}\right\|_{T}^{2}+\left\|\partial_{y}E_{12}\right\|_{T}^{2}\right)\right)^{\frac{1}{2}}
≤C​ε​(N−1⋅N​ε−2​∫xN/2−11e−2​(β1​x+β2​y)ε​𝑑x+∑T∈Ω2ε−2​N−1​hx​‖e−β1​x+β2​yε‖L∞​(T)2)12\displaystyle\leq C\varepsilon\left(N^{-1}\cdot N\varepsilon^{-2}\int_{x_{N/2-1}}^{1}e^{-\frac{2(\beta_{1}x+\beta_{2}y)}{\varepsilon}}\,dx+\sum_{T\in\Omega_{2}}\varepsilon^{-2}N^{-1}h_{x}\left\|e^{-\frac{\beta_{1}x+\beta_{2}y}{\varepsilon}}\right\|_{L^{\infty}(T)}^{2}\right)^{\frac{1}{2}}
≤C​ε​(ε−2​(ε​N−2​σ+N−2​σ))12\displaystyle\leq C\varepsilon\left(\varepsilon^{-2}\left(\varepsilon N^{-2\sigma}+N^{-2\sigma}\right)\right)^{\frac{1}{2}}
≤C​N−σ,\displaystyle\leq CN^{-\sigma},

Also it is easy to get similar result for ∂Ty\partial T_{y}. Combining the above estimates, We have completed the proof of (A.7).

Finally, for the last inequality (A.8), note that ϑT=ε​hj−1≤C​ε​h1−1≤C​N\vartheta_{T}=\varepsilon h_{j}^{-1}\leq C\varepsilon h_{1}^{-1}\leq CN in regions Ω1\Omega_{1}, Ω2\Omega_{2}, Ω12\Omega_{12} and ϑT=N\vartheta_{T}=N in region Ω0\Omega_{0}. Together with (A.6) we get

(∑T∈𝒯NϑT​‖u−𝒬0​u‖∂Ti2)12\displaystyle\left(\sum_{T\in\mathcal{T}_{N}}\vartheta_{T}\left\|u-\mathcal{Q}_{0}u\right\|_{\partial T_{i}}^{2}\right)^{\frac{1}{2}} ≤C​N−k.\displaystyle\leq CN^{-k}.

At this point, all the estimates are proved.

∎

References

  • [1] V. B. Andreev and N. V. Kopteva, On the convergence, uniform with respect to a small parameter, of monotone three-point difference schemes, Differ. Uravn., 34 (1998), pp. 921–928, 1006.
  • [2] N. S. Bahvalov, On the optimization of the methods for solving boundary value problems in the presence of a boundary layer, Ž. Vyčisl. Mat i Mat. Fiz., 9 (1969), pp. 841–859.
  • [3] M. Cui and S. Zhang, On the uniform convergence of the weak Galerkin finite element method for a singularly-perturbed biharmonic equation, J. Sci. Comput., 82 (2020), pp. Paper No. 5, 15.
  • [4] S. Franz and H.-G. Roos, The capriciousness of numerical methods for singular perturbations, SIAM Rev., 53 (2011), pp. 157–173.
  • [5] E. H. Georgoulis, Discontinuous Galerkin methods on shape-regular and anisotropic meshes, PhD thesis, University of Oxford D. Phil. Thesis, 2003.
  • [6] N. Kopteva, Uniform pointwise convergence of difference schemes for convection-diffusion problems on layer-adapted meshes, vol. 66, 2001, pp. 179–197. Archives for scientific computing. Numerical methods for transport-dominated and related problems (Magdeburg, 1999).
  • [7] R. Lin, Discontinuous discretization for least-squares formulation of singularly perturbed reaction-diffusion problems in one and two dimensions, SIAM J. Numer. Anal., 47 (2008/09), pp. 89–108.
  • [8]  , Discontinuous Galerkin least-squares finite element methods for singularly perturbed reaction-diffusion problems with discontinuous coefficients and boundary singularities, Numer. Math., 112 (2009), pp. 295–318.
  • [9] R. Lin, X. Ye, S. Zhang, and P. Zhu, A weak Galerkin finite element method for singularly perturbed convection-diffusion-reaction problems, SIAM J. Numer. Anal., 56 (2018), pp. 1482–1497.
  • [10] T. Linß, Uniform superconvergence of a Galerkin finite element method on Shishkin-type meshes, Numer. Methods Partial Differential Equations, 16 (2000), pp. 426–440.
  • [11] T. Linß, The necessity of Shishkin decompositions, Appl. Math. Lett., 14 (2001), pp. 891–896.
  • [12] T. Linß, Layer-adapted meshes for reaction-convection-diffusion problems, vol. 1985 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2010.
  • [13] T. Linßand M. Stynes, The SDFEM on Shishkin meshes for linear convection-diffusion problems, Numer. Math., 87 (2001), pp. 457–484.
  • [14] L.-B. Liu, H. Leng, and G. Long, Analysis of the SDFEM for singularly perturbed differential-difference equations, Calcolo, 55 (2018), pp. Paper No. 23, 17.
  • [15] L. Mu, J. Wang, and X. Ye, A stable numerical algorithm for the Brinkman equations by weak Galerkin finite element methods, J. Comput. Phys., 273 (2014), pp. 327–342.
  • [16] L. Mu, J. Wang, X. Ye, and S. Zhang, A weak Galerkin finite element method for the Maxwell equations, J. Sci. Comput., 65 (2015), pp. 363–386.
  • [17] H.-G. Roos, M. Stynes, and L. Tobiska, Robust numerical methods for singularly perturbed differential equations, vol. 24 of Springer Series in Computational Mathematics, Springer-Verlag, Berlin, second ed., 2008. Convection-diffusion-reaction and flow problems.
  • [18] H.-G. Roos and H. Zarin, A supercloseness result for the discontinuous Galerkin stabilization of convection-diffusion problems on Shishkin meshes, Numer. Methods Partial Differential Equations, 23 (2007), pp. 1560–1576.
  • [19] G. I. Shishkin, Grid approximation of singularly perturbed elliptic and parabolic equations, Keldysh Institute of Applied Mathematics, Moscow, In Russian, 1990. Second Doctoral thesis.
  • [20] S. Sumit, S. Kumar, and M. Kumar, Optimal fourth-order parameter-uniform convergence of a non-monotone scheme on equidistributed meshes for singularly perturbed reaction-diffusion problems, Int. J. Comput. Math., 99 (2022), pp. 1638–1653.
  • [21] c. Toprakseven, A weak Galerkin finite element method for time fractional reaction-diffusion-convection problems with variable coefficients, Appl. Numer. Math., 168 (2021), pp. 1–12.
  • [22]  , Optimal order uniform convergence of weak Galerkin finite element method on Bakhvalov-type meshes for singularly perturbed convection dominated problems, Hacet. J. Math. Stat., 52 (2023), pp. 850–875.
  • [23] c. Toprakseven and P. Zhu, Uniform convergent modified weak Galerkin method for convection-dominated two-point boundary value problems, Turkish J. Math., 45 (2021), pp. 2703–2730.
  • [24] S. Toprakseven, Superconvergence of a modified weak Galerkin method for singularly perturbed two-point elliptic boundary-value problems, Calcolo, 59 (2022), pp. Paper No. 1, 35.
  • [25] J. Wang and X. Ye, A weak Galerkin finite element method for second-order elliptic problems, J. Comput. Appl. Math., 241 (2013), pp. 103–115.
  • [26]  , A weak Galerkin finite element method for the stokes equations, Adv. Comput. Math., 42 (2016), pp. 155–174.
  • [27] R. Wang, X. Wang, Q. Zhai, and R. Zhang, A weak Galerkin finite element scheme for solving the stationary Stokes equations, J. Comput. Appl. Math., 302 (2016), pp. 171–185.
  • [28] X. Wang, Q. Zhai, and R. Zhang, The weak Galerkin method for solving the incompressible Brinkman flow, J. Comput. Appl. Math., 307 (2016), pp. 13–24.
  • [29] Z. Wang, R. Wang, and J. Liu, Robust weak Galerkin finite element solvers for Stokes flow based on a lifting operator, Comput. Math. Appl., 125 (2022), pp. 90–100.
  • [30] H. Zarin and H.-G. Roos, Interior penalty discontinuous approximations of convection-diffusion problems with parabolic layers, Numer. Math., 100 (2005), pp. 735–759.
  • [31] Q. Zhai, R. Zhang, and L. Mu, A new weak Galerkin finite element scheme for the Brinkman model, Commun. Comput. Phys., 19 (2016), pp. 1409–1434.
  • [32] J. Zhang and X. Liu, Optimal order of uniform convergence for finite element method on Bakhvalov-type meshes, J. Sci. Comput., 85 (2020), pp. Paper No. 2, 14.
  • [33] J. Zhang and X. Liu, Uniform convergence of a weak Galerkin finite element method on Shishkin mesh for singularly perturbed convection-diffusion problems in 2D, Appl. Math. Comput., 432 (2022), pp. Paper No. 127346, 12.
  • [34] J. Zhang and X. Liu, Uniform convergence of a weak Galerkin method for singularly perturbed convection-diffusion problems, Math. Comput. Simulation, 200 (2022), pp. 393–403.
  • [35] H. Zhu and Z. Zhang, Uniform convergence of the LDG method for a singularly perturbed problem with the exponential boundary layer, Math. Comp., 83 (2014), pp. 635–663.
  • [36] P. Zhu and S. Xie, A uniformly convergent weak Galerkin finite element method on Shishkin mesh for 1d convection-diffusion problem, J. Sci. Comput., 85 (2020), pp. Paper No. 34, 22.
  • [37] P. Zhu, Y. Yang, and Y. Yin, Higher order uniformly convergent NIPG methods for 1-d singularly perturbed problems of convection-diffusion type, Appl. Math. Model., 39 (2015), pp. 6806–6816.