Convergence and supercloseness of a finite element method for a two-parameter singularly perturbed problem on Shishkin triangular mesh

Highlights

• We consider a two-dimension singularly perturbed elliptic problem with two parameters.

• We combine the characteristics of the mesh and the layers to choose technical tools for error analysis.

• Using some important integral inequalities, we prove the convergence and supercloseness of finite element method on Shishkin triangular mesh.

Abstract

We consider a singularly perturbed elliptic problem with two parameters in two dimensions. Using linear finite element method on a Shishkin triangular mesh, we prove the uniform convergence and supercloseness in an energy norm. Some integral inequalities play an important role in our analysis. Numerical tests verify our theoretical results.

Introduction

We discuss the following singularly perturbed problem in the interval $\Theta :=(0,1)\times (0,1)$

$$\begin{array}{cc}& -{\epsilon }_1\Delta z+{\epsilon }_{2}b\left(x\right)z_x+c\left(x\right)z=f(x,y),\hfill \\ & {z|}_{\partial \Theta }=0,\hfill \end{array}$$

under the condition $0<{\epsilon }_1,{\epsilon }_2\ll 1$ and for all $x\in [0,1]$ there are

$$c\left(x\right)\ge \lambda >0,b\left(x\right)\ge \varphi >0,$$

and

$$-\frac{1}{2}{\epsilon }_2{b}^{\prime }\left(x\right)+c\left(x\right)\ge \psi >0,$$

where $\varphi$, $\lambda$, $\psi$ are positive constants. Assume $b$, $c$ and $f$ sufficiently smooth. Moreover, assume that the funciton $f$ satisfies the compatibility conditions

$$f(0,0)=f(0,1)=f(1,1)=f(1,0)=0.$$

From these assumptions and [1] we obtain that problem (1) has a unique solution $z\in C^{3,{\alpha }_0}\left(\overline{\Theta }\right)$ for $0<{\alpha }_0<1$.

Due to the influence of small parameters in the equation, boundary layers or inner layers often appear in singularly perturbed problems. Unfortunately, classical numerical methods are useless for these problems because the solution changes so quickly within the layer. In recent years, layer-adapted mesh has been developed to analyze singular perturbed problems. And one of the most popular layer-adapted meshes is Shishkin mesh, which is simple in structure and easy to analyze.

In general, one issue of numerical methods for singular perturbation problems is to obtain uniform convergence independent of perturbation parameters. It is worth noting that with the development of research, researchers have also discovered the importance of supercloseness, which is a useful tool to improve numerical solutions by postprocessing. For singularly perturbed problems with one parameter, there are plenty of supercloseness results; see [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13] and their bibliography. But, few studies have been done for singular perturbation problems with two parameters; see [14], [15].

In this article, for two-dimensional singular perturbation problems with two parameters, we prove the convergence and supercloseness of finite element method on Shishkin triangular mesh. Compared with the one-parameter problems, layers of two-parameter problems are more complex, as makes the analysis more difficult. The analysis shows that there are four corner layers, two exponential boundary layers and two parabolic layers in the solution of problem (1). Their properties are different and closely related to small parameters ${\epsilon }_1$, ${\epsilon }_2$ and mesh properties. Therefore, we combine the characteristics of the mesh and the layers to choose technical tools for error analysis. At the same time, some integral inequalities on Shishkin triangular mesh are also of great significance in error analysis. Finally, we obtain the main theoretical results of this paper.

The organization of this article is as follows. In Section 2, we present some information about the exact solution and some necessary assumptions. In Section 3, the Shishkin triangle mesh is built, and some information about the mesh is given. Besides, we provide the variational formulation and finite element method for the problem (1). In Section 4, the error results of exact solution and its interpolation in $L^{\infty }$ norm and energy norm are proved. We present some important integral inequalities in Section 5, while in Section 6 our main theoretical results on supercloseness and uniform convergence are given. In the last section, we illustrate and support the theoretical results with numerical examples.

For nonnegative integers $k$, $1\le p\le \infty$, and any measurable subset $G$ of $\Theta$, a standard notation for Banach spaces $L^p\left(G\right)$ is given, and we denote by $W^{k,p}\left(G\right)$ the Sobolev space and $H^k\left(G\right)=W^{k,2}\left(G\right)$ on $G$. We write by ${\parallel ·\parallel }_G$ the norm of ${\parallel ·\parallel }_{L^2\left(G\right)}$ and write by ${|·|}_{1,G}$ the seminorm of ${\parallel ·\parallel }_{H^1\left(G\right)}$, respectively. The notation ${(·,·)}_G$ stands for the scalar product in $L^2\left(G\right)$. Furthermore, we will omit the $G$ in the norm and seminorm when it is $\Theta$. Throughout this paper, $C$ represents a general constant independent of the perturbation parameters ${\epsilon }_1$, ${\epsilon }_2$ and the number of mesh points.