Convergence analysis of a weak Galerkin finite element method on a Shishkin mesh for a singularly perturbed fourth-order problem in 2D

This paper is concerned with a singularly perturbed fourth-order boundary value problem in a square region $\Omega$. The problem is solved by the weak Galerkin (WG) method with a Shishkin mesh, as follows:

with a positive parameter $\varepsilon$ satisfying $0<\varepsilon\ll 1$ and $f\in L^{2}(\Omega)$. This problem is used for thin elastic plates clamped in tension. $f$ represents the transverse load, $\varepsilon$ symbolizes the ratio of bending stiffness to tensile stiffness of the plate, and the function’s solution, denoted as $u$, represents the displacement of the plate. This problem emerges in the investigation of the linearization of the fourth-order perturbation associated with the fully nonlinear Monge-Ampère equation[2, 3].

A variational formulation for the fourth-order equation (1.1) with the boundary conditions (1.2) and (1.3) seeks $u\in H^{2}(\Omega)$ such that

where $H^{2}_{0}(\Omega)$ is a subspace of the Sobolev space $H^{2}(\Omega)$ consisting of functions with vanishing value and normal derivative on $\partial\Omega$.

Research into numerical methods for singularly perturbed differential equations commenced in the early 1970s, with the frontier of research continuously expanding ever since. Bakhvalov made an important early contribution to the optimization of numerical methods by means of special meshes [1] in 1969. In the early 1990s, G.I. Shishkin proposed piecewise-equidistant meshes [20, 26]. These meshes, characterized by their very simple structure, are usually easy to analyze. Shishkin meshes for various problems and numerical methods have been studied since and they are still popular. Numerous scholars have explored the numerical analysis of analogous problems using various finite element methods, such as the mixed finite element method in [10], the hp finite element method in [7], the continuous interior penalty finite element method in [11], the upwind finite volume element method in [36], the conforming finite element method in [24]. And also some papers consider finite element methods on quasi-uniform meshes: a continuous interior penalty finite element method in [3] and nonconforming finite element method in [5, 13, 15, 19, 23, 31, 34, 38].

In this paper, we employ the WG method with a Shishkin mesh to investigate the convergence behavior of a fourth-order boundary value problem that exhibits singular perturbation. The WG method proves to be an effective numerical technique for the partial differential equations(PDEs). The initial proposal for its application in solving second-order elliptic problems was made by Junping Wang and Xiu Ye in [28]. The core concept involves establishing distinct basis functions for the interior and boundary of each partitioned element, and substituting the traditional differential operator with a discretized weak differential operator. The WG method has been applied to Stokes equations [29, 32, 37], elasticity equations [4, 14, 27, 39], Maxwell’s equations [22], biharmonic equations [8, 44], Navier-Stokes equations [16, 18, 43], Brinkman equations [21, 35, 40], the multigrid approach [6], the incompressible flow [45], the maximum principle [17, 30], the post-processing technique [33] and so on. For singular perturbed value problems, the WG method has also yielded some results, such as the singularly perturbed convection-diffusion problems for WG in 1D [42, 46] and 2D [41], the singularly perturbed biharmonic equation for WG in uniform mesh [9].

This paper is organized as follows. In Section 2, we introduce the Shishkin mesh and the assumptions associated. In Section 3, we give the definitions of the weak Laplacian operator and weak gradient operator. We also present WG finite element schemes for the singularly perturbed value problem. In Section 4, we introduce some local $L^{2}$ projection operators and give some approximation properties. In Section 5, we establish error estimates for the WG scheme in a $H^{2}$-equivalent discrete norm. And in Section 6, we report the results of two numerical experiments.

To solve problem (1.1)-(1.3), we suppose the following assumption holds in [19], which involves structuring the solution and decomposing $u$ into smooth and layered components.

In order to solve the layer structure in the solution of problems (1.1)-(1.3), a well-suited layer-adapted Shishkin mesh be considered. This mesh is refined in the layers. For a comprehensive discussion on the construction of Shishkin meshes, please refer to [25].

Consider a positive integer $N\geq 4$ that is divisible by $4$. We introduce a mesh transition parameter $\lambda$ to determine the location at which the mesh switches from coarse to fine. This parameter is defined by

where $\alpha$ is a positive constant, selected to be $k+1$ for the subsequent analysis.

Create a piecewise equidistant mesh for the interval $[0,1]$ by dividing it as follows: divide $[0,\lambda]$ into $N/4$ subintervals, $[\lambda,1-\lambda]$ into $N/2$ subintervals, and $[1-\lambda,1]$ into $N/4$ subintervals. The Shishkin mesh for the problem (1.1)-(1.3) is formed by taking the tensor product of two such one-dimensional meshes, as illustrated in Figure 1. The fine meshwidth denoted as $h$ and the coarse meshwidth denoted as $H$ in the Shishkin mesh represented by $\mathcal{T}_{N}$ exhibit the following characteristics:

for some constant $C$. The domain $\Omega$ is divided into some subdomains, see Figure 1. Denote $\Omega_{l1}\cup\Omega_{s}\cup\Omega_{l3}$ by $\Omega_{r}^{1}$ and $\Omega_{l2}\cup\Omega_{s}\cup\Omega_{l4}$ by $\Omega_{r}^{2}$.

To propose the weak Galer-kin method, we introduce some key concepts. Consider a element $T$ belonging to the partition $\mathcal{T}_{N}$ with a boundary denoted by $\partial T$. Let the set of all edges in $\mathcal{T}_{N}$ be represented as $\mathcal{E}_{N}$. We define a weak function $v=\{v_{0},v_{b},\mathbf{v}_{g}\}$ on the element $T$, where $v_{0}\in L^{2}(T)$, $v_{b}\in H^{\frac{1}{2}}(\partial T)$, and $\mathbf{v}_{g}\cdot\mathbf{n}\in H^{\frac{1}{2}}(\partial T)$, with $\mathbf{n}$ representing the outward normal direction on $\partial T$. Furthermore, the first and second components, $v_{0}$ and $v_{b}$, correspond to values of $v$ in the interior and on the boundary of $T$. The third component $\mathbf{v}_{g}$ is employed to approximate the gradient of $v$ along the boundary of $T$. It’s important to note that each edge $e\in\mathcal{E}_{N}$ has a unique value for $v_{b}$ and $\mathbf{v}_{g}$. Additionally, it’s worth mentioning that $v_{b}$ and $\mathbf{v}_{g}$ may not necessarily be associated with the trace of $v_{0}$ and $\nabla v_{0}$ on $\partial T$.

For any integer $k\geq 3$, we establish a local discrete weak function space for any element $T\in\mathcal{T}_{N}$ denoted as $W_{k}(T)$:

where $e$ is the edge of $\partial T$, and $\mathbb{Q}_{k}$ is the space of polynomials which are of degree not exceeding $k$ with respect to each one of the variables $x$ and $y$. By extending $W_{k}(T)$ to encompass all element $T\in\mathcal{T}_{N}$, we introduce the definition of a weak Galerkin space:

Let $V_{N}^{0}$ represent the subspace of $V_{h}$ where the traces vanish:

For any $v=\{v_{0},v_{b},\mathbf{v}_{g}\}$ and a fixed integer $k\geq 3$, we define a discrete weak Laplacian operator $\Delta_{w,k}$ as a unique polynomial $\Delta_{w,k}v\in\mathbb{Q}_{k}(T)$ on $T$ satisfying the following equation:

where $\mathbf{n}$ is the unit outward normal vector to $\partial T$. Likewise, a discrete weak gradient $\nabla_{w,k}$ is defined on $T$ as a unique polynomial $\nabla_{w,k}v\in[\mathbb{Q}_{k}(T)]^{2}$ satisfying:

For simplicity, when there is no confusion, we drop the subscript $k$ in the notations $\Delta_{w,k}$ and $\nabla_{w,k}$ for the discrete weak Laplacian and the discrete weak gradient. Additionally, we introduce the following notations:

Let us introduce a stabilizer, which is a bilinear form for any $u_{N}=\{u_{0},u_{b},\mathbf{u}_{g}\}$ and $v=\{v_{0},v_{b},\mathbf{v}_{g}\}$ in the space $V_{h}$. It is defined as follows:

where $h$ and $H$ is defined as (2.2).

The following lemma provides a valuable result regarding the finite element space $V^{0}_{N}$.

In this section, we introduce some projection operators for each element $T\in\mathcal{T}_{N}$. Consider the $L^{2}$ projection operator $\mathcal{Q}_{0}$, which projects onto $\mathbb{Q}_{k}(T)$. Additionally, for each edge $e\in\partial T$, we consider the $L^{2}$ projection operators $\mathcal{Q}_{b}$ and $\mathbf{Q}_{g}$ onto local polynomial spaces $\mathbb{P}_{k}(e)$ and $[\mathbb{P}_{k}(e)]^{2}$, respectively. We define a projection $\mathcal{Q}_{N}$ of $u$ into the finite element space $V_{N}$ such that on each element $T$

Furthermore, let $\mathbf{Q}_{N}$ represent the local $L^{2}$ projection onto $[\mathbb{Q}_{k}(T)]^{2}$. The following lemma demonstrates that the weak Laplacian $\Delta_{w}$ is the polynomial projection of the classical Laplacian $\Delta$.

A similar lemma holds for the weak gradient $\nabla_{w}$, as indicated in the subsequent lemma.

Now, we introduce notation that will be employed in the following lemmas. Consider any element $T$ in the partition $\mathcal{T}_{N}$. We define $\partial T_{1}$ and $\partial T_{2}$ as the sets of element edges that are parallel to the $x$ and $y$ axes, respectively. The following lemmas are employed in the convergence analysis, and readers are directed to [12, chapter 3.1] for a detailed proof process.

By applying Lemma 4.3 and Lemma 4.4, we can deduce the following estimates which are valuable for the convergence analysis of the WG finite element schemes (3.4).

In this section, the objective is to provide error estimates for the WG solution $u_{N}$ obtained from (3.4).