A priori and a posteriori error estimates of the weak Galerkin finite element method for parabolic problems

Abstract

We derive the priori and a posteriori error estimates of the weak Galerkin finite element method with the Crank-Nicolson time discretization for the parabolic equation in this paper. The priori error estimates are deduced based on existing priori error results of the corresponding elliptic projection problem. For the a posteriori error estimates, the elliptic reconstruction technique is introduced to decompose the true error into elliptic error and parabolic error. Then the elliptic part is bounded by the a posteriori error estimates of the auxiliary elliptic reconstruction problem. The a posteriori error estimator is further used to develop the temporal and spatial adaptive algorithm. Numerical results in the uniform and adaptive meshes are provided to validate the proposed estimators.

Introduction

The weak Galerkin (WG) finite element method [30] is a generalization of the standard finite element method for numerically solving partial differential equations in which classical differential operators (such as gradient, divergence, curl) are approximated in weak sense. The WG method was initially introduced to solve elliptic equations [24], [30], [31]. Over the past decade, it has attracted much attention for its simplicity of the form and flexibility of the applicability in polygonal meshes. Now, it has been used to solve more complex equations, such as parabolic problems [13], [19], [25], biharmonic equations [28], Navier-Stokes equations [20], Maxwell's equations [27], eigenvalue problems [35], Helmholtz equations [8], elliptic Cauchy problems [29] and so on. At the same time, the superconvergence [15], [16], [34], the a posteriori error estimation [1], [5], [22], [23], [32], [36] and other research results involving the WG method are increasingly appearing.

Based on the reliable and efficient a posteriori error estimate, which provides a sound mathematical tool for the adaptive mesh refinement, adaptive finite element methods have been the object of intense study because of their ability to efficiently resolve problems with singularities and other rapid local variations. Since early 1990s [10], [11], in the context of parabolic equations, the a posteriori error estimates have been studied in the standard continuous finite element method [7], [17], [18], [21], [26], the discontinuous Galerkin finite element method (DG) [2], [3], [12], [14] and other numerical methods. The theory of a posteriori error estimates in the standard finite element and DG methods for parabolic equations is rather mature by now. However, it is known to us at least that there is no literature giving detailed analysis on it for the WG method. In this paper, we will give an available posteriori error estimates of the WG method for parabolic equations.

For parabolic equations, the elliptic reconstruction introduced in [21] for the semi-discrete finite element approximations and [17] for the fully discrete linear parabolic problems is the key tool in the process of deriving the a posteriori error estimates. It allows us to neatly separate the time error from the spatial one, and has expanded to the nonconforming methods for its general application. For the parabolic equation, u denotes its exact solution and $u_h$ denotes its numerical solution. In fact, the elliptic reconstruction $w=\mathcal{R}{u}_h$ of $u_h$ is the continuous solution of the elliptic problem whose numerical solution is $u_h$. The error $e=u_h-u$ can be divided as $e=(w-u)-(w-u_h)=\rho -ϵ$, where $\rho =w-u$ is the parabolic error, and $ϵ=w-u_h$ is the elliptic error. Different from obtaining a posteriori error estimates directly for each single method at hand, we can virtually use any a posteriori error estimates for elliptic equations to control the main part of the spatial error ϵ as long as it is reliable and efficient for the auxiliary elliptic problem. It is quite convenient for us to introduce the elliptic reconstruction ω in our analysis because it does not appear in the resulting a posteriori bounds.

For the WG scheme of parabolic problems in this paper, with the solving space $[P_k(K),P_{k-1}(\partial K),{[P_{k-1}(K)]}^2]$, no existing work gives results of the priori error estimates. We deduce the priori error estimates first, which will conduct us to analyze the subsequent numerical performance.

The framework of this paper is organized as follows. In Section 2, we introduce the Crank-Nicolson WG finite element algorithm for the parabolic problem. In Section 3, we present the priori error analysis of the energy norm based on the elliptic projection. In Section 4, using the elliptic reconstruction, we derive the a posteriori error estimates. In Section 5, some numerical experiments are provided to verify the reliability and efficiency of the estimates in uniform meshes. The adaptive algorithm and corresponding numerical performances in adaptive meshes are shown in Section 6.