On $L^2$ Error Estimate for Weak Galerkin Finite Element Methods for Parabolic Problems

Gao, Fuzheng; Mu, Lin

doi:10.4208/jcm.1401-m4385

Cited by 31 publications

(1 citation statement)

References 15 publications

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“…Experiments with the GrFEM have shown an increase in economy and in the nodal accuracy compared to FE solutions of the Burgers' equations [3,5,6] and for other problems [8,4,9,27,32] . Recently, the weak Galerkin finite element method has catched much consideration in the field of numericaln PDEs , The idea of the weak Galerkin method was first introduced by Wang and Ye [14], the method was applied to the second order elliptic equations [1,15,16,19,20,22,24,28], the Stokes equations [17,21], Parabolic equations [10,11,25], biharmonic equations [23,26], Navier-Stokes equations [7,18,30] and 1-D Burgers' equation [31], etc. In this paper we present (WGFEM) and (WGrFEM) for 2-D Burgers' problem with a fully-discrete approximation for the time variable, the backward-difference formula for the time variable is examined and the stability and error estimate are proved for these methods.…”

Section: Introductionmentioning

confidence: 99%

Weak Galerkin and Weak Group Finite Element Methods for 2-D Burgers' Problem

Noon¹

2020

ARFMTS

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In this paper, we apply weak Galerkin finite element method (WGFEM) and weak group finite element method (WGrFEM) for 2-D Burgers' problem by using the weak functions and their corresponding discrete weak derivatives in standard weak form. We consider the fully formulation using the backward-difference form for the time variable and we prove the stability and an error estimate for these methods. The numerical examples of our methods have been carried out through implementation in MATLAB programs and compared with the exact solutions and other available literature to illustrate the efficiency of the proposed methods, the WGFEM and the WGrFEM provide convergent approximations and handles the equation well in different cases, the results obtained of our methods are very acceptable and competent more than the results available in the other literature. Moreover, a WGrFEM is displayed to be better than the GFEM.

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Section: Introductionmentioning

confidence: 99%

Weak Galerkin and Weak Group Finite Element Methods for 2-D Burgers' Problem

Noon¹

2020

ARFMTS

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Numerical and theoretical study of weak Galerkin finite element solutions of Turing patterns in reaction–diffusion systems

Khaled-Abad

Salehi

2020

Numerical Methods Partial

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In this paper, we introduce numerical schemes and their analysis based on weak Galerkin finite element framework for solving 2-D reaction-diffusion systems. Weak Galerkin finite element method (WGFEM) for partial differential equations relies on the concept of weak functions and weak gradients, in which differential operators are approximated by weak forms through the Green's theorem. This method allows the use of totally discontinuous functions in the approximation space. In the current work, the WGFEM solves reaction-diffusion systems to find unknown concentrations (u, v) in element interiors and boundaries in the weak Galerkin finite element space WG(P 0 , P 0 , RT 0). The WGFEM is used to approximate the spatial variables and the time discretization is made by the backward Euler method. For reaction-diffusion systems, stability analysis and error bounds for semi-discrete and fully discrete schemes are proved. Accuracy and efficiency of the proposed method successfully tested on several numerical examples and obtained results satisfy the well-known result that for small values of diffusion coefficient, the steady state solution converges to equilibrium point. Acquired numerical results asserted the efficiency of the proposed scheme.

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A weak Galerkin method and its two‐grid algorithm for the quasi‐linear elliptic problems of non‐monotone type

Zhu¹,

Xie²

2022

Numerical Methods Partial

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In this article, a weak Galerkin (WG) method is first presented and analyzed for the quasi-linear elliptic problem of non-monotone type. By using Brouwer's fixed point technique, the existence of WG solution and error estimates in both the energy norm and the L 2 norm are derived. Then an efficient two-grid WG method is introduced to improve the computational efficiency. The convergence error of the two-grid WG method is analyzed in the energy norm. Numerical experiments are presented to verify our theoretical findings.

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On $L^2$ Error Estimate for Weak Galerkin Finite Element Methods for Parabolic Problems

Cited by 31 publications

References 15 publications

Weak Galerkin and Weak Group Finite Element Methods for 2-D Burgers' Problem

Weak Galerkin and Weak Group Finite Element Methods for 2-D Burgers' Problem

Numerical and theoretical study of weak Galerkin finite element solutions of Turing patterns in reaction–diffusion systems

A weak Galerkin method and its two‐grid algorithm for the quasi‐linear elliptic problems of non‐monotone type

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