A comparative study on the weak Galerkin, discontinuous Galerkin, and mixed finite element methods

Lin, Guang; Liu, Jiangguo; Sadre-Marandi, Farrah

doi:10.1016/j.cam.2014.06.024

Cited by 47 publications

(20 citation statements)

References 11 publications

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“…It has been demonstrated in [18,19] that the numerical pressures, velocities obtained from using the WGFEM and the MFEM have negligible differences, when the permeability in the Darcy equation is a piecewise constant scalar.…”

Section: Mfem For Darcymentioning

confidence: 98%

“…This is an obvious advantage over the classical mixed finite element method, which results in indefinite linear systems and special solvers, e.g., the Uzawa algorithm, need to be used. WGFEM holds also advantages over the discontinuous Galerkin finite element method [19], which proliferates in number of unknowns and involves problem-dependent penalty factors.…”

Section: )mentioning

confidence: 99%

“…The numerical pressure is then used to generate a discrete weak gradient and then a numerical flux that is both locally conservative and normally continuous across element interfaces. Compared to the MFEM [19], the WGFEM has the same number of unknowns but results in a definite system that is much easier to solve. Compared to the DGFEM, the WGFEM has less unknowns and has no need for choosing penalty factors.…”

Section: Introductionmentioning

confidence: 99%

“…WGFEM uses degrees of freedom in element interiors and on mesh skeleton to establish a new type of approximations to differential operators. When applied to the Darcy equation [19], the WGFEM solves for a numerical pressure that is defined both inside elements and on element interfaces. The numerical pressure is then used to generate a discrete weak gradient and then a numerical flux that is both locally conservative and normally continuous across element interfaces.…”

Section: Introductionmentioning

confidence: 99%

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On Application of the Weak Galerkin Finite Element Method to a Two-Phase Model for Subsurface Flow

2015

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This paper presents studies on applying the novel weak Galerkin finite element method (WGFEM) to a two-phase model for subsurface flow, which couples the Darcy equation for pressure and a transport equation for saturation in a nonlinear manner. The coupled problem is solved in the framework of operator decomposition. Specifically, the Darcy equation is solved by the WGFEM, whereas the saturation is solved by a finite volume method. The numerical velocity obtained from solving the Darcy equation by the WGFEM is locally conservative and has continuous normal components across element interfaces. This ensures accuracy and robustness of the finite volume solver for the saturation equation. Numerical experiments on benchmarks demonstrate that the combined methods can handle very well two-phase flow problems in high-contrast heterogeneous porous media.

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Section: Mfem For Darcymentioning

confidence: 98%

Section: )mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On Application of the Weak Galerkin Finite Element Method to a Two-Phase Model for Subsurface Flow

2015

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“…Recently, the weak Galerkin finite element method has attracted much attention in the field of numerical partial differential equations [7,9,10,11,13,15,16,17,18,19,20].…”

Section: Introductionmentioning

confidence: 99%

A stable weak Galerkin finite element method for Stokes problem

Zhang

Lin

2018

Journal of Computational and Applied Mathematics

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We study the weak Galerkin finite element method for Stokes problem. A new weak Galerkin finite element velocity-pressure space pair is presented which satisfies the discrete inf-sup condition. Based on this space pair, we establish a stable weak Galerkin approximation scheme without adding any stability term or penalty term. Then, we further derive the optimal error estimates for velocity and pressure approximations, respectively. Numerical experiments are provided to illustrate the theoretical analysis.

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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 comparative study on the weak Galerkin, discontinuous Galerkin, and mixed finite element methods

Cited by 47 publications

References 11 publications

On Application of the Weak Galerkin Finite Element Method to a Two-Phase Model for Subsurface Flow

On Application of the Weak Galerkin Finite Element Method to a Two-Phase Model for Subsurface Flow

A stable weak Galerkin finite element method for Stokes problem

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

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