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the modified weak galerkin finite element method for Solving Brinkman Equations

Journal of Mathematical Research with Applications 2019年第6期39卷 657-676页

作者： Li-na SUN Yue FENG Yuanyuan LIU Ran ZHANG Department of Mathematics Jilin University Laboratory of Computational Physics Institute of Applied Physics and Computational Mathematics

A modified weak galerkin(MWG) finite element method is introduced for the Brinkman equations in this paper. We approximate the model by the variational formulation based on two discrete weak gradient operators. In the MWG finite element method, discontinuous piecewise polynomials of degree k and k-1 are used to approximate the velocity u and the pressure p, respectively. The main idea of the MWG finite element method is to replace the boundary functions by the average of the interior functions. Therefore, the MWG finite element method has fewer degrees of freedom than the WG finite element method without loss of accuracy. The MWG finite element method satisfies the stability conditions for any polynomial with degree no more than k-1. The MWG finite element method is highly flexible by allowing the use of discontinuous functions on arbitrary polygons or polyhedra with certain shape *** order error estimates are established for the velocity and pressure approximations in H~1 and L~2 norms. Some numerical examples are presented to demonstrate the accuracy, convergence and stability of the method.

关键词： the Brinkman equations the modified weak galerkin finite element method discrete weak gradient

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A modified weak galerkin finite elementmethod FOR SINGULARLY PERTURBED PARABOLIC CONVECTION-DIFFUSION-REACTION PROBLEMS

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Journal of Computational Mathematics 2023年第6期41卷 1246-1280页

作者： Suayip Toprakseven Fuzheng Gao Faculty of Engineering Department of Computer ScienceArtvin Coruh UniversityArtvin08100Turkey School of Mathematics Shandong UniversityJinan 250100China

In this work,a modified weak galerkin finite element method is proposed for solving second order linear parabolic singularly perturbed convection-diffusion *** key feature of the proposed method is to replace the classical gradient and divergence operators by the modified weak gradient and modified divergence operators,*** apply the backward finite difference method in time and the modified weak galerkin finite element method in space on uniform *** stability analyses are presented for both semi-discrete and fully-discrete modified weak galerkin finite element *** order of convergences are obtained in suitable *** have achieved the same accuracy with the weak galerkin method while the degrees of freedom are reduced in our *** numerical examples are presented to support the theoretical *** is theoretically and numerically shown that the method is quite stable.

关键词： the modified weak galerkin finite element method Backward Euler method Parabolic convection-diffusion problems Error estimates

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