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A Stabilizer Free Weak Galerkin Finite Element Method for Elliptic Equation with Lower Regularity

A Stabilizer Free Weak Galerkin Finite Element Method for Elliptic Equation with Lower Regularity

Year:    2024

Author:    Yiying Wang, Yongkui Zou, Xuan Liu, Chenguang Zhou

Numerical Mathematics: Theory, Methods and Applications, Vol. 17 (2024), Iss. 2 : pp. 514–533

Abstract

This paper presents error analysis of a stabilizer free weak Galerkin finite element method (SFWG-FEM) for second-order elliptic equations with low regularity solutions. The standard error analysis of SFWG-FEM requires additional regularity on solutions, such as $H^2$-regularity for the second-order convergence. However, if the solutions are in $H^{1+s}$ with $0 < s < 1,$ numerical experiments show that the SFWG-FEM is also effective and stable with the $(1+s)$-order convergence rate, so we develop a theoretical analysis for it. We introduce a standard $H^2$ finite element approximation for the elliptic problem, and then we apply the SFWG-FEM to approach this smooth approximating finite element solution. Finally, we establish the error analysis for SFWG-FEM with low regularity in both discrete $H^1$-norm and standard $L^2$-norm. The $(P_k(T ), P_{k−1}(e), [P_{k+1}(T)]^d)$ elements with dimensions of space $d = 2, 3$ are employed and the numerical examples are tested to confirm the theory.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/nmtma.OA-2023-0163

Numerical Mathematics: Theory, Methods and Applications, Vol. 17 (2024), Iss. 2 : pp. 514–533

Published online:    2024-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    20

Keywords:    Stabilizer free weak Galerkin FEM weak gradient error estimate lower regularity second-order elliptic equation.

Author Details

Yiying Wang

Yongkui Zou

Xuan Liu

Chenguang Zhou

  1. Numerical Analysis of Stabilizer‐Free Weak Galerkin Finite Element Method for Time‐Dependent Differential Equation Under Low Regularity

    Liu, Xuan

    Zou, Yongkui

    Wang, Yiying

    Zhou, Chenguang

    Wang, Huimin

    Numerical Methods for Partial Differential Equations, Vol. 41 (2025), Iss. 1

    https://doi.org/10.1002/num.23165 [Citations: 0]