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
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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]