A Posteriori Error Estimator for Weak Galerkin Finite Element Method for Stokes Problem Using Diagonalization Techniques-Reference-Cited by-同舟云学术

A Posteriori Error Estimator for Weak Galerkin Finite Element Method for Stokes Problem Using Diagonalization Techniques

Published:2022-12-06 Issue:0 Volume:0 Page:
ISSN:1609-4840
Container-title:Computational Methods in Applied Mathematics
language:en
Short-container-title:

Author:

Zhang Jiachuan¹,Zhang Ran²,Li Jingzhi³

Affiliation:

1. School of Physical and Mathematical Sciences , Nanjing Tech University , Nanjing 211816 , P. R. China

2. School of Mathematics , Jilin University , Changchun 130012, Jilin , P. R. China

3. Department of Mathematics & National Center for Applied Mathematics , Shenzhen & SUSTech International Center for Mathematics , Southern University of Science and Technology , Shenzhen 518055 , P. R. China

Abstract

Abstract Based on a hierarchical basis a posteriori error estimator, an adaptive weak Galerkin finite element method (WGFEM) is proposed for the Stokes problem in two and three dimensions. In this paper, we propose two novel diagonalization techniques for velocity and pressure, respectively. Using diagonalization techniques, we need only to solve two diagonal linear algebraic systems corresponding to the degree of freedom to get the error estimator. The upper bound and lower bound of the error estimator are also shown to address the reliability of the adaptive method. Numerical simulations are provided to demonstrate the effectiveness and robustness of our algorithm.

Publisher

Walter de Gruyter GmbH

Subject

Applied Mathematics,Computational Mathematics,Numerical Analysis

Link

https://www.degruyter.com/document/doi/10.1515/cmam-2022-0087/pdf

Reference39 articles.

1. S. Adjerid, M. Aiffa and J. E. Flaherty, Hierarchical finite element bases for triangular and tetrahedral elements, Comput. Methods Appl. Mech. Engrg. 190 (2001), no. 22–23, 2925–2941.

2. J. H. Adler, X. Hu, L. Mu and X. Ye, An a posteriori error estimator for the weak Galerkin least-squares finite-element method, J. Comput. Appl. Math. 362 (2019), 383–399.

3. P. F. Antonietti, L. Beirão da Veiga, C. Lovadina and M. Verani, Hierarchical a posteriori error estimators for the mimetic discretization of elliptic problems, SIAM J. Numer. Anal. 51 (2013), no. 1, 654–675.

4. R. Araya, G. R. Barrenechea and A. Poza, An adaptive stabilized finite element method for the generalized Stokes problem, J. Comput. Appl. Math. 214 (2008), no. 2, 457–479.

5. R. Araya, A. H. Poza and F. Valentin, On a hierarchical error estimator combined with a stabilized method for the Navier–Stokes equations, Numer. Methods Partial Differential Equations 28 (2012), no. 3, 782–806.