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Volume 33, Issue 2
A Posteriori Error Estimate of Weak Galerkin FEM for Stokes Problem Using Auxiliary Subspace Techniques

Jiachuan Zhang, Ran Zhang & Xiaoshen Wang

DOI: 10.4208/cicp.OA-2022-0207

Commun. Comput. Phys., 33 (2023), pp. 511-537.

Published online: 2023-03

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  • Abstract

Based on the auxiliary subspace techniques, a posteriori error estimator of nonconforming weak Galerkin finite element method (WGFEM) for Stokes problem in two and three dimensions is presented. Without saturation assumption, we prove that the WGFEM approximation error is bounded by the error estimator up to an oscillation term. The computational cost of the approximation and the error problems is considered in terms of size and sparsity of the system matrix. To reduce the computational cost of the error problem, an equivalent error problem is constructed by using diagonalization techniques, which needs to solve only two diagonal linear algebraic systems corresponding to the degree of freedom $(d.o.f)$ to get the error estimator. Numerical experiments are provided to demonstrate the effectiveness and robustness of the a posteriori error estimator.

  • Keywords

Auxiliary subspace techniques, diagonalization techniques, weak Galerkin, A posteriori error estimate, Stokes problem.

  • AMS Subject Headings

65N15, 65N30, 76D07

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-33-511, author = {Zhang , JiachuanZhang , Ran and Wang , Xiaoshen}, title = {A Posteriori Error Estimate of Weak Galerkin FEM for Stokes Problem Using Auxiliary Subspace Techniques}, journal = {Communications in Computational Physics}, year = {2023}, volume = {33}, number = {2}, pages = {511--537}, abstract = {

Based on the auxiliary subspace techniques, a posteriori error estimator of nonconforming weak Galerkin finite element method (WGFEM) for Stokes problem in two and three dimensions is presented. Without saturation assumption, we prove that the WGFEM approximation error is bounded by the error estimator up to an oscillation term. The computational cost of the approximation and the error problems is considered in terms of size and sparsity of the system matrix. To reduce the computational cost of the error problem, an equivalent error problem is constructed by using diagonalization techniques, which needs to solve only two diagonal linear algebraic systems corresponding to the degree of freedom $(d.o.f)$ to get the error estimator. Numerical experiments are provided to demonstrate the effectiveness and robustness of the a posteriori error estimator.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2022-0207}, url = {http://global-sci.org/intro/article_detail/cicp/21498.html} }
TY - JOUR T1 - A Posteriori Error Estimate of Weak Galerkin FEM for Stokes Problem Using Auxiliary Subspace Techniques AU - Zhang , Jiachuan AU - Zhang , Ran AU - Wang , Xiaoshen JO - Communications in Computational Physics VL - 2 SP - 511 EP - 537 PY - 2023 DA - 2023/03 SN - 33 DO - http://doi.org/10.4208/cicp.OA-2022-0207 UR - https://global-sci.org/intro/article_detail/cicp/21498.html KW - Auxiliary subspace techniques, diagonalization techniques, weak Galerkin, A posteriori error estimate, Stokes problem. AB -

Based on the auxiliary subspace techniques, a posteriori error estimator of nonconforming weak Galerkin finite element method (WGFEM) for Stokes problem in two and three dimensions is presented. Without saturation assumption, we prove that the WGFEM approximation error is bounded by the error estimator up to an oscillation term. The computational cost of the approximation and the error problems is considered in terms of size and sparsity of the system matrix. To reduce the computational cost of the error problem, an equivalent error problem is constructed by using diagonalization techniques, which needs to solve only two diagonal linear algebraic systems corresponding to the degree of freedom $(d.o.f)$ to get the error estimator. Numerical experiments are provided to demonstrate the effectiveness and robustness of the a posteriori error estimator.

Zhang , JiachuanZhang , Ran and Wang , Xiaoshen. (2023). A Posteriori Error Estimate of Weak Galerkin FEM for Stokes Problem Using Auxiliary Subspace Techniques. Communications in Computational Physics. 33 (2). 511-537. doi:10.4208/cicp.OA-2022-0207
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