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.
Funding statement: The work of Jiachuan Zhang was supported by the Natural Science Foundation of Jiangsu Province (No. BK20210540) and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (No. 21KJB110015). The work of Ran Zhang was supported by the National Key Research and Development Program of China (No. 2020YFA0713601). The work of Jingzhi Li was partially supported by the National Natural Science Foundation of China (No. 11971221), Guangdong NSF Major Fund (No. 2021ZDZX1001), the Shenzhen Sci-Tech Fund (No. RCJC20200714114556020, JCYJ20200109115422828 and JCYJ20190809150413261).
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