Based on a posteriori error estimator with hierarchical bases, an adaptive weak Galerkin finite element method (WGFEM) is proposed for the elliptic problem with mixed boundary conditions. For the posteriori error estimator, we are only required to solve a linear algebraic system with diagonal entries corresponding to the degree of freedoms, which significantly reduces the computational cost. The upper and lower bounds of the error estimator are shown to addresses the reliability and efficiency of the adaptive approach. Numerical simulations are provided to demonstrate the effectiveness and robustness of the proposed method.

中文翻译：

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具有层次基础的椭圆问题的自适应弱Galerkin有限元方法

基于具有分层基础的后验误差估计器，针对混合边界条件的椭圆问题，提出了一种自适应的弱Galerkin有限元方法（WGFEM）。对于后验误差估计器，我们只需要求解具有与自由度相对应的对角线项的线性代数系统，就可以大大降低计算成本。示出了误差估计器的上限和下限以解决自适应方法的可靠性和效率。数值仿真表明了该方法的有效性和鲁棒性。