An adaptive modified weak Galerkin method (AmWG) for an elliptic problem is studied in this paper, in addition to its convergence and optimality. The modified weak Galerkin bilinear form is simplified without the need of the skeletal variable, and the approximation space is chosen as the discontinuous polynomial space as in the discontinuous Galerkin method. Upon a reliable residual-based a posteriori error estimator, an adaptive algorithm is proposed together with its convergence and quasi-optimality proved for the lowest order case. The primary tool is to bridge the connection between the modified weak Galerkin method and the Crouzeix-Raviart nonconforming finite element. Unlike the traditional convergence analysis for methods with a discontinuous polynomial approximation space, the convergence of AmWG is penalty parameter free. Numerical results are presented to support the theoretical results.

翻译：本文研究了一种用于椭圆问题的自适应修正弱Galerkin方法（AmWG），并分析了其收敛性与最优性。该修正弱Galerkin双线性形式通过省略骨架变量得以简化，近似空间选用间断多项式空间（与间断Galerkin方法相同）。基于可靠的残差型后验误差估计器，我们提出了一种自适应算法，并针对最低阶情形证明了其收敛性与拟最优性。核心工具在于建立修正弱Galerkin方法与Crouzeix-Raviart非协调有限元之间的关联。与传统采用间断多项式近似空间的收敛性分析不同，AmWG的收敛性无需依赖罚参数。数值实验结果支持了理论分析结论。