Published

Convergence and Optimality of an adaptive modified weak Galerkin finite element method

Yingying Xie, Shuhao Cao, Long Chen and Liuqiang Zhong

Numerical Methods for Partial Differential Equations, 39(5), 3847-3873, 2023

PDF   Bibtex   doi: https://doi.org/10.1002/num.23027

ABSTRACT:

 An adaptive modified weak Galerkin method (AmWG) for an
elliptic problem is studied in this paper, in addition to its
convergence and optimality. The 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 major tool is to bridge the connection between 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.