Adaptive Modified Weak Galerkin Method for Obstacle Problem

TL;DR

This paper develops a new adaptive finite element method using the Modified Weak Galerkin approach for obstacle problems, introducing residual-based error estimators that are proven reliable and efficient.

Contribution

It is the first to integrate MWG into an adaptive framework for variational inequalities, with rigorous error estimation and validation through numerical experiments.

Findings

Error estimators are reliable and efficient.

Numerical experiments confirm theoretical results.

First integration of MWG with adaptive obstacle problem methods.

Abstract

This article introduces a novel residual-based a posteriori error estimators for the Modified Weak Galerkin (MWG) finite element method applied to the obstacle problem. To the best of the author's knowledge, this work represents the first integration of the MWG method into an adaptive finite element framework for variational inequalities. The proposed error estimators is rigorously proven to be both reliable and efficient in quantifying the approximation error, measured in a natural energy norm. A key aspect of the analysis involves decomposing the discrete solution into conforming and non-conforming components, which plays a central role in the error estimation process. Numerical experiments are conducted to validate the theoretical findings, demonstrating the reliability and efficiency of the proposed error estimator.