Adaptive Modified Weak Galerkin Method for Obstacle Problem
Tanvi Wadhawan
TL;DR
This work addresses the obstacle problem framed as a variational inequality and develops an adaptive Modified Weak Galerkin (MWG) finite element method. A residual-based a posteriori error estimator is derived and proven to be reliable and efficient, leveraging a space-decomposition of the discrete space and a discrete Lagrange multiplier to handle the contact condition. The method integrates the continuous and discrete Lagrange multipliers $\lambda(u)$ and $\lambda_h(u_h)$, with a conforming/nonconforming decomposition that yields rigorous error bounds in the energy norm and $H^{-1}$ for the multiplier. Numerical experiments in two dimensions corroborate the theoretical results, showing optimal convergence rates and refined meshes concentrated near the evolving free boundary, thereby advancing adaptive MWG techniques for variational inequalities.
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.