A Posteriori Error Estimation for Parabolic Equations with Enriched Galerkin Finite Element Methods

TL;DR

This paper develops a new a posteriori error estimation framework for the enriched Galerkin finite element method applied to linear parabolic equations, enhancing adaptive mesh refinement and error control.

Contribution

It introduces the first mathematical analysis of residual-based a posteriori error estimators for the EG method in parabolic problems, proving reliability and efficiency.

Findings

Error estimators are reliable and efficient for EG methods.

Adaptive mesh refinement improves solution accuracy.

Numerical examples confirm the effectiveness of the proposed approach.

Abstract

This paper introduces a novel a posteriori error estimation framework for the enriched Galerkin (EG) finite element method applied to linear parabolic equations. While the EG method has been recognized for its local conservation property and computational efficiency compared to discontinuous Galerkin methods, its mathematical analysis in the context of a posteriori error estimation for parabolic problems remains unexplored. In this work, we prove reliability and efficiency using the residual-based approach. Furthermore, we integrate these error estimators into an adaptive mesh refinement strategy, demonstrating their effectiveness in achieving efficient and reliable error control through several numerical examples. The proposed approach provides a significant advancement in the mathematical foundation and practical applicability of the EG method for time-dependent problems.