A bound-preserving and conservative enriched Galerkin method for elliptic problems

TL;DR

This paper introduces a novel enriched Galerkin method for elliptic problems that ensures physical bounds, maintains local conservation, and achieves optimal convergence through a specialized splitting approach to address ill-conditioning.

Contribution

The paper presents a new bound-preserving, conservative enriched Galerkin scheme with a splitting approach to improve stability and accuracy in elliptic problem solutions.

Findings

Proves existence of discrete solutions.

Establishes optimal error estimates.

Numerical validation confirms theoretical results.

Abstract

We propose a locally conservative enriched Galerkin scheme that preserves the physical bounds for an elliptic problem. To this end, we use a substantial over-penalization of the discrete solution's jumps to obtain optimal convergence. To avoid the ill-conditioning issues that arise in over-penalized schemes, we introduce an involved splitting approach that separates the system of equations for the discontinuous solution part from the system of equations for the continuous solution part, yielding well-behaved subproblems. We prove the existence of discrete solutions and optimal error estimates, which are validated numerically.