Weighted and unweighted enrichment strategies for solving the Poisson problem with Dirichlet boundary conditions

TL;DR

This paper introduces weighted and unweighted enrichment strategies to improve the accuracy of linear Lagrangian finite element methods for solving the Poisson problem with Dirichlet boundary conditions, especially in capturing sharp gradients.

Contribution

The paper proposes two new three-parameter families of enrichment functions and derives explicit error bounds, enhancing finite element approximation accuracy.

Findings

Numerical experiments show improved accuracy with the proposed enrichment strategies.

The explicit error bounds validate the effectiveness of the new enrichment functions.

The methods are applicable to a wide range of boundary value problems.

Abstract

In this paper, we propose weighted and unweighted enrichment strategies to enhance the accuracy of the linear lagrangian finite element for solving the Poisson problem with Dirichlet boundary conditions. We first recall key examples of admissible enrichment functions, specifically designed to overcome the limitations of the linear lagrangian finite element in capturing solution features such as sharp gradients and boundary-layer phenomena. We then introduce two novel three-parameter families of weighted enrichment functions and derive an explicit error bound in $L^2$-norm. Numerical experiments confirm the effectiveness of the proposed approach in improving approximation accuracy, demonstrating its potential for a wide range of applications.