A micromorphic-based artificial diffusion method for stabilized finite element approximation of convection-diffusion problems

TL;DR

This paper introduces a micromorphic-based artificial diffusion method that stabilizes finite element solutions for convection-diffusion equations by using an auxiliary variable, improving accuracy over existing methods.

Contribution

The paper proposes a novel stabilized finite element method based on micromorphic theory, introducing an auxiliary variable to enhance solution stability and accuracy.

Findings

Outperforms existing methods in accuracy

Effective in 1D and 2D convection-diffusion problems

Provides conditions for well-posedness

Abstract

We present a novel artificial diffusion method to circumvent the instabilities associated with the standard finite element approximation of convection-diffusion equations. Motivated by the micromorphic approach, we introduce an auxiliary variable, which is related to the gradient of the field of interest, and which leads to a coupled problem. Conditions for well-posedness of the resulting formulation are established. We carry out a comprehensive numerical study to compare the proposed methodology against some well-established approaches in one- and two-dimensional settings. The proposed method outperforms established approaches in general in approximating accurately the solutions to pertinent and challenging problems.