Simplex-averaged finite element methods for $H({\rm grad})$, $H({\rm curl})$ and $H({\rm div})$ convection-diffusion problems

TL;DR

This paper develops and analyzes finite element methods for convection-diffusion problems in 3D, using coefficient averaging on sub-simplexes to create exponential fitting schemes that are robust across different diffusion regimes.

Contribution

It introduces a novel finite element approach that properly averages coefficients on sub-simplexes, ensuring stability and convergence for convection-diffusion problems in $H(D)$ spaces.

Findings

Methods are well-posed for small mesh sizes.

First-order convergence is achieved under minimal smoothness.

Numerical examples confirm robustness and effectiveness.

Abstract

This paper is devoted to the construction and analysis of the finite element approximations for the $H(D)$ convection-diffusion problems, where $D$ can be chosen as ${\rm grad}$, ${\rm curl}$ or ${\rm div}$ in 3D case. An essential feature of these constructions is to properly average the PDE coefficients on the sub-simplexes. The schemes are of the class of exponential fitting methods that result in special upwind schemes when the diffusion coefficient approaches to zero. Their well-posedness are established for sufficiently small mesh size assuming that the convection-diffusion problems are uniquely solvable. Convergence of first order is derived under minimal smoothness of the solution. Some numerical examples are given to demonstrate the robustness and effectiveness for general convection-diffusion problems.