An exponential fitting scheme for general convection-diffusion equations on tetrahedral meshes

TL;DR

This paper develops an exponentially fitted finite element scheme for convection-diffusion equations on tetrahedral meshes, which is stable, monotone, and convergent without relying on flow direction checks.

Contribution

It introduces a new exponential fitting scheme for general convection-diffusion problems on simplicial meshes, extending previous methods to full coefficient matrices.

Findings

The scheme is stable for small step sizes.

It is monotone under certain mesh conditions.

First-order convergence is achieved with minimal solution smoothness.

Abstract

This paper contains construction and analysis a finite element approximation for convection dominated diffusion problems with full coefficient matrix on general simplicial partitions in $R^d$, $d=2,3$. This construction is quite close to the scheme of Xu and Zikatanov (Math. Comp. 1999) where a diagonal coefficient matrix has been considered. The scheme is of the class of exponentially fitted methods that does not use upwind or checking the flow direction. It is stable for sufficiently small discretization step-size assuming that the boundary value problem for the convection-diffusion equation is uniquely solvable. Further, it is shown that, under certain conditions on the mesh the scheme is monotone. Convergence of first order is derived under minimal smoothness of the solution.