Edge-based nonlinear diffusion for finite element approximations of convection-diffusion equations and its relation to algebraic flux-correction schemes

TL;DR

This paper introduces an edge-based nonlinear diffusion operator for finite element methods solving convection-diffusion equations, ensuring stability, maximum principle adherence, and optimal convergence, and relates it to algebraic flux correction schemes.

Contribution

It proposes a novel edge-based nonlinear diffusion operator that guarantees stability and maximum principle compliance, and links it to algebraic flux correction methods.

Findings

The method satisfies a discrete maximum principle.

It is Lipschitz continuous and linearity preserving.

Numerical tests demonstrate its effectiveness.

Abstract

For the case of approximation of convection--diffusion equations using piecewise affine continuous finite elements a new edge-based nonlinear diffusion operator is proposed that makes the scheme satisfy a discrete maximum principle. The diffusion operator is shown to be Lipschitz continuous and linearity preserving. Using these properties we provide a full stability and error analysis, which, in the diffusion dominated regime, shows existence, uniqueness and optimal convergence. Then the algebraic flux correction method is recalled and we show that the present method can be interpreted as an algebraic flux correction method for a particular definition of the flux limiters. The performance of the method is illustrated on some numerical test cases in two space dimensions.