An algebraically stabilized method for convection-diffusion-reaction problems with optimal experimental convergence rates on general meshes

TL;DR

This paper introduces the SMUAS method, an algebraically stabilized finite element approach for convection-diffusion-reaction problems, achieving optimal convergence rates on general meshes while preserving the maximum principle.

Contribution

The paper develops the SMUAS stabilization technique, ensuring DMP satisfaction and optimal convergence on arbitrary meshes, addressing accuracy issues in existing methods.

Findings

SMUAS preserves linearity and satisfies DMP on arbitrary meshes.

Numerical results show SMUAS achieves optimal convergence rates.

Theoretical analysis explains the stability and accuracy improvements.

Abstract

Algebraically stabilized finite element discretizations of scalar steady-state convection-diffusion-reaction equations often provide accurate approximate solutions satisfying the discrete maximum principle (DMP). However, it was observed that a deterioration of the accuracy and convergence rates may occur for some problems if meshes without local symmetries are used. The paper investigates these phenomena both numerically and analytically and the findings are used to design a new algebraic stabilization called Symmetrized Monotone Upwind-type Algebraically Stabilized (SMUAS) method. It is proved that the SMUAS method is linearity preserving and satisfies the DMP on arbitrary simplicial meshes. Numerical results indicate that the SMUAS method leads to optimal convergence rates on general meshes.