Adaptive Grids in the Context of Algebraic Stabilizations for Convection-Diffusion-Reaction Equations

TL;DR

This paper evaluates three algebraically stabilized finite element schemes on adaptively refined grids for convection-diffusion-reaction equations, focusing on their stability, accuracy, and computational efficiency.

Contribution

It compares the performance of AFC with Kuzmin and BJK limiters and the MUAS method on various grid types, including hanging vertices, highlighting their suitability for complex problems.

Findings

All schemes satisfy the discrete maximum principle under certain conditions.

MUAS shows superior accuracy in capturing sharp layers.

Hanging vertex grids require a special algorithmic step for scheme application.

Abstract

Three algebraically stabilized finite element schemes for discretizing convection-diffusion-reaction equations are studied on adaptively refined grids. These schemes are the algebraic flux correction (AFC) scheme with Kuzmin limiter, the AFC scheme with BJK limiter, and the recently proposed Monotone Upwind-type Algebraically Stabilized (MUAS) method. Both, conforming closure of the refined grids and grids with hanging vertices are considered. A non-standard algorithmic step becomes necessary before these schemes can be applied on grids with hanging vertices. The assessment of the schemes is performed with respect to the satisfaction of the global discrete maximum principle (DMP), the accuracy, e.g., smearing of layers, and the efficiency in solving the corresponding nonlinear problems.