Multilevel Preconditioners for Reaction-Diffusion Problems with Discontinuous Coefficients

TL;DR

This paper analyzes the effectiveness of multilevel preconditioners, like BPX and multigrid V-cycle, for solving reaction-diffusion equations with discontinuous coefficients, extending previous convergence results.

Contribution

It extends multilevel convergence analysis to reaction-diffusion problems with discontinuous coefficients, focusing on finite element discretizations.

Findings

Performance of BPX and multigrid preconditioners is influenced by coefficient discontinuities.

Theoretical convergence results are adapted for piecewise-constant coefficients.

Insights into preconditioner efficiency for complex reaction-diffusion systems.

Abstract

In this paper, we extend some of the multilevel convergence results obtained by Xu and Zhu in [Xu and Zhu, M3AS 2008], to the case of second order linear reaction-diffusion equations. Specifically, we consider the multilevel preconditioners for solving the linear systems arising from the linear finite element approximation of the problem, where both diffusion and reaction coefficients are piecewise-constant functions. We discuss in detail the influence of both the discontinuous reaction and diffusion coefficients to the performance of the classical BPX and multigrid V-cycle preconditioners.