Efficient discretization and preconditioning of the singularly perturbed Reaction Diffusion problem

TL;DR

This paper introduces efficient discretization and preconditioning techniques for reaction diffusion problems with dominant reaction terms, ensuring robustness across various perturbation parameters and mesh types.

Contribution

It develops a novel preconditioning strategy and discretization methods based on optimal test norms and saddle point reformulation for singularly perturbed reaction diffusion equations.

Findings

Preconditioning strategy effective over a wide range of parameters

Methods work on both uniform and non-uniform meshes

Numerical examples demonstrate efficiency and robustness

Abstract

We consider the reaction diffusion problem and present efficient ways to discretize and precondition in the singular perturbed case when the reaction term dominates the equation. Using the concepts of optimal test norm and saddle point reformulation, we provide efficient discretization processes for uniform and non-uniform meshes. We present a preconditioning strategy that works for a large range of the perturbation parameter. Numerical examples to illustrate the efficiency of the method are included for a problem on the unit square.