First-order system least squares finite-elements for singularly perturbed reaction-diffusion equations

TL;DR

This paper introduces a novel symmetric first-order-system least squares finite-element method for singularly perturbed reaction-diffusion equations, effectively capturing layer phenomena with proven stability and robustness.

Contribution

It develops a new FOSLS finite-element scheme with a balanced norm for better approximation of reaction-diffusion layers, including stability analysis and numerical validation.

Findings

Method is stable and coercive.

Achieves accurate approximation on layer-adapted meshes.

Numerical experiments confirm robustness and effectiveness.

Abstract

We propose a new first-order-system least squares (FOSLS) finite-element discretization for singularly perturbed reaction-diffusion equations. Solutions to such problems feature layer phenomena, and are ubiquitous in many areas of applied mathematics and modelling. There is a long history of the development of specialized numerical schemes for their accurate numerical approximation. We follow a well-established practice of employing a priori layer-adapted meshes, but with a novel finite-element method that yields a symmetric formulation while also inducing a so-called "balanced" norm. We prove continuity and coercivity of the FOSLS weak form, present a suitable piecewise uniform mesh, and report on the results of numerical experiments that demonstrate the accuracy and robustness of the method.