Condition number for finite element discretisation of nonlocal PDE systems with applications to biology

TL;DR

This paper analyzes the condition number of finite element discretizations for coupled non-local PDE systems in biological wound healing, providing bounds and insights on parameter effects to ensure numerical stability.

Contribution

It establishes bounds for the condition number of discretized non-local PDE systems and examines how model parameters influence system conditioning.

Findings

Bounds for the condition number are derived.

Model parameters significantly affect system conditioning.

Guidelines for parameter choices to maintain numerical stability.

Abstract

In this work, we investigate the condition number for a system of coupled non-local reaction-diffusion-advection equations developed in the context of modelling normal and abnormal wound healing. Following a finite element discretisation of the coupled non-local system, we establish bounds for this condition number. We further discuss how model parameter choices affect the conditioning of the system. Finally, we discuss how the step size of the chosen time-stepping scheme and the spatial grid size of the finite element methods affect the bound for the condition number, while also suggesting possible parameter ranges that could keep the model well conditioned.