Convergence of the finite difference scheme for a general class of the spatial segregation of reaction-diffusion systems

TL;DR

This paper proves that a finite difference scheme converges to the stationary solutions of a broad class of reaction-diffusion systems with multiple components, including multi-phase obstacle problems, as the mesh size approaches zero.

Contribution

It establishes the convergence of a finite difference scheme for spatially segregated reaction-diffusion systems with multiple components, extending to multi-phase obstacle problems.

Findings

Proved convergence of the finite difference scheme for reaction-diffusion systems.

Established the scheme's convergence to stationary states as mesh size tends to zero.

Included convergence results for multi-phase obstacle problems.

Abstract

In this work we prove convergence of the finite difference scheme for equations of stationary states of a general class of the spatial segregation of reaction-diffusion systems with $m\geq 2$ components. More precisely, we show that the numerical solution $u_h^l$, given by the difference scheme, converges to the $l^{th}$ component $u_l,$ when the mesh size $h$ tends to zero, provided $u_l\in C^2(\Omega),$ for every $l=1,2,\dots,m.$ In particular, our proof provides convergence of a difference scheme for the multi-phase obstacle problem.