Discrete reaction-diffusion system with stochastic dynamical boundary conditions: convergence results

TL;DR

This paper proves that a space-discrete approximation of a nonlinear stochastic reaction-diffusion system with dynamic boundary conditions converges to the true continuum solution, using a splitting strategy and compactness arguments.

Contribution

It introduces a novel convergence analysis for a discrete scheme approximating a complex stochastic reaction-diffusion system with boundary noise.

Findings

Established convergence of the discrete scheme to the continuum solution.

Developed a splitting strategy to handle stochastic boundary conditions.

Discussed convergence of a fully discrete approximation.

Abstract

A space discrete approximation to a highly nonlinear reaction-diffusion system endowed with a stochastic dynamical boundary condition is analyzed and the convergence of the discrete scheme to the solution to the corresponding continuum random system is established. A splitting strategy allows us to decompose the random system into a space-discrete heat equation with a stochastic boundary condition, and a nonlinear and nonlocal space-discrete differential system coupled with the first one and with deterministic initial and boundary conditions. The convergence result is obtained by first establishing some a priori estimates for both space-discrete splitted variables and then exploiting compact embedding theorems for time-space Besov spaces on the positive lattice. The convergence of a fully discrete approximation of the random system is also discussed.