Error estimates for a splitting integrator for abstract semilinear boundary coupled systems

TL;DR

This paper introduces a splitting integrator for abstract semilinear boundary coupled systems, providing convergence analysis and numerical experiments demonstrating its effectiveness in handling coupled parabolic problems.

Contribution

The paper develops a novel operator splitting method for boundary coupled systems and proves its convergence using error estimates based on semigroup properties.

Findings

The method achieves expected convergence rates in numerical experiments.

It effectively decouples linear and nonlinear components for efficient computation.

Numerical results confirm the theoretical error estimates.

Abstract

We derive a numerical method, based on operator splitting, to abstract parabolic semilinear boundary coupled systems. The method decouples the linear components which describe the coupling and the dynamics in the bulk and on the surface, and treats the nonlinear terms by approximating the integral in the variation of constants formula. The convergence proof is based on estimates for a recursive formulation of the error, using the parabolic smoothing property of analytic semigroups and a careful comparison of the exact and approximate flows. Numerical experiments, including problems with dynamic boundary conditions, reporting on convergence rates are presented.