Parallel Schwarz Waveform Relaxation Algorithm for an N-Dimensional Semilinear Heat Equation

TL;DR

This paper proves the well-posedness and convergence of a parallel Schwarz waveform relaxation algorithm for solving N-dimensional semilinear heat equations, introducing exponential decay error estimates for convergence analysis.

Contribution

It provides a new convergence proof for the algorithm applied to N-dimensional semilinear heat equations, including a novel exponential decay error estimate technique.

Findings

Proved well-posedness of the algorithm.

Established convergence with exponential decay estimates.

Applicable to multisubdomain N-dimensional problems.

Abstract

We present in this paper a proof of well-posedness and convergence for the parallel Schwarz Waveform Relaxation Algorithm adapted to an N-dimensional semilinear heat equation. Since the equation we study is an evolution one, each subproblem at each step has its own local existence time, we then determine a common existence time for every problem in any subdomain at any step. We also introduce a new technique: Exponential Decay Error Estimates, to prove the convergence of the Schwarz Methods, with multisubdomains, and then apply it to our problem.