A numerical algorithm based on probing to find optimized transmission conditions

TL;DR

This paper introduces a numerical algorithm that efficiently finds optimized transmission conditions for Schwarz Methods, enhancing their applicability and convergence speed across various problems by requiring minimal offline computations.

Contribution

The paper presents a novel numerical algorithm to determine optimized transmission conditions for any problem, reducing the strong theoretical assumptions of previous methods.

Findings

Algorithm requires few subdomain solves in offline phase

Results in faster convergence of Schwarz Methods

Applicable to a wide range of problems

Abstract

Optimized Schwarz Methods (OSMs) are based on optimized transmission conditions along the interfaces between the subdomains. Optimized transmission conditions are derived at the theoretical level, using techniques developed in the last decades. The hypothesis behind these analyses are quite strong, so that the applicability of OSMs is still limited. In this manuscript, we present a numerical algorithm to obtain optimized transmission conditions for any given problem at hand. This algorithm requires few subdomain solves to be performed in an offline phase. This additional cost is usually negligible due to the resulting faster convergence, even in a single-query context.