# Natural domain decomposition algorithms for the solution of time-harmonic elastic waves

## TL;DR

This paper introduces and analyzes new Schwarz domain decomposition algorithms with improved transmission conditions for solving time-harmonic elastic wave equations, demonstrating enhanced convergence and efficiency over classical methods.

## Contribution

The paper proves classical Schwarz methods do not converge for Navier equations and proposes new transmission conditions that ensure convergence and better performance.

## Key findings

- Classical Schwarz method is not convergent for Navier equations.
- New Schwarz method with adapted transmission conditions converges when overlap is sufficient.
- Numerical experiments show improved solver efficiency with the new method.

## Abstract

We study for the first time Schwarz domain decomposition methods for the solution of the Navier equations modeling the propagation of elastic waves. These equations in the time harmonic regime are difficult to solve by iterative methods, even more so than the Helmholtz equation. We first prove that the classical Schwarz method is not convergent when applied to the Navier equations, and can thus not be used as an iterative solver, only as a preconditioner for a Krylov method. We then introduce more natural transmission conditions between the subdomains, and show that if the overlap is not too small, this new Schwarz method is convergent. We illustrate our results with numerical experiments, both for situations covered by our technical two subdomain analysis, and situations that go far beyond, including many subdomains, cross points, heterogeneous materials in a transmission problem, and Krylov acceleration. Our numerical results show that the Schwarz method with adapted transmission conditions leads systematically to a better solver for the Navier equations than the classical Schwarz method.