Functional Analysis for Helmholtz Equation in the Framework of Domain Decomposition

TL;DR

This paper explores the geometric structure of functional spaces in domain decomposition methods for Helmholtz equations, introduces algorithms with proven convergence, and offers new insights into classical methods through spectral analysis of boundary operators.

Contribution

It provides a geometric interpretation of domain decomposition operators for Helmholtz equations, leading to new algorithms and convergence proofs.

Findings

Spectral properties of Despres operators are established.

Convergence of the proposed algorithms is proved.

Numerical tests validate the theoretical results.

Abstract

This paper gives a geometric description of functional spaces related to Domain Decomposition techniques for computing solutions of Laplace and Helmholtz equations. Understanding the geometric structure of these spaces leads to algorithms for solving the equations. It leads also to a new interpretation of classical algorithms, enhancing convergence. The algorithms are given and convergence is proved. Numerical tests are given. This is done by building tools enabling geometric interpretations of the operators related to Domain Decomposition technique. The Despres operators, expressing conservation of energy for Helmholtz equation, are defined on the fictitious boundary and their spectral properties proved.It turns to be the key for proving convergence of the given algorithm for Helmholtz equation in a non-dissipating cavity. Using these tools, one can prove that the Domain Decomposition setting for the Helmholtz equation leads to an ill-posed problem. Nevertheless, one can prove that if a solution exists, it is unique. And that the algorithm do converge to the solution.