A Trefftz-like coarse space for the two-level Schwarz method on perforated domains

TL;DR

This paper introduces a Trefftz-like coarse space for two-level Schwarz methods that improves convergence and robustness in solving elliptic PDEs on complex perforated urban domains, outperforming traditional Nicolaides spaces.

Contribution

A novel Trefftz-like coarse space using polygonal subdomains and harmonic basis functions enhances the efficiency of Schwarz methods on perforated domains, addressing limitations of existing approaches.

Findings

Significantly reduces Krylov iterations compared to Nicolaides space.

Demonstrates robustness across complex urban topographies.

Scales effectively with problem complexity.

Abstract

We consider a new coarse space for the ASM and RAS preconditioners to solve elliptic partial differential equations on perforated domains, where the numerous polygonal perforations represent structures such as walls and buildings in urban data. With the eventual goal of modelling urban floods by means of the nonlinear Diffusive Wave equation, this contribution focuses on the solution of linear problems on perforated domains. Our coarse space uses a polygonal subdomain partitioning and is spanned by Trefftz-like basis functions that are piecewise linear on the boundary of a subdomain and harmonic inside it. It is based on nodal degrees of freedom that account for the intersection between the perforations and the subdomain boundaries. As a reference, we compare this coarse space to the well-studied Nicolaides coarse space with the same subdomain partitioning. It is known that the Nicolaides space is unable to prevent stagnation in convergence when the subdomains are not connected; we work around this issue by separating each subdomain by disconnected component. Scalability and robustness are tested for multiple data sets based on realistic urban topography. Numerical results show that the new coarse space is very robust and accelerates the number of Krylov iterations when compared to Nicolaides, independent of the complexity of the data.