A GenEO Domain Decomposition method for Saddle Point problems

TL;DR

This paper presents an adaptive domain decomposition method based on GenEO theory for efficiently solving large-scale saddle point problems without prior knowledge of the constrained space, demonstrated on complex elasticity simulations.

Contribution

It extends the GenEO domain decomposition approach to saddle point problems with an adaptive coarse space, enabling scalable solutions without requiring constrained space knowledge.

Findings

Successfully applied to 3D elasticity problems with up to one billion degrees of freedom.

Achieves efficient and scalable solutions for large saddle point systems.

Demonstrates robustness and adaptability of the method in complex simulations.

Abstract

We introduce an adaptive element-based domain decomposition (DD) method for solving saddle point problems defined as a block two by two matrix. The algorithm does not require any knowledge of the constrained space. We assume that all sub matrices are sparse and that the diagonal blocks are spectrally equivalent to a sum of positive semi definite matrices. The latter assumption enables the design of adaptive coarse space for DD methods that extends the GenEO theory to saddle point problems. Numerical results on three dimensional elasticity problems for steel-rubber structures discretized by a finite element with continuous pressure are shown for up to one billion degrees of freedom.