A Geometric Multigrid Preconditioning Strategy for DPG System Matrices

TL;DR

This paper introduces a geometric multigrid preconditioning strategy for DPG system matrices, enhancing the efficiency of iterative solvers for large-scale problems, including adaptive and complex mesh scenarios.

Contribution

It presents a novel multigrid preconditioning approach for DPG matrices, implemented in Camellia, with demonstrated effectiveness and scalability in various variational and adaptive mesh problems.

Findings

Preconditioner effectiveness depends on test space enrichment.

Method scales well to many MPI ranks.

Applicable to adaptive meshes with hanging nodes.

Abstract

The discontinuous Petrov-Galerkin (DPG) methodology of Demkowicz and Gopalakrishnan [15,17] guarantees the optimality of the solution in an energy norm, and provides several features facilitating adaptive schemes. A key question that has not yet been answered in general - though there are some results for Poisson, e.g. - is how best to precondition the DPG system matrix, so that iterative solvers may be used to allow solution of large-scale problems. In this paper, we detail a strategy for preconditioning the DPG system matrix using geometric multigrid which we have implemented as part of Camellia [26], and demonstrate through numerical experiments its effectiveness in the context of several variational formulations. We observe that in some of our experiments, the behavior of the preconditioner is closely tied to the discrete test space enrichment. We include experiments involving adaptive meshes with hanging nodes for lid-driven cavity flow, demonstrating that the preconditioners can be applied in the context of challenging problems. We also include a scalability study demonstrating that the approach - and our implementation - scales well to many MPI ranks.