Work

Subspace and auxiliary space preconditioners for high-order interior penalty discretizations in H(div)

Year: 2025

Type: article

Abstract: In this paper, we construct and analyze preconditioners for the interior penalty discontinuous Galerkin discretization posed in the space H(div). These discretizations are used as one component in exactly divergence-free pressure-robust discretizations for the Stokes problem. Three preconditioners are presently considered: a subspace correction preconditioner using vertex patches and the lowest-order H1-conforming space as a coarse space, a fictitious space preconditioner using the degree-p discontinuous Galerkin space, and an auxiliary space preconditioner using the degree-(p − 1) discontinuous Galerkin space and a block Jacobi smoother. On certain classes of meshes, the subspace and fictitious space preconditioners result in provably well-conditioned systems, independent of the mesh size h, polynomial degree p, and penalty parameter η. All three preconditioners are shown to be robust with respect to h on general meshes, and numerical results indicate that the iteration counts grow only mildly with respect to p in the general case. Numerical examples illustrate the convergence properties of the preconditioners applied to structured and unstructured meshes. These solvers are used to construct block-diagonal preconditioners for the Stokes problem, which result in uniform convergence when used with MINRES.

Source: ESAIM Mathematical Modelling and Numerical Analysis

DOI: 10.1051/m2an/2025045

Author: Will Pazner

Institution: Portland State University

Language: English

Topic: Advanced Numerical Methods in Computational Mathematics

Subfield: Computational Mechanics

Field: Engineering

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid:

Funder: Oak Ridge Associated Universities