2024/11/21 by Will Pazner, Pazner, Will
Computer Science · Engineering · #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Numerical methods in engineering
paper · pdf · doi:10.48550/arxiv.2411.14561
openalex publication_date 2024/11/21 · openalex created_date 2024/12/04 · openalex updated_date 2026/07/28
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.