An analysis of sparsity preserving pivot strategies for discontinuous Galerkin methods applied to acoustic scattering

TL;DR

This work analyzes the sparsity pattern of the IP-DG discretization of the acoustic Helmholtz problem and systematically compares three sparsity-preserving pivot strategies—AMD, ND, and RCM—for reducing LU factorization fill-in. Through extensive numerical experiments on uniform, nonuniform, and scattering-domain meshes, the study highlights regime-dependent performance: AMD and ND often excel for piecewise linear spaces ($V^1_h$), with ND favoring finer meshes, while RCM can outperform AMD/ND for piecewise quadratic spaces ($V^2_h$). The findings provide practical guidance on selecting pivot strategies to manage memory and computation in LU-based solvers for high-frequency Helmholtz problems. Overall, the results underscore the importance of mesh structure and solution space in determining optimal sparsity-preserving pivoting for IP-DG discretizations.

Abstract

In this paper we discuss and analyze the sparse structure of matrices associated to the interior penalty discontinuous Galerkin (IP-DG) method applied to the Helmholtz equation. It is well-known that LU-factorization causes fill-in and this paper discusses three pivoting strategies: approximate minimal degree (AMD), nested dissection, and reverse Cuthill-McKee, that can reduce fill-in associated to the LU-factorization. Numerical experiments are included that demonstrate the performance of these pivoting strategies. These experiments include both uniform and non-uniform mesh structures, the inclusion of a scattering boundary, and both piecewise linear and quadratic solution spaces.

Figures (24)

• Figure 4.1: \newlabelfig:Exp1Mesh Mesh used in experiment 1 with $n = 5$ (left) and $n = 10$ (right).
• Figure 4.2: \newlabelfig:Exp1NoPivot5 (Left) sparsity structure of the global matrix A produced by the IP-DG method with $n = 5$. (Right) sparsity structure of the combined LU decomposition of A.
• Figure 4.3: \newlabelfig:Exp1AMD5 (Left) sparsity structure of the global matrix after AMD pivoting. (Right) sparsity structure of the combined LU decomposition the global matrix after AMD pivoting.
• Figure 4.4: \newlabelfig:Exp1ND5 (Left) sparsity structure of the global matrix after ND pivoting. (Right) sparsity structure of the combined LU decomposition the global matrix after ND pivoting.
• Figure 4.5: \newlabelfig:Exp1RCM5 (Left) sparsity structure of the global matrix after RCM pivoting. (Right) sparsity structure of the combined LU decomposition the global matrix after RCM pivoting.