On the convergence of iterative solvers for polygonal discontinuous Galerkin discretizations

TL;DR

This paper analyzes how the shape of polygonal mesh elements affects the convergence of iterative solvers in discontinuous Galerkin methods for hyperbolic systems, showing that certain mesh types lead to faster convergence.

Contribution

It provides a comparative analysis of iterative solver convergence on polygonal meshes versus traditional triangular meshes, including theoretical and numerical results.

Findings

Hexagonal and square meshes yield faster convergence of Jacobi's method.

Polygonal meshes influence the eigenvalues and preconditioner performance.

Numerical experiments confirm theoretical predictions across various flow conditions.

Abstract

We study the convergence of iterative linear solvers for discontinuous Galerkin discretizations of systems of hyperbolic conservation laws with polygonal mesh elements compared with that of traditional triangular elements. We solve the semi-discrete system of equations by means of an implicit time discretization method, using iterative solvers such as the block Jacobi method and GMRES. We perform a von Neumann analysis to analytically study the convergence of the block Jacobi method for the two-dimensional advection equation on four classes of regular meshes: hexagonal, square, equilateral-triangular, and right-triangular. We find that hexagonal and square meshes give rise to smaller eigenvalues, and thus result in faster convergence of Jacobi's method. We perform numerical experiments with variable velocity fields, irregular, unstructured meshes, and the Euler equations of gas dynamics to confirm and extend these results. We additionally study the effect of polygonal meshes on the performance of block ILU(0) and Jacobi preconditioners for the GMRES method.