On the convergence of iterative solvers for polygonal discontinuous Galerkin discretizations
For computational scientists using DG methods, this work provides guidance on mesh selection to improve solver efficiency, though the findings are incremental extensions of known mesh effects.
The paper analyzes the convergence of iterative solvers (block Jacobi and GMRES) for discontinuous Galerkin discretizations on polygonal meshes, finding that hexagonal and square meshes yield faster convergence than triangular meshes due to smaller eigenvalues.
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.