Vol. 13, No. 1, 2018

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On the convergence of iterative solvers for polygonal discontinuous Galerkin discretizations

Will Pazner and Per-Olof Persson

Vol. 13 (2018), No. 1, 27–51

We study the convergence of iterative linear solvers for discontinuous Galerkin discretizations of systems of hyperbolic conservation laws with polygonal mesh elements compared with traditional triangular elements. We solve the semidiscrete 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.

discontinuous Galerkin, iterative solvers, preconditioners
Mathematical Subject Classification 2010
Primary: 65F10, 65M60, 65N22
Received: 8 August 2016
Revised: 25 September 2017
Accepted: 30 October 2017
Published: 17 February 2018
Will Pazner
Division of Applied Mathematics
Brown University
Providence, RI
United States
Per-Olof Persson
Department of Mathematics
University of California, Berkeley
Berkeley, CA
United States