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Abstract
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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.
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Keywords
discontinuous Galerkin, iterative solvers, preconditioners
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Mathematical Subject Classification 2010
Primary: 65F10, 65M60, 65N22
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Milestones
Received: 8 August 2016
Revised: 25 September 2017
Accepted: 30 October 2017
Published: 17 February 2018
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