Communications in Applied Mathematics and Computational Science
- Commun. Appl. Math. Comput. Sci.
- Volume 13, Number 1 (2018), 27-51.
On the convergence of iterative solvers for polygonal discontinuous Galerkin discretizations
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.
Commun. Appl. Math. Comput. Sci., Volume 13, Number 1 (2018), 27-51.
Received: 8 August 2016
Revised: 25 September 2017
Accepted: 30 October 2017
First available in Project Euclid: 28 March 2018
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 65F10: Iterative methods for linear systems [See also 65N22] 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65N22: Solution of discretized equations [See also 65Fxx, 65Hxx]
Pazner, Will; Persson, Per-Olof. On the convergence of iterative solvers for polygonal discontinuous Galerkin discretizations. Commun. Appl. Math. Comput. Sci. 13 (2018), no. 1, 27--51. doi:10.2140/camcos.2018.13.27. https://projecteuclid.org/euclid.camcos/1522202438