## Communications in Applied Mathematics and Computational Science

### On the convergence of iterative solvers for polygonal discontinuous Galerkin discretizations

#### 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 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.

#### Article information

Source
Commun. Appl. Math. Comput. Sci., Volume 13, Number 1 (2018), 27-51.

Dates
Revised: 25 September 2017
Accepted: 30 October 2017
First available in Project Euclid: 28 March 2018

https://projecteuclid.org/euclid.camcos/1522202438

Digital Object Identifier
doi:10.2140/camcos.2018.13.27

Mathematical Reviews number (MathSciNet)
MR3778319

Zentralblatt MATH identifier
06864865

#### Citation

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

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