Communications in Applied Mathematics and Computational Science

On the convergence of iterative solvers for polygonal discontinuous Galerkin discretizations

Will Pazner and Per-Olof Persson

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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]

discontinuous Galerkin iterative solvers preconditioners


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.

Export citation


  • R. Alexander, Diagonally implicit Runge–Kutta methods for stiff O.D.E.'s, SIAM J. Numer. Anal. 14 (1977), no. 6, 1006–1021.
  • G. Balafas, Polyhedral mesh generation for CFD-analysis of complex structures, master's thesis, Technische Universität München, 2014.
  • F. Bassi and S. Rebay, GMRES discontinuous Galerkin solution of the compressible Navier–Stokes equations, Discontinuous Galerkin methods (B. Cockburn, G. E. Karniadakis, and C.-W. Shu, eds.), Lect. Notes Comput. Sci. Eng., no. 11, Springer, 2000, pp. 197–208.
  • M. Benzi, W. Joubert, and G. Mateescu, Numerical experiments with parallel orderings for ILU preconditioners, Electron. Trans. Numer. Anal. 8 (1999), 88–114.
  • M. Berggren, A vertex-centered, dual discontinuous Galerkin method, J. Comput. Appl. Math. 192 (2006), no. 1, 175–181.
  • B. Cockburn and C.-W. Shu, Runge–Kutta discontinuous Galerkin methods for convection-dominated problems, J. Sci. Comput. 16 (2001), no. 3, 173–261.
  • E. Cuthill and J. McKee, Reducing the bandwidth of sparse symmetric matrices, ACM '69: proceedings of the 1969 24th National Conference, Association for Computing Machinery, 1969, pp. 157–172.
  • L. T. Diosady and D. L. Darmofal, Preconditioning methods for discontinuous Galerkin solutions of the Navier–Stokes equations, J. Comput. Phys. 228 (2009), no. 11, 3917–3935.
  • B. Diskin and J. L. Thomas, Comparison of node-centered and cell-centered unstructured finite-volume discretizations: inviscid fluxes, AIAA J. 49 (2011), no. 4, 836–854.
  • B. Diskin, J. L. Thomas, E. J. Nielsen, H. Nishikawa, and J. A. White, Comparison of node-centered and cell-centered unstructured finite-volume discretizations: viscous fluxes, AIAA J. 48 (2010), no. 7, 1326–1338.
  • I. S. Duff and G. A. Meurant, The effect of ordering on preconditioned conjugate gradients, BIT 29 (1989), no. 4, 635–657.
  • R. V. Garimella, J. Kim, and M. Berndt, Polyhedral mesh generation and optimization for non-manifold domains, Proceedings of the 22nd International Meshing Roundtable (J. Sarrate and M. Staten, eds.), Springer, 2014, pp. 313–330.
  • A. George, Nested dissection of a regular finite element mesh, SIAM J. Numer. Anal. 10 (1973), 345–363.
  • R. Hartmann, Discontinuous Galerkin methods for compressible flows: higher order accuracy, error estimation and adaptivity, 34th CFD: higher order discretization methods (H. Deconinck and M. Ricchiuto, eds.), Von Karman Institute Lecture Series, no. 2006-01, Von Karman Institute, 2006.
  • J. S. Hesthaven and T. Warburton, Nodal discontinuous Galerkin methods: algorithms, analysis, and applications, Texts in Applied Mathematics, no. 54, Springer, 2008.
  • E. J. Kubatko, C. Dawson, and J. J. Westerink, Time step restrictions for Runge–Kutta discontinuous Galerkin methods on triangular grids, J. Comput. Phys. 227 (2008), no. 23, 9697–9710.
  • R. J. LeVeque, Finite volume methods for hyperbolic problems, Cambridge University, 2002.
  • H. Luo, J. D. Baum, and R. Löhner, A discontinuous Galerkin method based on a Taylor basis for the compressible flows on arbitrary grids, J. Comput. Phys. 227 (2008), no. 20, 8875–8893.
  • G. Manzini, A. Russo, and N. Sukumar, New perspectives on polygonal and polyhedral finite element methods, Math. Models Methods Appl. Sci. 24 (2014), no. 8, 1665–1699.
  • H. M. Markowitz, The elimination form of the inverse and its application to linear programming, Management Sci. 3 (1957), 255–269.
  • W. Oaks and S. Paoletti, Polyhedral mesh generation, 9th International Meshing Roundtable, Sandia National Laboratories, 2000, pp. 57–67.
  • J. Peraire, M. Vahdati, K. Morgan, and O. C. Zienkiewicz, Adaptive remeshing for compressible flow computations, J. Comput. Phys. 72 (1987), no. 2, 449–466.
  • M. Peric, Flow simulation using control volumes of arbitrary polyhedral shape, ERCOFTAC Bull. 62 (2004), 25–29.
  • P.-O. Persson and J. Peraire, Newton–GMRES preconditioning for discontinuous Galerkin discretizations of the Navier–Stokes equations, SIAM J. Sci. Comput. 30 (2008), no. 6, 2709–2733.
  • P.-O. Persson, Mesh generation for implicit geometries, Ph.D. thesis, Massachusetts Institute of Technology, 2005.
  • ––––, Scalable parallel Newton–Krylov solvers for discontinuous Galerkin discretizations, 47th AIAA Aerospace Sciences Meeting, American Institute of Aeronautics and Astronautics, 2009.
  • W. H. Reed and T. R. Hill, Triangular mesh methods for the neutron transport equation, technical report LA-UR-73-479, Los Alamos Scientific Laboratory, 1973.
  • J. Ruppert, A Delaunay refinement algorithm for quality $2$-dimensional mesh generation, J. Algorithms 18 (1995), no. 3, 548–585.
  • Y. Saad, Iterative methods for sparse linear systems, 2nd ed., Society for Industrial and Applied Mathematics, 2003.
  • J. R. Shewchuk, Delaunay refinement algorithms for triangular mesh generation, Comput. Geom. 22 (2002), no. 1–3, 21–74.
  • C. Talischi, G. H. Paulino, A. Pereira, and I. F. M. Menezes, PolyMesher: a general-purpose mesh generator for polygonal elements written in Matlab, Struct. Multidiscip. Optim. 45 (2012), no. 3, 309–328.
  • Z. J. Wang, K. Fidkowski, R. Abgrall, and et al., High-order CFD methods: current status and perspective, Internat. J. Numer. Methods Fluids 72 (2013), no. 8, 811–845.