Families of interior penalty hybridizable discontinuous Galerkin methods for second order elliptic problems

Abstract The focus of this paper is the analysis of families of hybridizable interior penalty discontinuous Galerkin methods for second order elliptic problems. We derive a priori error estimates in the energy norm that are optimal with respect to the mesh size. Suboptimal L2-norm error estimates are proven. These results are valid in two and three dimensions. Numerical results support our theoretical findings, and we illustrate the computational cost of the method.

[1]  Bernardo Cockburn,et al.  Static Condensation, Hybridization, and the Devising of the HDG Methods , 2016 .

[2]  Béatrice Rivière,et al.  Sub-optimal Convergence of Non-symmetric Discontinuous Galerkin Methods for Odd Polynomial Approximations , 2009, J. Sci. Comput..

[3]  Robert Michael Kirby,et al.  To CG or to HDG: A Comparative Study , 2012, J. Sci. Comput..

[4]  Martin Kronbichler,et al.  ExaDG: High-Order Discontinuous Galerkin for the Exa-Scale , 2020, Software for Exascale Computing.

[5]  Fumio Kikuchi,et al.  Discontinuous Galerkin FEM of hybrid type , 2010, JSIAM Lett..

[6]  B. Rivière,et al.  Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. Part I , 1999 .

[7]  Anthony T. Patera,et al.  Nonconforming mortar element methods: Application to spectral discretizations , 1988 .

[8]  Blanca Ayuso de Dios,et al.  L2‐estimates for the DG IIPG‐0 scheme , 2012 .

[9]  H. Egger,et al.  hp analysis of a hybrid DG method for Stokes flow , 2013 .

[10]  D. Arnold,et al.  Discontinuous Galerkin Methods for Elliptic Problems , 2000 .

[11]  Mary F. Wheeler,et al.  A DYNAMIC, ADAPTIVE, LOCALLY CONSERVATIVE, AND NONCONFORMING SOLUTION STRATEGY FOR TRANSPORT PHENOMENA IN CHEMICAL ENGINEERING , 2004 .

[12]  Issei Oikawa.  Hybridized Discontinuous Galerkin Methods for Elliptic Problems , 2012 .

[13]  L. M. Delves,et al.  An Implicit Matching Principle for Global Element Calculations , 1979 .

[14]  Richard T. Mills,et al.  Manycore Parallel Computing for a Hybridizable Discontinuous Galerkin Nested Multigrid Method , 2019, SIAM J. Sci. Comput..

[15]  J. Peraire,et al.  Efficiency of high‐order elements for continuous and discontinuous Galerkin methods , 2013 .

[16]  Sukru Guzey,et al.  The embedded discontinuous Galerkin method: application to linear shell problems , 2007 .

[17]  Ivo Babuška,et al.  The optimal convergence rate of the p-version of the finite element method , 1987 .

[18]  Vít Dolejší,et al.  On the discontinuous Galerkin method for the numerical solution of the Navier–Stokes equations , 2004 .

[19]  Craig Michoski,et al.  A parallel and adaptive hybridized discontinuous Galerkin method for anisotropic nonhomogeneous diffusion , 2016 .

[20]  Ke Shi,et al.  Conditions for superconvergence of HDG methods for second-order elliptic problems , 2012, Math. Comput..

[21]  Bernardo Cockburn,et al.  Hybridizable Discontinuous Galerkin Methods , 2011 .

[22]  Ke Shi,et al.  Conditions for superconvergence of HDG methods for Stokes flow , 2013, Math. Comput..

[23]  Mary F. Wheeler,et al.  Compatible algorithms for coupled flow and transport , 2004 .

[24]  Béatrice Rivière,et al.  Penalty-free discontinuous Galerkin methods for incompressible Navier–Stokes equations , 2014 .

[25]  T. Wihler.  Locking‐free DGFEM for elasticity problems in polygons , 2004 .

[26]  Francisco-Javier Sayas,et al.  Superconvergence by $M$-decompositions. Part I: General theory for HDG methods for diffusion , 2016, Mathematics of Computation.

[27]  Chi-Wang Shu,et al.  Discontinuous Galerkin Methods: Theory, Computation and Applications , 2011 .

[28]  C. Farhat,et al.  The Discontinuous Enrichment Method , 2000 .

[29]  Richard E. Ewing,et al.  A stabilized discontinuous finite element method for elliptic problems , 2003, Numer. Linear Algebra Appl..

[30]  Daniele Antonio Di Pietro,et al.  A Review of Hybrid High-Order Methods: Formulations, Computational Aspects, Comparison with Other Methods , 2016 .

[31]  Douglas N. Arnold,et al.  Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems , 2001, SIAM J. Numer. Anal..

[32]  Thomas P. Wihler,et al.  Locking-free DGFEM for elasticity problems in polygons , 2002 .

[33]  Bernardo Cockburn,et al.  Superconvergence by M-decompositions. Part III: Construction of three-dimensional finite elements∗ , 2017 .

[34]  Bo Dong,et al.  A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems , 2008, Math. Comput..

[35]  Soheil Hajian.  Analysis of Schwarz methods for discontinuous Galerkin discretizations , 2015 .

[36]  Mary F. Wheeler,et al.  A Priori Error Estimates for Finite Element Methods Based on Discontinuous Approximation Spaces for Elliptic Problems , 2001, SIAM J. Numer. Anal..

[37]  Raytcho D. Lazarov,et al.  Unified Hybridization of Discontinuous Galerkin, Mixed, and Continuous Galerkin Methods for Second Order Elliptic Problems , 2009, SIAM J. Numer. Anal..

[38]  Haiying Wang,et al.  Superconvergent discontinuous Galerkin methods for second-order elliptic problems , 2009, Math. Comput..

[39]  I. Babuska,et al.  A DiscontinuoushpFinite Element Method for Diffusion Problems , 1998 .

[40]  Mary F. Wheeler,et al.  Symmetric and Nonsymmetric Discontinuous Galerkin Methods for Reactive Transport in Porous Media , 2005, SIAM J. Numer. Anal..

[41]  R. Guyan.  Reduction of stiffness and mass matrices , 1965 .

[42]  Bernardo Cockburn,et al.  An Analysis of the Embedded Discontinuous Galerkin Method for Second-Order Elliptic Problems , 2009, SIAM J. Numer. Anal..

[43]  Bernardo Cockburn,et al.  Superconvergence by M-decompositions. Part II: Construction of two-dimensional finite elements , 2017 .

[44]  Ivo Babuška,et al.  The h-p version of the finite element method , 1986 .