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Journal of Numerical Mathematics

Editor-in-Chief: Hoppe, Ronald H. W. / Kuznetsov, Yuri

Managing Editor: Olshanskii, Maxim

Editorial Board: Benzi, Michele / Brenner, Susanne C. / Carstensen, Carsten / Dryja, M. / Feistauer, Miloslav / Glowinski, R. / Lazarov, Raytcho / Nataf, Frédéric / Neittaanmaki, P. / Bonito, Andrea / Quarteroni, Alfio / Guzman, Johnny / Rannacher, Rolf / Repin, Sergey I. / Shi, Zhong-ci / Tyrtyshnikov, Eugene E. / Zou, Jun / Simoncini, Valeria / Reusken, Arnold


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Families of Interior Penalty Hybridizable discontinuous Galerkin methods for second order elliptic problems

Maurice S. Fabien / Matthew G. Knepley
  • Department of Computer Science and Engineering, State University of New York at Buffalo, Buffalo, New York, 14260, USA
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Beatrice M. Riviere
Published Online: 2019-09-06 | DOI: https://doi.org/10.1515/jnma-2019-0027

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.

Keywords: error estimates; discontinuous Galerkin; hybridization; embedded; non-symmetric; elliptic equations

References

  • [1]

    D. N. Arnold, F. Brezzi, B. Cockburn, and L. D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM journal on numerical analysis, 39 (2002), pp. 1749–1779.CrossrefGoogle Scholar

  • [2]

    I. Babuška and M. Suri, The hp version of the finite element method with quasiuniform meshes, ESAIM: Mathematical Modelling and Numerical Analysis, 21 (1987), pp. 199–238.CrossrefGoogle Scholar

  • [3]

    L. Babuška and M. Suri, The optimal convergence rate of the p-version of the finite element method, SIAM Journal on Numerical Analysis, 24 (1987), pp. 750–776.CrossrefGoogle Scholar

  • [4]

    P. Bastian, C. Engwer, D. Göddeke, O. Iliev, O. Ippisch, M. Ohlberger, S. Turek, J. Fahlke, S. Kaulmann, S. Müthing, and D. Ribbrock, EXA-DUNE: flexible PDE solvers, numerical methods and applications, in European Conference on Parallel Processing, Springer, 2014, pp. 530–541.Google Scholar

  • [5]

    S. Brenner and R. Scott, The Mathematical Theory of Finite Element Methods, vol. 15, Springer Science & Business Media, 2007.Google Scholar

  • [6]

    B. Cockburn, The hybridizable discontinuous Galerkin methods, in Proceedings of the International Congress of Mathematicians 2010 (ICM 2010) (In 4 Volumes) Vol. I: Plenary Lectures and Ceremonies Vols. II–IV: Invited Lectures, World Scientific, 2010, pp. 2749–2775.Google Scholar

  • [7]

    B. Cockburn, Static condensation, hybridization, and the devising of the HDG methods, in Building bridges: connections and challenges in modern approaches to numerical partial differential equations, Springer, 2016, pp. 129–177.Google Scholar

  • [8]

    B. Cockburn, B. Dong, and J. Guzmán, A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems, Mathematics of Computation, 77 (2008), pp. 1887–1916.CrossrefWeb of ScienceGoogle Scholar

  • [9]

    B. Cockburn and G. Fu, Superconvergence by M-decompositions. Part II: Construction of two-dimensional finite elements, ESAIM: Mathematical Modelling and Numerical Analysis, 51 (2017), pp. 165–186.Web of ScienceCrossrefGoogle Scholar

  • [10]

    B. Cockburn and G. Fu, Superconvergence by M-decompositions. Part III: Construction of three-dimensional finite elements, ESAIM: Mathematical Modelling and Numerical Analysis, 51 (2017), pp. 365–398.Web of ScienceCrossrefGoogle Scholar

  • [11]

    B. Cockburn, G. Fu, and F. Sayas, Superconvergence by M-decompositions. Part I: General theory for HDG methods for diffusion, Mathematics of Computation, 86 (2017), pp. 1609–1641.Web of ScienceGoogle Scholar

  • [12]

    B. Cockburn, J. Gopalakrishnan, and R. Lazarov, Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems, SIAM Journal on Numerical Analysis, 47 (2009), pp. 1319–1365.Web of ScienceCrossrefGoogle Scholar

  • [13]

    B. Cockburn, J. Guzmán, S.-C. Soon, and H. K. Stolarski, An analysis of the embedded discontinuous Galerkin method for second-order elliptic problems, SIAM Journal on Numerical Analysis, 47 (2009), pp. 2686–2707.CrossrefWeb of ScienceGoogle Scholar

  • [14]

    B. Cockburn, J. Guzmán, and H. Wang, Superconvergent discontinuous Galerkin methods for second-order elliptic problems, Mathematics of Computation, 78 (2009), pp. 1–24.CrossrefGoogle Scholar

  • [15]

    B. Cockburn, G. E. Karniadakis, C.-W. Shu, and M. Griebel, Discontinuous Galerkin Methods Theory, Computation and Applications, Lectures Notes in Computational Science and Engineering, Inc. Marzo del, (2000).Google Scholar

  • [16]

    B. Cockburn, W. Qiu, and K. Shi, Conditions for superconvergence of HDG methods for second-order elliptic problems, Mathematics of Computation, 81 (2012), pp. 1327–1353.Web of ScienceCrossrefGoogle Scholar

  • [17]

    B. Cockburn and K. Shi, Conditions for superconvergence of HDG methods for Stokes flow, Mathematics of Computation, 82 (2013), pp. 651–671.Google Scholar

  • [18]

    C. Dawson, S. Sun, and M. F. Wheeler, Compatible algorithms for coupled flow and transport, Computer Methods in Applied Mechanics and Engineering, 193 (2004), pp. 2565–2580.CrossrefGoogle Scholar

  • [19]

    B. A. de Dios, F. Brezzi, O. Havle, and L. D. Marini, L2-estimates for the DG IIPG-0 scheme, Numerical Methods for Partial Differential Equations, 28 (2012), pp. 1440–1465.CrossrefWeb of ScienceGoogle Scholar

  • [20]

    L. Delves and C. Hall, An implicit matching principle for global element calculations, IMA Journal of Applied Mathematics, 23 (1979), pp. 223–234.CrossrefGoogle Scholar

  • [21]

    D. A. Di Pietro, A. Ern, and S. Lemaire, A review of hybrid high-order methods: formulations, computational aspects, comparison with other methods, in Building bridges: connections and challenges in modern approaches to numerical partial differential equations, Springer, 2016, pp. 205–236.Google Scholar

  • [22]

    V. Dolejší, On the discontinuous Galerkin method for the numerical solution of the Navier–Stokes equations, International Journal for Numerical Methods in Fluids, 45 (2004), pp. 1083–1106.CrossrefGoogle Scholar

  • [23]

    H. Egger and C. Waluga, hp analysis of a hybrid DG method for Stokes flow, IMA Journal of Numerical Analysis, 33 (2012), pp. 687–721.Web of ScienceGoogle Scholar

  • [24]

    R. E. Ewing, J. Wang, and Y. Yang, A stabilized discontinuous finite element method for elliptic problems, Numerical linear algebra with applications, 10 (2003), pp. 83–104.CrossrefGoogle Scholar

  • [25]

    M. S. Fabien, M. G. Knepley, R. T. Mills, and B. Riviere, Manycore parallel computing for a hybridizable discontinuous Galerkin nested multigrid method, SIAM Journal on Scientific Computing, 41 (2019), pp. C73–C96.Web of ScienceCrossrefGoogle Scholar

  • [26]

    C. Farhat, I. Harari, and L. P. Franca, The discontinuous enrichment method, Computer methods in applied mechanics and engineering, 190 (2001), pp. 6455–6479.Web of ScienceCrossrefGoogle Scholar

  • [27]

    R. J. Guyan, Reduction of stiffness and mass matrices, AIAA Journal, 3 (1965), pp. 380–380.CrossrefGoogle Scholar

  • [28]

    S. Güzey, B. Cockburn, and H. Stolarski, The embedded discontinuous Galerkin method: application to linear shell problems, International journal for numerical methods in engineering, 70 (2007), pp. 757–790.Web of ScienceCrossrefGoogle Scholar

  • [29]

    J. Guzmán and B. Riviere, Sub-optimal convergence of non-symmetric discontinuous Galerkin methods for odd polynomial approximations, Journal of Scientific Computing, 40 (2009), pp. 273–280.CrossrefWeb of ScienceGoogle Scholar

  • [30]

    S. Hajian, Analysis of Schwarz methods for discontinuous Galerkin discretizations, PhD thesis, University of Geneva, 2015.Google Scholar

  • [31]

    G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge university press, 1952.Google Scholar

  • [32]

    A. Huerta, A. Angeloski, X. Roca, and J. Peraire, Efficiency of high-order elements for continuous and discontinuous Galerkin methods, International Journal for numerical methods in Engineering, 96 (2013), pp. 529–560.CrossrefWeb of ScienceGoogle Scholar

  • [33]

    G. Kanschat, K. Kormann, M. Kronbichler, and W. A. Wall, ExaDG: Highorder discontinuous Galerkin for the exa-scale, in SPPEXA, 2016.Google Scholar

  • [34]

    R. M. Kirby, S. J. Sherwin, and B. Cockburn, To CG or to HDG: a comparative study, Journal of Scientific Computing, 51 (2012), pp. 183–212.CrossrefGoogle Scholar

  • [35]

    C. Lehrenfeld, Hybrid Discontinuous Galerkin methods for solving incompressible flow problems, PhD thesis, Rheinisch-Westfalischen Technischen Hochschule Aachen, 2010.Google Scholar

  • [36]

    Y. Maday, C. Mavriplis, and A. Patera, Nonconforming mortar element methods: Application to spectral discretizations, Domain decomposition methods (Los Angeles, CA, 1988), SIAM, Philadelphia, PA, (1988), pp. 392–418.Google Scholar

  • [37]

    N. C. Nguyen, J. Peraire, and B. Cockburn, Hybridizable discontinuous Galerkin methods, in Spectral and High Order Methods for Partial Differential Equations, Springer, 2011, pp. 63–84.Google Scholar

  • [38]

    J. T. Oden, I. Babuŝka, and C. E. Baumann, A discontinuous hp finite element method for diffusion problems, Journal of computational physics, 146 (1998), pp. 491– 519.CrossrefGoogle Scholar

  • [39]

    I. Oikawa, Hybridized Discontinuous Galerkin Methods for Elliptic Problems, PhD thesis, The University of Tokyo, 2012.Google Scholar

  • [40]

    I. Oikawa and F. Kikuchi, Discontinuous Galerkin FEM of hybrid type, Japan Society for Industrial and Applied Mathematics Letters, 2 (2010), pp. 49–52.Google Scholar

  • [41]

    T. H. Pian and C.-C. Wu, Hybrid and Incompatible Finite Element Methods, Chapman and Hall/CRC, 2005.Google Scholar

  • [42]

    B. Riviere, Discontinuous Galerkin methods for solving elliptic and parabolic equations: theory and implementation, SIAM, 2008.Google Scholar

  • [43]

    B. Riviere and S. Sardar, Penalty-free discontinuous Galerkin methods for incompressible Navier–Stokes equations, Mathematical Models and Methods in Applied Sciences, 24 (2014), pp. 1217–1236.Web of ScienceCrossrefGoogle Scholar

  • [44]

    B. Riviere and M. F. Wheeler, Discontinuous finite element methods for acoustic and elastic wave problems, Contemporary Mathematics, 329 (2003), pp. 271–282.CrossrefGoogle Scholar

  • [45]

    B. Riviere, M. F. Wheeler, and V. Girault, Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. Part I, Computational Geosciences, 3 (1999), pp. 337–360.CrossrefGoogle Scholar

  • [46]

    B. Riviere, M. F. Wheeler, and V. Girault, A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems, SIAM Journal on Numerical Analysis, 39 (2001), pp. 902–931.CrossrefGoogle Scholar

  • [47]

    A. Samii, C. Michoski, and C. Dawson, A parallel and adaptive hybridized discontinuous Galerkin method for anisotropic nonhomogeneous diffusion, Computer Methods in Applied Mechanics and Engineering, 304 (2016), pp. 118–139.Web of ScienceCrossrefGoogle Scholar

  • [48]

    S. Sun and M. F. Wheeler, Symmetric and nonsymmetric discontinuous Galerkin methods for reactive transport in porous media, SIAM Journal on Numerical Analysis, 43 (2005), pp. 195–219.CrossrefGoogle Scholar

  • [49]

    S. Sun and M. F. Wheeler, A dynamic, adaptive, locally conservative, and nonconforming solution strategy for transport phenomena in chemical engineering, Chemical Engineering Communications, 193 (2006), pp. 1527–1545.CrossrefGoogle Scholar

  • [50]

    T. P. Wihler, Locking-free DGFEM for elasticity problems in polygons, IMA Journal of Numerical Analysis, 24 (2004), pp. 45–75.CrossrefGoogle Scholar

About the article

Published Online: 2019-09-06


Citation Information: Journal of Numerical Mathematics, ISSN (Online) 1569-3953, ISSN (Print) 1570-2820, DOI: https://doi.org/10.1515/jnma-2019-0027.

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