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

References

[1] D. N. Arnold, F. Brezzi, B. Cockburn, and L. D. Marini, Uniied analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal. 39 (2002), No. 5, 1749–1779.10.1137/S0036142901384162Search in Google Scholar

[2] I. Babuška and M. Suri, The hp version of the inite element method with quasiuniform meshes, ESAIM: Math. Modelling Numer. Anal. 21 (1987), No. 2, 199–238.10.1051/m2an/1987210201991Search in Google Scholar

[3] L. Babuška and M. Suri, The optimal convergence rate of the p-version of the inite element method, SIAM J. Numer. Anal. 24 (1987), No. 4, 750–776.10.1137/0724049Search in Google Scholar

[4] P. Bastian, Ch. 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 Conf. on Parallel Processing, Springer, pp. 530–541, 2014.10.1007/978-3-319-14313-2_45Search in Google Scholar

[5] S. Brenner and R. Scott, The Mathematical Theory of Finite Element Methods, 15, Springer Science & Business Media, 2007.Search in Google Scholar

[6] B. Cockburn, The hybridizable discontinuous Galerkin methods, In: Proc. of the Int. Congress of Mathematicians 2010 (ICM 2010), Vol. I: Plenary Lectures and Ceremonies, Vols. II–IV: Invited Lectures, World Scientiic, pp. 2749–2775, 2010.Search in Google Scholar

[7] B. Cockburn, Static Condensation, Hybridization, and the Devising of the HDG Methods, Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Diferential Equations, Springer, 2016, pp. 129–177.10.1007/978-3-319-41640-3_5Search in Google Scholar

[8] B. Cockburn, B. Dong, and J. Guzmán, A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems, Math. Comp. 77 (2008), No. 264, 1887–1916.10.1090/S0025-5718-08-02123-6Search in Google Scholar

[9] B. Cockburn and G. Fu, Superconvergence by M-decompositions. Part II: Construction of two-dimensional inite elements, ESAIM: Math. Modelling Numer. Anal. 51 (2017), No. 1, 165–186.10.1051/m2an/2016016Search in Google Scholar

[10] B. Cockburn and G. Fu, Superconvergence by M-decompositions. Part III: Construction of three-dimensional inite elements, ESAIM: Math. Modelling Numer. Anal. 51 (2017), No. 1, 365–398.10.1051/m2an/2016023Search in Google Scholar

[11] B. Cockburn, G. Fu, and F. Sayas, Superconvergence by M-decompositions. Part I: General theory for HDG methods for difusion, Math. Comp. 86 (2017), No. 306, 1609–1641.10.1090/mcom/3140Search in Google Scholar

[12] B. Cockburn, J. Gopalakrishnan, and R. Lazarov, Uniied hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems, SIAM J. Numer. Anal. 47 (2009), No. 2, 1319–1365.10.1137/070706616Search in Google Scholar

[13] B. Cockburn, J. Guzmán, S.-Ch. Soon, and H. K. Stolarski, An analysis of the embedded discontinuous Galerkin method for second-order elliptic problems, SIAM J. Numer. Anal. 47 (2009), No. 4, 2686–2707.10.1137/080726914Search in Google Scholar

[14] B. Cockburn, J. Guzmán, and H. Wang, Superconvergent discontinuous Galerkin methods for second-order elliptic problems, Math. Comp. 78 (2009), No. 265, 1–24.10.1090/S0025-5718-08-02146-7Search in Google Scholar

[15] B. Cockburn, G. E. Karniadakis, Ch.-W. Shu, and M. Griebel, Discontinuous Galerkin Methods Theory, Computation and Applications, Lectures Notes in Computational Science and Engineering 11, 2000.10.1007/978-3-642-59721-3Search in Google Scholar

[16] B. Cockburn, W. Qiu, and K. Shi, Conditions for superconvergence of HDG methods for second-order elliptic problems, Math. Comp. 81 (2012), No. 279, 1327–1353.10.1090/S0025-5718-2011-02550-0Search in Google Scholar

[17] B. Cockburn and K. Shi, Conditions for superconvergence of HDG methods for Stokes flow, Math. Comp. 82 (2013), No. 282, 651–671.10.1090/S0025-5718-2012-02644-5Search in Google Scholar

[18] C. Dawson, Sh. Sun, and M. F. Wheeler, Compatible algorithms for coupled flow and transport, Comp. Methods Appl. Mech. Engrg. 193 (2004), No. 23-26, 2565–2580.10.1016/j.cma.2003.12.059Search in Google Scholar

[19] B. A. de Dios, F. Brezzi, O. Havle, and L. D. Marini, L2-estimates for the DG IIPG-0 scheme, Numer. Meth. Partial Dif. Equ. 28 (2012), No. 5, 1440–1465.10.1002/num.20687Search in Google Scholar

[20] L. M. Delves and C. A. Hall, An implicit matching principle for global element calculations, IMA J. Appl. Math. 23 (1979), No. 2, 223–234.10.1093/imamat/23.2.223Search in Google 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, Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Diferential Equations, Springer, 2016, pp. 205–236.10.1007/978-3-319-41640-3_7Search in Google Scholar

[22] V. Dolejší, On the discontinuous Galerkin method for the numerical solution of the Navier–Stokes equations, Int. J. Numer. Methods Fluids45 (2004), No. 10, 1083–1106.10.1002/fld.730Search in Google Scholar

[23] H. Egger and Ch. Waluga, hp analysis of a hybrid DG method for Stokes flow, IMA J. Numer. Anal. 33 (2012), No. 2, 687–721.10.1093/imanum/drs018Search in Google Scholar

[24] R. E. Ewing, J. Wang, and Y. Yang, A stabilized discontinuous inite element method for elliptic problems, Numer. Linear Algebra Appl. 10 (2003), No. 1-2, 83–104.10.1002/nla.313Search in Google 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 J. Sci. Comp. 41 (2019), No. 2, C73–C96.10.1137/17M1128903Search in Google Scholar

[26] Ch. Farhat, I. Harari, and L. P. Franca, The discontinuous enrichment method, Comp. Methods Appl. Mech. Engrg. 190 (2001), No. 48, 6455–6479.10.1016/S0045-7825(01)00232-8Search in Google Scholar

[27] R. J. Guyan, Reduction of stifness and mass matrices, AIAA Journal3 (1965), No. 2, 380–380.10.2514/3.2874Search in Google Scholar

[28] S. Güzey, B. Cockburn, and H. K. Stolarski, The embedded discontinuous Galerkin method: application to linear shell problems, Int. J. Numer. Methods Engrg. 70 (2007), No. 7, 757–790.10.1002/nme.1893Search in Google Scholar

[29] J. Guzmán and B. Riviere, Sub-optimal convergence of non-symmetric discontinuous Galerkin methods for odd polynomial approximations, J. Sci. Comp. 40 (2009), No. 1-3, 273–280.10.1007/s10915-008-9255-zSearch in Google Scholar

[30] S. Hajian, Analysis of Schwarz Methods for Discontinuous Galerkin Discretizations, Ph.D. thesis, University of Geneva, 2015.Search in Google Scholar

[31] G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge University Press, 1952.Search in Google Scholar

[32] A. Huerta, A. Angeloski, X. Roca, and J. Peraire, Eiciency of high-order elements for continuous and discontinuous Galerkin methods, Int. J. Numer. Methods Engrg. 96 (2013), No. 9, 529–560.10.1002/nme.4547Search in Google Scholar

[33] G. Kanschat, K. Kormann, M. Kronbichler, and W. A. Wall, ExaDG: High-order discontinuous Galerkin for the exa-scale, In: SPPEXA, 2016.Search in Google Scholar

[34] R. M. Kirby, S. J. Sherwin, and B. Cockburn, To CG or to HDG: a comparative study, J. Sci. Comp. 51 (2012), No. 1, 183–212.10.1007/s10915-011-9501-7Search in Google Scholar

[35] Ch. Lehrenfeld, Hybrid Discontinuous Galerkin Methods for Solving Incompressible Flow Problems, Ph.D. thesis, Rheinisch-Westfalischen Technischen Hochschule Aachen, 2010.Search in 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), 392–418.Search in Google Scholar

[37] N. C. Nguyen, J. Peraire, and B. Cockburn, Hybridizable Discontinuous Galerkin Methods, Spectral and High Order Methods for Partial Diferential Equations, Springer, 2011, pp. 63–84.10.1007/978-3-642-15337-2_4Search in Google Scholar

[38] J. T. Oden, I. Babuška, and C. E. Baumann, A discontinuous hp inite element method for difusion problems, J. Comput. Phys. 146 (1998), No. 2, 491–519.10.1006/jcph.1998.6032Search in Google Scholar

[39] I. Oikawa, Hybridized Discontinuous Galerkin Methods for Elliptic Problems, Ph.D. thesis, The University of Tokyo, 2012.Search in Google Scholar

[40] I. Oikawa and F. Kikuchi, Discontinuous Galerkin FEM of hybrid type, Japan Soc. Industr. Appl. Math. Letters2 (2010), 49–52.10.14495/jsiaml.2.49Search in Google Scholar

[41] Th. H. H. Pian and Ch.-Ch. Wu, Hybrid and Incompatible Finite Element Methods, Chapman and Hall/CRC, 2005.10.1201/9780203487693Search in Google Scholar

[42] B. Riviere, Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation, SIAM, 2008.10.1137/1.9780898717440Search in Google Scholar

[43] B. Riviere and Sh. Sardar, Penalty-free discontinuous Galerkin methods for incompressible Navier–Stokes equations, Math. Models Methods Appl. Sci. 24 (2014), No. 6, 1217–1236.10.1142/S0218202513500826Search in Google Scholar

[44] B. Riviere and M. F. Wheeler, Discontinuous inite element methods for acoustic and elastic wave problems, Contemporary Math. 329 (2003), 271–282.10.1090/conm/329/05862Search in Google Scholar

[45] B. Riviere, M. Fi. Wheeler, and V. Girault, Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. Part I, Comp. Geosci. 3 (1999), No. 3-4, 337–360.10.1023/A:1011591328604Search in Google Scholar

[46] B. Riviere, M. Fi. Wheeler, and V. Girault, A priori error estimates for inite element methods based on discontinuous approximation spaces for elliptic problems, SIAM J. Numer. Anal. 39 (2001), No. 3, 902–931.10.1137/S003614290037174XSearch in Google Scholar

[47] A. Samii, C. Michoski, and C. Dawson, A parallel and adaptive hybridized discontinuous Galerkin method for anisotropic nonhomogeneous difusion, Comp. Methods Appl. Mech. Engrg. 304 (2016), 118–139.10.1016/j.cma.2016.02.009Search in Google Scholar

[48] Sh. Sun and M. F. Wheeler, Symmetric and nonsymmetric discontinuous Galerkin methods for reactive transport in porous media, SIAM J. Numer. Anal. 43 (2005), No. 1, 195–219.10.1137/S003614290241708XSearch in Google Scholar

[49] Sh. Sun and M. F. Wheeler, A dynamic, adaptive, locally conservative, and nonconforming solution strategy for transport phenomena in chemical engineering, Chem. Engrg. Commun. 193 (2006), No. 12, 1527–1545.10.1080/00986440600584284Search in Google Scholar

[50] Th. P. Wihler, Locking-free DGFEM for elasticity problems in polygons, IMA J. Numer. Anal. 24 (2004), No. 1, 45–75.10.1093/imanum/24.1.45Search in Google Scholar