[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

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