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Hybridizable discontinuous Galerkin methods for second-order elliptic problems: overview, a new result and open problems

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Abstract

We describe, in the framework of steady-state diffusion problems, the history of the development of the so-called hybridizable discontinuous Galerkin (HDG) methods, since their inception in 2009 until now. We show how it runs parallel to the development of the so-called hybridized mixed (HM) methods and how, a few years ago, it prompted the introduction of the \(\textsf{M}\)-decompositions as a novel tool for the construction of superconvergent HM and HDG methods for elements of quite general shapes. We then uncover a new link between HM and HDG methods, namely, that any HM method can be rewritten as an HDG method by a suitable transformation of a subspace of the approximate fluxes of the HM method into a stabilization function. We end by listing several open problems which are a direct consequence of this result.

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References

  1. Arbogast, T., Xiao, H.: Two-level mortar domain decomposition mortar preconditioners for heterogeneous elliptic problems. Comput. Methods Appl. Mech. Engrg. 292, 221–242 (2015)

    MathSciNet  MATH  Google Scholar 

  2. Arnold, D., Falk, R., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numer. 1–155 (2006)

  3. Arnold, D., Brezzi, F.: Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. RAIRO Modél. Math. Anal. Numér. 19, 7–32 (1985)

    MathSciNet  MATH  Google Scholar 

  4. Arnold, D., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39, 1749–1779 (2002)

    MathSciNet  MATH  Google Scholar 

  5. Balis, J., Jacobs, F., May, G.: Aerodynamic shape optimization with hybridized discontinuous Galerkin schemes. In: AIAA SCITECH 2023 Forum, p. 1422 (2023)

  6. Bassi, F., Rebay, S.: A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. J. Comput. Phys. 131, 267–279 (1997)

    MathSciNet  MATH  Google Scholar 

  7. Boffi, D., Brezzi, F., Fortin, M.: Mixed Finite Element Methods and Applications. Springer Series in Computational Mathematics, vol. 44. Springer, Heidelberg (2013)

    MATH  Google Scholar 

  8. Bramble, J.H., Schatz, A.H.: Higher order local accuracy by averaging in the finite element method. Math. Comp. 31, 94–111 (1977)

    MathSciNet  MATH  Google Scholar 

  9. Brezzi, F., Douglas, J., Jr., Marini, L.D.: Two families of mixed finite elements for second order elliptic problems. Numer. Math. 47, 217–235 (1985)

    MathSciNet  MATH  Google Scholar 

  10. Bui-Thanh, T.: From Godunov to a unified hybridized discontinuous Galerkin framework for partial differential equations. J. Comput. Phys. 295, 114–146 (2015)

    MathSciNet  MATH  Google Scholar 

  11. Castillo, P.: Performance of discontinuous Galerkin methods for elliptic PDE’s. SIAM J. Sci. Comput. 24, 524–547 (2002)

    MathSciNet  MATH  Google Scholar 

  12. Castillo, P., Cockburn, B., Perugia, I., Schötzau, D.: An a priori error analysis of the local discontinuous Galerkin method for elliptic problems. SIAM J. Numer. Anal. 38, 1676–1706 (2000)

    MathSciNet  MATH  Google Scholar 

  13. Chung, E.T., Engquist, B.: Optimal discontinuous Galerkin methods for the acoustic wave equation in higher dimensions. SIAM J. Numer. Anal. 47, 3820–3848 (2009)

    MathSciNet  MATH  Google Scholar 

  14. Chung, E.T., Cockburn, B., Fu, G.: The staggered DG method is the limit of a hybridizable DG method. SIAM J. Numer. Anal. 52, 915–932 (2014)

    MathSciNet  MATH  Google Scholar 

  15. Chung, E.T., Cockburn, B., Fu, G.: The staggered DG method is the limit of a hybridizable DG method. Part II: the Stokes flow. J. Sci. Comput. 66, 870–887 (2016)

    MathSciNet  MATH  Google Scholar 

  16. Cicuttin, M., Ern, A., Pignet, N.: Hybrid High-Order Methods. A Primer with Applications to Solid Mechanics. Springer Briefs in Mathematics, Springer, Berlin (2021)

    MATH  Google Scholar 

  17. Ciuca, C., Fernandez, P., Christophe, A., Nguyen, N.C., Peraire, J.: Implicit hybridized discontinuous Galerkin methods for compressible magnetohydrodynamics. J. Comput. Phys.: X 5, 100042 (2020)

    MathSciNet  Google Scholar 

  18. Cockburn, B., Fu, G., Shi, K.: An introduction to the theory of \(M\)-decompositions. In: Numerical methods for PDEs, vol. 15 of SEMA SIMAI Springer Ser., Springer, Cham, pp. 5–29 (2018)

  19. Cockburn, B., Karniadakis, G., Shu, C.-W.: The development of discontinuous Galerkin methods. In: Cockburn, B., Karniadakis, G., and Shu, C.-W. (eds.) Discontinuous Galerkin Methods. Theory, Computation and Applications, vol. 11 of Lect. Notes Comput. Sci. Engrg., Berlin, Springer Verlag, pp. 3–50 (2000)

  20. Cockburn, B., Nguyen, N., Peraire, J.: HDG methods for hyperbolic problems, in Handbook of numerical methods for hyperbolic problems, vol. 17 of Handb. Numer. Anal., Elsevier/North-Holland, Amsterdam, pp. 173–197 (2016)

  21. Cockburn, B., Shu,C.-W.: eds., Special issue on discontinuous Galerkin methods, vol. 22 and 23 of J. Sci. Comput., Springer, (2005)

  22. Cockburn, B.: Static condensation, hybridization, and the devising of the HDG methods. In: Barrenechea, G., Brezzi, F., Cagniani, A. and Georgoulis, E. (eds.) Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations, vol. 114 of Lect. Notes Comput. Sci. Engrg., Springer Verlag, Berlin, 2016, pp. 129–177. LMS Durham Symposia funded by the London Mathematical Society. Durham, U.K., on July 8–16, 2014

  23. Cockburn, B.: The hybridizable discontinuous Galerkin methods. In: Bhatia, R., Pal, A., Rangarajan, G., Srinivas, V., and Vanninathan, M. (eds) Proceedings of the International Congress of Mathematicians 2010, Hyderabad. Hindustan Book Agency, pp. 2749–2775 (2010)

  24. Cockburn, B.: Discontinuous Galerkin methods. ZAMM Z. Angew. Math. Mech. 83, 731–754 (2003)

    MathSciNet  MATH  Google Scholar 

  25. Cockburn, B.: Discontinuous Galerkin methods for computational fluid dynamics. In: Stein, E., Borst, R., Hughes, T. (eds.) Encyclopedia of Computational Mechanics, vol. 5, 2nd edn., pp. 141–203. John Wiley & Sons Ltd, Chichester (2018)

    Google Scholar 

  26. Cockburn, B.: The pursuit of a dream, Francisco Javier Sayas and the HDG methods. SeMA J. 79, 37–56 (2022)

    MathSciNet  MATH  Google Scholar 

  27. Cockburn, B., Fu, G.: Superconvergence by \(M\)-decompositions. Part II: construction of two-dimensional finite elements. ESAIM Math. Model. Numer. Anal. 51, 165–186 (2017)

    MathSciNet  MATH  Google Scholar 

  28. Cockburn, B., Fu, G.: Superconvergence by \(M\)-decompositions. Part III: construction of three-dimensional finite elements. ESAIM Math. Model. Numer. Anal. 51, 365–398 (2017)

    MathSciNet  MATH  Google Scholar 

  29. Cockburn, B., Fu, G.: A systematic construction of finite element commuting exact sequences. SIAM J. Numer. Anal. 55, 1650–1688 (2017)

    MathSciNet  MATH  Google Scholar 

  30. Cockburn, B., Fu, G.: Devising superconvergent HDG methods with symmetric approximate stresses for linear elasticity by M-decompositions. IMA J. Numer. Anal. 38, 566–604 (2018)

    MathSciNet  MATH  Google Scholar 

  31. Cockburn, B., Gopalakrishnan, J.: A characterization of hybridized mixed methods for second order elliptic problems. SIAM J. Numer. Anal. 42, 283–301 (2004)

    MathSciNet  MATH  Google Scholar 

  32. Cockburn, B., Shen, J.: A Hybridizable discontinuous Galerkin method for the \(p\)-Laplacian. SIAM J. Sci. Comput. 38, A545–A566 (2016)

    MathSciNet  MATH  Google Scholar 

  33. Cockburn, B., Shu, C.-W.: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for scalar conservation laws II: general framework. Math. Comput. 52, 411–435 (1989)

    MATH  Google Scholar 

  34. Cockburn, B., Shu, C.-W.: The Runge-Kutta local projection \({P}^1\)-discontinuous Galerkin method for scalar conservation laws. RAIRO Modél. Math. Anal. Numér. 25, 337–361 (1991)

    MathSciNet  MATH  Google Scholar 

  35. Cockburn, B., Shu, C.-W.: The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35, 2440–2463 (1998)

    MathSciNet  MATH  Google Scholar 

  36. Cockburn, B., Shu, C.-W.: The Runge-Kutta discontinuous Galerkin finite element method for conservation laws V: multidimensional systems. J. Comput. Phys. 141, 199–224 (1998)

    MathSciNet  MATH  Google Scholar 

  37. Cockburn, B., Shu, C.-W.: Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16, 173–261 (2001)

    MathSciNet  MATH  Google Scholar 

  38. Cockburn, B., Wang, Z.: Adjoint-based, superconvergent Galerkin approximations of linear functionals. J. Sci. Comput. 73, 644–666 (2017)

    MathSciNet  MATH  Google Scholar 

  39. Cockburn, B., Xia, S.: An a priori error analysis of adjoint-based super-convergent Galerkin approximations of linear functionals. IMA J. Numer. Anal. 42, 1050–1086 (2021)

    MathSciNet  MATH  Google Scholar 

  40. Cockburn, B., Xia, S.: An adjoint-based super-convergent Galerkin approximation of eigenvalues. J. Comput. Phys. 449, 110816 (2022)

    MathSciNet  MATH  Google Scholar 

  41. Cockburn, B., Lin, S., Shu, C.-W.: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one dimensional systems. J. Comput. Phys. 84, 90–113 (1989)

    MathSciNet  MATH  Google Scholar 

  42. Cockburn, B., Hou, S., Shu, C.-W.: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case. Math. Comput. 54, 545–581 (1990)

    MATH  Google Scholar 

  43. Cockburn, B., Luskin, M., Shu, C.-W., Süli, E.: Enhanced accuracy by post-processing for finite element methods for hyperbolic equations. Math. Comput. 72, 577–606 (2003). (electronic)

    MathSciNet  MATH  Google Scholar 

  44. Cockburn, B., Dong, B., Guzmán, J.: Optimal convergence of the original DG method for the transport-reaction equation on special meshes. SIAM J. Numer. Anal. 46, 1250–1265 (2008)

    MathSciNet  MATH  Google Scholar 

  45. Cockburn, B., Dong, B., Guzmán, J.: A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems. Math. Comput. 77, 1887–1916 (2008)

    MathSciNet  MATH  Google Scholar 

  46. Cockburn, B., Gopalakrishnan, J., Lazarov, R.: Unified hybridization of discontinuous Galerkin, mixed and continuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. 47, 1319–1365 (2009)

    MathSciNet  MATH  Google Scholar 

  47. Cockburn, B., Guzmán, J., Wang, H.: Superconvergent discontinuous Galerkin methods for second-order elliptic problems. Math. Comput. 78, 1–24 (2009)

    MathSciNet  MATH  Google Scholar 

  48. Cockburn, B., Dong, B., Guzmán, J., Qian, J.: Optimal convergence of the original DG method on special meshes for variable convective velocity. SIAM J. Numer. Anal. 48, 133–146 (2010)

    MathSciNet  MATH  Google Scholar 

  49. Cockburn, B., Gopalakrishnan, J., Li, F., Nguyen, N.-C., Peraire, J.: Hybridization and postprocessing techniques for mixed eigenfunctions. SIAM J. Numer. Anal. 48, 857–881 (2010)

    MathSciNet  MATH  Google Scholar 

  50. Cockburn, B., Gopalakrishnan, J., Sayas, F.-J.: A projection-based error analysis of HDG methods. Math. Comput. 79, 1351–1367 (2010)

    MathSciNet  MATH  Google Scholar 

  51. Cockburn, B., Qiu, W., Shi, K.: Conditions for superconvergence of HDG methods for second-order eliptic problems. Math. Comput. 81, 1327–1353 (2012)

    MATH  Google Scholar 

  52. Cockburn, B., Di-Pietro, D., Ern, A.: Bridging the hybrid high-order and hybridizable discontinuous Galerkin methods. ESAIM Math. Model. Numer. Anal. 50, 635–650 (2016)

    MathSciNet  MATH  Google Scholar 

  53. Cockburn, B., Nochetto, R.H., Zhang, W.: Contraction property of adaptive hybridizable discontinuous Galerkin methods. Math. Comput. 85, 1113–1141 (2016)

    MathSciNet  MATH  Google Scholar 

  54. Cockburn, B., Fu, G., Qiu, W.: A note on the devising of superconvergent HDG methods for Stokes flow by \(M\)-decompositions. IMA J. Numer. Anal. 37, 730–749 (2017)

    MathSciNet  MATH  Google Scholar 

  55. Cockburn, B., Fu, G., Sayas, F.: Superconvergence by \(M\)-decompositions. Part I: general theory for HDG methods for diffusion. Math. Comput. 86, 1609–1641 (2017)

    MathSciNet  MATH  Google Scholar 

  56. Cockburn, B., Fu, G., Qiu, W.: Discrete H\(^1\)-inequalities for spaces admitting \(m\)-decompositions. SIAM J. Numer. Anal. 56, 3407–3429 (2018)

    MathSciNet  MATH  Google Scholar 

  57. Cockburn, B., Sánchez, M., Xiong, C.: Supercloseness of Primal-Dual Galerkin approximations for second order elliptic problems. J. Sci. Comput. 75, 376–394 (2018)

    MathSciNet  MATH  Google Scholar 

  58. Cockburn, B., Du, S., Sánchez, M.: Discontinuous Galerkin methods with time-operators in their numerical traces for time-dependent electromagnetics. Comput. Methods Appl. Mech. Engrg. 22, 775–796 (2022)

    MathSciNet  MATH  Google Scholar 

  59. Dawson, C. (ed.): Special issue on discontinuous Galerkin methods. Comput, vol. 195. Methods Appl. Mech. Engrg, Elsevier (2006)

  60. Di-Pietro, D., Droniou, J.: The hybrid high-order method for polytopal meshes, vol. 19 of MS &A. Modeling, Simulation and Applications, Springer, Cham, [2020] 2020. Design, analysis, and applications

  61. Di-Pietro, D., Ern, A.: A hybrid high-order locking-free method for linear elasticity on general meshes. Comput. Method Appl. Mech. Engrg. 283, 1–21 (2015)

    MathSciNet  MATH  Google Scholar 

  62. Di-Pietro, D., Ern, A., Lemaire, S.: An arbitrary-order and compact-stencil discretization of diffusion on general meshes based on local reconstruction operators, Comput. Meth. Appl. Math. 14, 461–472 (2014)

    MathSciNet  MATH  Google Scholar 

  63. Du, S., Sayas, F.-J.: An Invitation to the Theory of the Hybridizable Discontinuous Galerkin Method: Projections, Estimates, Tools. SpringerBriefs in Mathematics, Springer (2019)

  64. Du, S., Sayas, F.-J.: New analytical tools for HDG in elasticity, with applications to elastodynamics. Math. Comput. 89, 1745–1782 (2020)

    MathSciNet  MATH  Google Scholar 

  65. Fraejis de Veubeke, B.M.: Displacement and equilibrium models in the finite element method. In: Zienkiewicz, O., Holister, G. (eds.) Stress Analysis, pp. 145–197. Wiley, New York (1977)

    Google Scholar 

  66. Gastaldi, L., Nochetto, R.: Sharp maximum norm error estimates for general mixed finite element approximations to second order elliptic equations. RAIRO Modél. Math. Anal. Numér. 23, 103–128 (1989)

    MathSciNet  MATH  Google Scholar 

  67. Gopalakrishnan, J., Li, F., Nguyen, N.-C., Peraire, J.: Spectral approximations by the HDG method. Math. Comput. 84, 1037–1059 (2015)

    MathSciNet  MATH  Google Scholar 

  68. Guyan, R.J.: Reduction of stiffness and mass matrices. J. Am. Inst. Aron. Astro. 3, 380 (1965)

    Google Scholar 

  69. Johnson, C., Pitkäranta, J.: An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Math. Comput. 46, 1–26 (1986)

    MathSciNet  MATH  Google Scholar 

  70. Kercher, A., Corrigan, A., Kessler, D.: Aeroheating predictions of hypersonic flight geometries with high-order discontinuous Galerkin methods. AIAA SCITECH 2021 Forum, p. 1315 (2021)

  71. Klaij, C., van der Vegt, J.J.W., van der Ven, H.: Space-time discontinuous Galerkin method for the compressible Navier-Stokes equations. J. Comput. Phys. 217, 589–611 (2006)

    MathSciNet  MATH  Google Scholar 

  72. Lal, Z., Candler, G., Cockburn, B.: Aeroheating predictions of hypersonic flight geometries with high-order discontinuous Galerkin methods. AIAA SCITECH 2023 Forum, p. 0854 (2023)

  73. Lehrenfeld, C.: Hybrid discontinuous Galerkin methods for solving incompressible flow problems, PhD thesis, Diplomigenieur Rheinisch-Westfalishen Technischen Hochchule Aachen (2010)

  74. Lesaint, P., Raviart, P.: On a finite element method for solving the neutron transport equation. In: de Boor, C. (ed.) Mathematical Aspects of Finite Elements in Partial Differential Equations, pp. 89–145. Academic Press (1974)

    Google Scholar 

  75. Luo, H., Absillis, G., Nourgaliev, R.: A moving discontinuous Galerkin finite element method with interface condition enforcement for compressible flows. J. Comput. Phys. 445 (2021)

  76. MacCormack, R.: The carbuncle CFD problem, in 49th. AIAA Aerospace Sciences Meeting, Orlando, Florida (2011)

  77. May, G.: On the connection between the spectral difference method and the discontinuous Galerkin method. Commun. Comput. Phys. 9, 1071–1080 (2011)

    MathSciNet  MATH  Google Scholar 

  78. May, G., Devesse, K., Rangarajan, A., Magin, T.: A hybridized discontinuous Galerkin solver for high-speed compressible flow. Aerospace 8, 322 (2021)

    Google Scholar 

  79. Nguyen, N., Peraire, J., Cockburn, B.: Hybridizable discontinuous Galerkin methods. In: Hesthaven, J. and Ronquist, E. (eds.) Spectral and High Order Methods for Partial Differential Equations, vol. 76 of Lect. Notes Comput. Sci. Engrg., Springer Verlag, Berlin Heidelberg, pp. 63–84 (2011)

  80. Nguyen, N., Persson, P.-O., Peraire, J.: RANS solutions using high order discontin- uous Galerkin methods, in 45th. AAIA Aerospace Sciences Meeting and Exhibit, Reno, Nevada (2007)

  81. Nguyen, N. C., Terrana, S., Peraire, J.: An adaptive shock-capturing HDG method for compressible flows, in AIAA 2011 Conference (2011)

  82. Nguyen, N. C., Terrana, S., Peraire, J.: Implicit large eddy simulation of hypersonic boundary-layer transition for a flared cone. In: AIAA Scitech 2023 Forum, p. 1062 (2022)

  83. Oikawa, I.: A hybridized discontinuous Galerkin method with reduced stabilization. J. Sci. Comput. 65, 327–340 (2015)

    MathSciNet  MATH  Google Scholar 

  84. Oikawa, I.: Analysis of a reduced-Order HDG method for the stokes equations. J. Sci. Comput. 67, 475–492 (2016)

    MathSciNet  MATH  Google Scholar 

  85. Oikawa, I., Kikuchi, F.: Discontinuous Galerkin FEM of hybrid type. JSIAM Lett. 2, 49–52 (2010)

    MathSciNet  MATH  Google Scholar 

  86. Oikawa, I., Ishii, K., Kikuchi, F.: Discontinuous Galerkin FEM of hybrid displacement type- development of polygonal elements. Theoret. Appl. Mech. 57, 395–404 (2009)

    Google Scholar 

  87. Peraire, J., Nguyen, N. C., Cockburn, B.: A hybridizable discontinuous Galerkin method for the compressible Euler and Navier-Stokes equations (AIAA Paper 2010-363). In Proceedings of the 48th AIAA Aerospace Sciences Meeting and Exhibit, Orlando, Florida, January (2010)

  88. Peraire, J., Persson,P.-O.: High-order discontinuous Galerkin methods for CFD. In: Adaptive high-order methods in Computational fluid dynamics, vol. 2 of Adv. Comput. Fluid Dyn., World Sci. Publ., Hackensack, NJ, pp. 119–152 (2011)

  89. Peterson, T.: A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation. SIAM J. Numer. Anal. 28, 133–140 (1991)

    MathSciNet  MATH  Google Scholar 

  90. Qiu, W., Shi, K.: A superconvergent HDG method for the incompressible NavierĐStokes equations on general polyhedral meshes. IMA J. Num. Anal. 36, 1943–1967 (2016)

    MathSciNet  MATH  Google Scholar 

  91. Qiu, W., Shen, J., Shi, K.: An HDG method for linear elasticity with strong symmetric stresses. Math. Comput. 87, 69–93 (2018)

    MathSciNet  MATH  Google Scholar 

  92. Raviart, P., Thomas, J. M.: A mixed finite element method for second order elliptic problems. In: Galligani, I. and Magenes, E. (eds.) Mathematical Aspects of Finite Element Method, Lecture Notes in Math. 606, Springer-Verlag, New York, pp. 292–315 (1977)

  93. Reed, W., Hill, T.: Triangular mesh methods for the neutron transport equation. Tech. Rep. LA-UR-73-479, Los Alamos Scientific Laboratory (1973)

  94. Richter, G.: On the order of convergence of the discontinuous Galerkin method for hyperbolic equations. Math. Comput. 77, 1871–1885 (2008)

    MathSciNet  MATH  Google Scholar 

  95. Stenberg, R.: A family of mixed finite elements for the elasticity problem. Numer. Math. 53, 513–538 (1988)

    MathSciNet  MATH  Google Scholar 

  96. Stenberg, R.: Postprocessing schemes for some mixed finite elements. RAIRO Modél. Math. Anal. Numér. 25, 151–167 (1991)

    MathSciNet  MATH  Google Scholar 

  97. Stoter, S., Cockburn, B., Hughes, T., Schillinger, D.: Discontinuous Galerkin methods through the lens of variational multiscale analysis. Comput. Methods Appl. Mech. Engrg. 399, 115310 (2022)

    MathSciNet  MATH  Google Scholar 

  98. Vieira, L.M., Giacomini, M., Sevilla, R., Huerta, A.: A second-order face-centred finite volume method for elliptic problems. Comput. Methods Appl. Mech. Engrg. 358, 112655 (2020)

    MathSciNet  MATH  Google Scholar 

  99. Vila-Pérez, J., Giacomini, M., Sevilla, R., Huerta, A.: Hybridisable discontinuous Galerkin formulation of compressible flows. Arch. Comput. Methods Eng. 28 (2020)

  100. Voralík, M.: Unified primal formulation-based a priori and a posteriori error analysis of mixed methods. Math. Comput. 79, 2001–2032 (2010)

    MathSciNet  Google Scholar 

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Acknowledgements

The author would like to thank Sreevatsa Anantharamu, for his comments leading to a better presentation of Sect. 4; Shukai Du, for his excellent proofreading leading to the clarification of several computations and for underlining the relevance of reference [64]; Alexandre Ern, for bringing to the author’s attention reference [60]; Guosheng Fu, for suggesting the open problem (11), and for recommending to display the stabilization function for the case in which \(\varvec{V}_a(K)=\{\varvec{0}\}\) for the Raviart-Thomas method of lowest index; Jay Gopalakrishnan, for conversations leading to clarifications of the main result; Zubin Lal, for the simulations of his DG method for Mach 17.61 hypersonic flows; N.-Cuong Nguyen, for the statements (1) and (10) of the open problems; Manuel Sánchez and Stein Stoter, for valuable feedback on the first version of this manuscript; and Martin Voralík, for bringing his work on the relation of the mixed and the contiuous Galerkin method [100] to the author’s attention. Last, but not least, the author would like to thank the referee for a careful reading of the manuscript.

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    Bernardo Cockburn

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Bernardo Cockburn’s research was supported in part by the Advanced Computational Center for Entry Systems Simulation (ACCESS) through NASA grant 80NSSC21K1117.

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Cockburn, B. Hybridizable discontinuous Galerkin methods for second-order elliptic problems: overview, a new result and open problems. Japan J. Indust. Appl. Math. 40, 1637–1676 (2023). https://doi.org/10.1007/s13160-023-00603-9

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