Open Access
Volume 57, Number 3, May-June 2023
Page(s) 1553 - 1587
Published online 26 May 2023
  1. D.N. Arnold and F. Brezzi, Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. RAIRO Modél. Math. Anal. Numér. 19 (1985) 7–32. [Google Scholar]
  2. D.N. Arnold, R.S. Falk and R. Winther, Multigrid in H(div) and H(curl). Numer. Math. 85 (2000) 197–217. [Google Scholar]
  3. J. Betteridge, T.H. Gibson, I.G. Graham and E.H. Müller, Multigrid preconditioners for the hybridised discontinuous Galerkin discretisation of the shallow water equations. J. Comput. Phys. 426 (2021) 34. [Google Scholar]
  4. D. Braess and R. Verfürth, Multigrid methods for nonconforming finite element methods. SIAM J. Numer. Anal. 27 (1990) 979–986. [Google Scholar]
  5. J.H. Bramble and J.E. Pasciak, New convergence estimates for multigrid algorithms. Math. Comp. 49 (1987) 311–329. [Google Scholar]
  6. J.H. Bramble, J.E. Pasciak and J. Xu, The analysis of multigrid algorithms with nonnested spaces or noninherited quadratic forms. Math. Comp. 56 (1991) 1–34. [Google Scholar]
  7. C. Brennecke, A. Linke, C. Merdon and J. Schöberl, Optimal and pressure-independent L2 velocity error estimates for a modified Crouzeix-Raviart element with BDM reconstructions, in Finite volumes for Complex Applications VII. Methods and Theoretical Aspects. Vol. 77 of Springer Proc. Math. Stat. Springer, Cham (2014) 159–167. [Google Scholar]
  8. S.C. Brenner, An optimal-order multigrid method for P1 nonconforming finite elements. Math. Comput. 52 (1989) 1–15. [Google Scholar]
  9. S.C. Brenner, An optimal-order nonconforming multigrid method for the biharmonic equation. SIAM J. Numer. Anal. 26 (1989) 1124–1138. [Google Scholar]
  10. S.C. Brenner, A nonconforming multigrid method for the stationary Stokes equations. Math. Comput. 55 (1990) 411–437. [Google Scholar]
  11. S.C. Brenner, A multigrid algorithm for the lowest-order Raviart-Thomas mixed triangular finite element method. SIAM J. Numer. Anal. 29 (1992) 647–678. [Google Scholar]
  12. S.C. Brenner, A nonconforming mixed multigrid method for the pure displacement problem in planar linear elasticity. SIAM J. Numer. Anal. 30 (1993) 116–135. [Google Scholar]
  13. S.C. Brenner, A nonconforming mixed multigrid method for the pure traction problem in planar linear elasticity. Math. Comput. 63435–460 (1994) S1–S5. [Google Scholar]
  14. S.C. Brenner, Convergence of nonconforming multigrid methods without full elliptic regularity. Math. Comput. 68 (1999) 25–53. [Google Scholar]
  15. S.C. Brenner, Poincaré-Friedrichs inequalities for piecewise H1 functions. SIAM J. Numer. Anal. 41 (2003) 306–324. [Google Scholar]
  16. S.C. Brenner, Convergence of nonconforming V-cycle and F-cycle multigrid algorithms for second order elliptic boundary value problems. Math. Comput. 73 (2004) 1041–1066. [Google Scholar]
  17. S.C. Brenner, Forty years of the Crouzeix-Raviart element. Numer. Methods Part. Differ. Equ. 31 (2015) 367–396. [Google Scholar]
  18. Z. Cai and S. Zhang, Recovery-based error estimators for interface problems: mixed and nonconforming finite elements. SIAM J. Numer. Anal. 48 (2010) 30–52. [Google Scholar]
  19. Z. Chen, Equivalence between and multigrid algorithms for nonconforming and mixed methods for second-order elliptic problems. East-West J. Numer. Math. 4 (1996) 1–33. [Google Scholar]
  20. Z. Chen and D.Y. Kwak, The analysis of multigrid algorithms for nonconforming and mixed methods for second order elliptic problems, in IMA Preprints Series, Institute for Mathematics and Its Applications, University of Minnesota (1994). [Google Scholar]
  21. H. Chen, P. Lu and X. Xu, A robust multilevel method for hybridizable discontinuous Galerkin method for the Helmholtz equation. J. Comput. Phys. 264 (2014) 133–151. [Google Scholar]
  22. L. Chen, J. Wang, Y. Wang and X. Ye, An auxiliary space multigrid preconditioner for the weak Galerkin method. Comput. Math. Appl. 70 (2015) 330–344. [Google Scholar]
  23. P.G. Ciarlet, The Finite Element Method for Elliptic Problems. Vol. 40 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2002). Reprint of the 1978 original [North-Holland, Amsterdam; MR0520174 (58 #25001)]. [Google Scholar]
  24. 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. Vol. 114 of Lect. Notes Comput. Sci. Eng. Springer, [Cham], (2016) . 129–177. [Google Scholar]
  25. B. Cockburn, Discontinuous galerkin methods for computational fluid dynamics, in Encyclopedia of Computational Mechanics, 2nd edition. John Wiley & Sons, Ltd (2017) 1–63. [Google Scholar]
  26. B. Cockburn, J. Gopalakrishnan and R. Lazarov, Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. 47 (2009) 1319–1365. [Google Scholar]
  27. B. Cockburn, O. Dubois, J. Gopalakrishnan and S. Tan, Multigrid for an HDG method. IMA J. Numer. Anal. 34 (2014) 1386–1425. [Google Scholar]
  28. B. Cockburn, N.C. Nguyen and J. Peraire, HDG methods for hyperbolic problems, in Handbook of Numerical Methods for Hyperbolic Problems. Vol. 17 of Handb. Numer. Anal. Elsevier/North-Holland, Amsterdam (2016) 173–197. [Google Scholar]
  29. D.A. Di Pietro, F. Hülsemann, P. Matalon, P. Mycek, U. Rüde and D. Ruiz, An h-multigrid method for hybrid high-order discretizations. SIAM J. Sci. Comput. 43 (2021) S839–S861. [Google Scholar]
  30. D.A. Di Pietro, F. Hülsemann, P. Matalon, P. Mycek, U. Rüde and D. Ruiz, Towards robust, fast solutions of elliptic equations on complex domains through hybrid high-order discretizations and non-nested multigrid methods. Int. J. Numer. Methods Eng. 122 (2021) 6576–6595. [Google Scholar]
  31. H.-Y. Duan, S.-Q. Gao, R.C.E. Tan and S. Zhang, A generalized BPX multigrid framework covering nonnested V-cycle methods. Math. Comput. 76 (2007) 137–152. [Google Scholar]
  32. M.S. Fabien, M.G. Knepley, R.T. Mills and B.M. Rivière, Manycore parallel computing for a hybridizable discontinuous Galerkin nested multigrid method. SIAM J. Sci. Comput. 41 (2019) C73–C96. [Google Scholar]
  33. P.E. Farrell, L. Mitchell and F. Wechsung, An augmented Lagrangian preconditioner for the 3D stationary incompressible Navier-Stokes equations at high Reynolds number. SIAM J. Sci. Comput. 41 (2019) A3073–A3096. [Google Scholar]
  34. P.E. Farrell, L. Mitchell, L.R. Scott and F. Wechsung, Robust multigrid methods for nearly incompressible elasticity using macro elements. IMA J. Numer. Anal. 42 (2022) 3306–3329. [Google Scholar]
  35. P. Fernandez, A. Christophe, S. Terrana, N.C. Nguyen and J. Peraire, Hybridized discontinuous Galerkin methods for wave propagation. J. Sci. Comput. 77 (2018) 1566–1604. [Google Scholar]
  36. M. Fortin and R. Glowinski, Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary Value Problems, translated from the French by B. Hunt and D. C. Spicer. Vol. 15 of Studies in Mathematics and its Applications. North-Holland Publishing Co., Amsterdam (1983). [Google Scholar]
  37. G. Fu, C. Lehrenfeld, A. Linke and T. Streckenbach, Locking-free and gradient-robust H(div)-conforming HDG methods for linear elasticity. J. Sci. Comput. 86 (2021) 30. [Google Scholar]
  38. W. Hackbusch, Multigrid Methods and Applications. Vol. 4 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin (1985). [Google Scholar]
  39. Q. Hong, J. Kraus, J. Xu and L. Zikatanov, A robust multigrid method for discontinuous Galerkin discretizations of Stokes and linear elasticity equations. Numer. Math. 132 (2016) 23–49. [Google Scholar]
  40. X. Huang, Nonconforming finite element stokes complexes in three dimensions. Sci. China Math. (2023). DOI: 10.1007/s11425-021-2026-7. [Google Scholar]
  41. V. John, A. Linke, C. Merdon, M. Neilan and L.G. Rebholz, On the divergence constraint in mixed finite element methods for incompressible flows. SIAM Rev. 59 (2017) 492–544. [Google Scholar]
  42. G. Kanschat and Y. Mao, Multigrid methods for Hdiv-conforming discontinuous Galerkin methods for the Stokes equations. J. Numer. Math. 23 (2015) 51–66. [Google Scholar]
  43. T.V. Kolev, J. Xu and Y. Zhu, Multilevel preconditioners for reaction-diffusion problems with discontinuous coefficients. J. Sci. Comput. 67 (2016) 324–350. [Google Scholar]
  44. Y.-J. Lee, J. Wu, J. Xu and L. Zikatanov, Robust subspace correction methods for nearly singular systems. Math. Models Methods Appl. Sci. 17 (2007) 1937–1963. [Google Scholar]
  45. Y.-J. Lee, J. Wu and J. Chen, Robust multigrid method for the planar linear elasticity problems. Numer. Math. 113 (2009) 473–496. [Google Scholar]
  46. C. Lehrenfeld, Hybrid Discontinuous Galerkin methods for solving incompressible flow problems. Ph.D. thesis, RWTH Aachen University (2010). [Google Scholar]
  47. A. Linke, On the role of the Helmholtz decomposition in mixed methods for incompressible flows and a new variational crime. Comput. Methods Appl. Mech. Eng. 268 (2014) 782–800. [Google Scholar]
  48. A. Linke and C. Merdon, Pressure-robustness and discrete Helmholtz projectors in mixed finite element methods for the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 311 (2016) 304–326. [Google Scholar]
  49. P. Lu, A. Rupp and G. Kanschat, Homogeneous multigrid for HDG. IMA J. Numer. Anal. 42 (2021) 3135–3153. [Google Scholar]
  50. L.D. Marini, An inexpensive method for the evaluation of the solution of the lowest order Raviart-Thomas mixed method. SIAM J. Numer. Anal. 22 (1985) 493–496. [Google Scholar]
  51. L.S.D. Morley, The triangular equilibrium element in the solution of plate bending problems. Aeronaut. Q. 19 (1968) 149–169. [Google Scholar]
  52. N.C. Nguyen and J. Peraire, Hybridizable discontinuous Galerkin methods for partial differential equations in continuum mechanics. J. Comput. Phys. 231 (2012) 5955–5988. [Google Scholar]
  53. Y. Notay, A new algebraic multigrid approach for Stokes problems. Numer. Math. 132 (2016) 51–84. [Google Scholar]
  54. Y. Notay, Algebraic multigrid for Stokes equations. SIAM J. Sci. Comput. 39 (2017) S88–S111. [Google Scholar]
  55. I. Oikawa, A hybridized discontinuous Galerkin method with reduced stabilization. J. Sci. Comput. 65 (2015) 327–340. [Google Scholar]
  56. I. Oikawa, Analysis of a reduced-order HDG method for the Stokes equations. J. Sci. Comput. 67 (2016) 475–492. [Google Scholar]
  57. W. Qiu and K. Shi, An HDG method for convection diffusion equation. J. Sci. Comput. 66 (2016) 346–357. [Google Scholar]
  58. W. Qiu and K. Shi, A superconvergent HDG method for the incompressible Navier-Stokes equations on general polyhedral meshes. IMA J. Numer. Anal. 36 (2016) 1943–1967. [Google Scholar]
  59. W. Qiu, J. Shen and K. Shi, An HDG method for linear elasticity with strong symmetric stresses. Math. Comput. 87 (2018) 69–93. [Google Scholar]
  60. J. Schöberl, Multigrid methods for a parameter dependent problem in primal variables. Numer. Math. 84 (1999) 97–119. [Google Scholar]
  61. J. Schöberl, Robust multigrid methods for parameter dependent problems. Ph.D. thesis, Johannes Kepler University Linz (1999). [Google Scholar]
  62. J. Schöberl, C++11 implementation of finite elements in NGSolve, ASC Report 30/2014, Institute for Analysis and Scientific Computing, Vienna University of Technology (2014). [Google Scholar]
  63. R. Stevenson, Nonconforming finite elements and the cascadic multi-grid method. Numer. Math. 91 (2002) 351–387. [Google Scholar]
  64. S. Tan, Iterative solvers for hybridized finite element methods. Ph.D. thesis, University of Florida (2009). [Google Scholar]
  65. S. Turek, Multigrid techniques for a divergence-free finite element discretization. East-West J. Numer. Math. 2 (1994) 229–255. [Google Scholar]
  66. H. Uzawa, Iterative methods for concave programming, in Studies in Linear and Non-Linear Programming. Stanford University Press, Stanford, CA (1958) 154–165. [Google Scholar]
  67. T. Wildey, S. Muralikrishnan and T. Bui-Thanh, Unified geometric multigrid algorithm for hybridized high-order finite element methods. SIAM J. Sci. Comput. 41 (2019) S172–S195. [Google Scholar]
  68. J. Xu, The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids. Computing 56 (1996) 215–235. [Google Scholar]
  69. B. Zheng, Q. Hu and J. Xu, A nonconforming finite element method for fourth order curl equations in ℝ3. Math. Comput. 80 (2011) 1871–1886. [Google Scholar]
  70. Y. Zhu, Analysis of a multigrid preconditioner for Crouzeix-Raviart discretization of elliptic partial differential equation with jump coefficients. Numer. Linear Algebra Appl. 21 (2014) 24–38. [Google Scholar]

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