Advertisement

Mimetic Spectral Element Method for Anisotropic Diffusion

  • Marc Gerritsma
  • Artur Palha
  • Varun Jain
  • Yi Zhang
Chapter
Part of the SEMA SIMAI Springer Series book series (SEMA SIMAI, volume 15)

Abstract

This chapter addresses the topological structure of steady, anisotropic, inhomogeneous diffusion problems. Differential operators are represented by sparse incidence matrices, while weighted mass matrices play the role of metric-dependent Hodge matrices. The resulting mixed formulation is point-wise divergence-free if the right hand side function f = 0. The method is inf-sup stable; no stabilization is required and the method displays optimal convergence on orthogonal and deformed grids.

References

  1. 1.
    Aarnes, J.E., Krogstad, S., Lie, K.A.: Multiscale mixed/mimetic methods on corner-point grids. Comput. Geosci. 12, 297–315 (2008). https://doi.org/10.1007/s10596-007-9072-8 MathSciNetzbMATHGoogle Scholar
  2. 2.
    Aavatsmark, I.: An introduction to multipoint flux approximations for quadrilateral grids. Comput. Geosci. 6, 405–432 (2002). https://doi.org/10.1023/A:1021291114475 MathSciNetzbMATHGoogle Scholar
  3. 3.
    Aavatsmark, I.: Interpretation of a two-point flux stencil for skew parallelogram grids. Comput. Geosci. 11, 199–206 (2007). https://doi.org/10.1007/s10596-007-9042-1 zbMATHGoogle Scholar
  4. 4.
    Aavatsmark, I., Barkve, T., Bøe, O., Mannseth, T.: Discretization on unstructured grids for inhomogeneous, anisotropic media. Part I: derivation of the methods. SIAM J. Sci. Comput. 19, 1700–1716 (1998). https://doi.org/10.1137/S1064827595293582 MathSciNetzbMATHGoogle Scholar
  5. 5.
    Aavatsmark, I., Barkve, T., Bøe, O., Mannseth, T.: Discretization on unstructured grids for inhomogeneous, anisotropic media. Part II: discussion and numerical results. SIAM J. Sci. Comput. 19, 1717–1736 (1998). https://doi.org/10.1137/S1064827595293594 MathSciNetzbMATHGoogle Scholar
  6. 6.
    Alpak, F.O.: A mimetic finite volume discretization operator for reservoir simulation. In: SPE Reservoir Simulation Symposium. Society of Petroleum Engineers, Richardson (2007). https://doi.org/10.2118/106445-MS
  7. 7.
    Alpak, F.O.: A mimetic finite volume discretization method for reservoir simulation. SPE J. 15, 436–453 (2010). https://doi.org/10.2118/106445-PA Google Scholar
  8. 8.
    Arnold, D.N., Boffi, D., Falk, R.S.: Quadrilateral H(div) finite elements. SIAM J. Numer. Anal. 42, 2429–2451 (2005). https://doi.org/10.1137/S0036142903431924 MathSciNetzbMATHGoogle Scholar
  9. 9.
    Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numer. 15, 1–155 (2006)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus: from Hodge theory to numerical stability. Bull. Am. Math. Soc. 47, 281–354 (2010)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Aziz, K.: Reservoir simulation grids: opportunities and problems. J. Pet. Technol. 45, 658–663 (1993). https://doi.org/10.2118/25233-PA Google Scholar
  12. 12.
    Babuska, I., Suri, M.: On locking and robustness in the finite element method. SIAM J. Numer. Anal. 29, 1261–1293 (1992). https://doi.org/10.1137/0729075 MathSciNetzbMATHGoogle Scholar
  13. 13.
    Bastian, P., Ippisch, O., Marnach, S.: Benchmark 3D: a mimetic finite difference method. In: Finite Volumes for Complex Applications VI: Problems and Perspectives, pp. 961–968. Springer, Berlin (2011). https://doi.org/10.1007/978-3-642-20671-9_93 zbMATHGoogle Scholar
  14. 14.
    Bauer, W., Gay-Balmaz, F.: Variational integrators for an elastic and pseudo-incompressible flows (2017). http://arxiv.org/abs/1701.06448. ArXiv preprint n.1701.06448
  15. 15.
    Bergman, T.L., Incropera, F.P.: Fundamentals of Heat and Mass Transfer. Wiley, Hoboken (2011)Google Scholar
  16. 16.
    Bochev, P.B., Gerritsma, M.: A spectral mimetic least-squares method. Comput. Math. Appl. 68, 1480–1502 (2014). https://doi.org/10.1016/j.camwa.2014.09.014 MathSciNetzbMATHGoogle Scholar
  17. 17.
    Bochev, P.B., Gunzburger, M.D.: Least-Squares Finite Element Methods. Springer Series in Applied Mathematical Sciences. Springer, New York (2009)Google Scholar
  18. 18.
    Bochev, P.B., Hyman, J.M.: Principles of mimetic discretizations of differential operators. IMA Vol. Math. Appl. 142, 89 (2006)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Bochev, P.B., Ridzal, D.: Rehabilitation of the lowest-order Raviart–Thomas element on quadrilateral grids. SIAM J. Numer. Anal. 47, 487–507 (2008). https://doi.org/10.1137/070704265 MathSciNetzbMATHGoogle Scholar
  20. 20.
    Boffi, D., Gastaldi, L.: Some remarks on quadrilateral mixed finite elements. Comput. Struct. 87, 751–757 (2009)Google Scholar
  21. 21.
    Boffi, D., Brezzi, F., Fortin, M.: Mixed Finite Element Methods and Applications. Springer Series in Computational Mathematics. Springer, Berlin (2013)zbMATHGoogle Scholar
  22. 22.
    Bonelle, J., Ern, A.: Analysis of compatible discrete operator schemes for elliptic problems on polyhedral meshes. ESAIM: Math. Model. Numer. Anal. 48, 553–581 (2014). https://doi.org/10.1051/m2an/2013104 MathSciNetzbMATHGoogle Scholar
  23. 23.
    Bonelle, J., Ern, A.: Analysis of compatible discrete operator schemes for the Stokes equations on polyhedral meshes. IMA J. Numer. Anal. 35, 1672–1697 (2015).  https://doi.org/10.1093/imanum/dru051 MathSciNetzbMATHGoogle Scholar
  24. 24.
    Bonelle, J., Di Pietro, D.A., Ern, A.: Low-order reconstruction operators on polyhedral meshes: application to compatible discrete operator schemes. Comput. Aided Geom. Des. 35–36, 27–41 (2015). https://doi.org/10.1016/j.cagd.2015.03.015 MathSciNetGoogle Scholar
  25. 25.
    Bossavit, A.: Computational electromagnetism and geometry: (1) network equations. J. Jpn. Soc. Appl. Electromagn. 7, 150–159 (1999)Google Scholar
  26. 26.
    Bossavit, A.: Computational electromagnetism and geometry: (2) network constitutive laws. J. Jpn. Soc. Appl. Electromagn. 7, 294–301 (1999)Google Scholar
  27. 27.
    Bossavit, A.: Computational electromagnetism and geometry: (3) convergence. J. Jpn. Soc. Appl. Electromagn. 7, 401–408 (1999)Google Scholar
  28. 28.
    Bossavit, A.: Computational electromagnetism and geometry: (4) from degrees of freedom to fields. J. Jpn. Soc. Appl. Electromagn. 8, 102–109 (2000)Google Scholar
  29. 29.
    Bossavit, A.: Computational electromagnetism and geometry: (5) the “Galerkin Hodge”. J. Jpn. Soc. Appl. Electromagn. 8, 203–209 (2000)Google Scholar
  30. 30.
    Bouman, M., Palha, A., Kreeft, J., Gerritsma, M.: A conservative spectral element method for curvilinear domains. In: Spectral and High Order Methods for Partial Differential Equations. Lecture Notes in Computational Science and Engineering, vol. 76, pp. 111–119. Springer, Berlin (2011). https://doi.org/10.1007/978-3-642-15337-2_8 Google Scholar
  31. 31.
    Brezzi, F., Buffa, A.: Innovative mimetic discretizations for electromagnetic problems. J. Comput. Appl. Math. 234, 1980–1987 (2010)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics, vol. 15. Springer, New York (1991)Google Scholar
  33. 33.
    Brezzi, F., Lipnikov, K., Simoncini, V.: A family of mimetic finite difference methods on polygonal and polyhedral meshes. Math. Models Methods Appl. Sci. 15, 1533–1551 (2005). https://doi.org/10.1142/S0218202505000832 MathSciNetzbMATHGoogle Scholar
  34. 34.
    Brezzi, F., Lipnikov, K., Shashkov, M.: Convergence of mimetic finite difference method for diffusion problems on polyhedral meshes with curved faces. Math. Models Methods Appl. Sci. 16, 275–297 (2006). https://doi.org/10.1142/S0218202506001157 MathSciNetzbMATHGoogle Scholar
  35. 35.
    Brezzi, F., Lipnikov, K., Shashkov, M., Simoncini, V.: A new discretization methodology for diffusion problems on generalized polyhedral meshes. Comput. Methods Appl. Mech. Eng. 196, 3682–3692 (2007). https://doi.org/10.1016/j.cma.2006.10.028 MathSciNetzbMATHGoogle Scholar
  36. 36.
    Brezzi, F., Buffa, A., Lipnikov, K.: Mimetic finite differences for elliptic problems. Math. Model. Numer. Anal. 43, 277–296 (2009)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Brezzi, F., Falk, R.S., Donatella Marini, L.: Basic principles of mixed virtual element methods. ESAIM: Math. Model. Numer. Anal. 48, 1227–1240 (2014). https://doi.org/10.1051/m2an/2013138 MathSciNetzbMATHGoogle Scholar
  38. 38.
    Budd, C., Piggott, M.: Geometric Integration and its Applications. In: Handbook of Numerical Analysis, vol. 11, pp. 35–139. North-Holland, Amsterdam (2003). https://doi.org/10.1016/S1570-8659(02)11002-7 Google Scholar
  39. 39.
    Buffa, A., de Falco, C., Sangalli, G.: Isogeometric analysis: stable elements for the 2D Stokes equation. Int. J. Numer. Methods Fluids 65, 1407–1422 (2011).  https://doi.org/10.1002/fld.2337 MathSciNetzbMATHGoogle Scholar
  40. 40.
    Cantin, P., Bonelle, J., Burman, E., Ern, A.: A vertex-based scheme on polyhedral meshes for advection-reaction equations with sub-mesh stabilization. Comput. Math. Appl. 72(9), 2057–2071 (2016)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods in Fluid Dynamics. Springer, Berlin (1988)zbMATHGoogle Scholar
  42. 42.
    Christiansen, S.H., Munthe-Kaas, H.Z., Owren, B.: Topics in structure-preserving discretization. Acta Numer. 20, 1–119 (2011). https://doi.org/10.1017/S096249291100002X MathSciNetzbMATHGoogle Scholar
  43. 43.
    Cockburn, B.: Static Condensation, Hybridization, and the Devising of the HDG Methods, pp. 129–177. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-41640-3_5 Google Scholar
  44. 44.
    da Veiga, L.B., Brezzi, F., Marini, L.D., Russo, A.: The Hitchhiker’s guide to the virtual element method. Math. Models Methods Appl. Sci. 24, 1541–1573 (2014). https://doi.org/10.1142/S021820251440003X MathSciNetzbMATHGoogle Scholar
  45. 45.
    da Veiga, L.B., Lipnikov, K., Manzini, G.: The Mimetic Finite Difference Method for Elliptic Problems. Springer, Basel (2014). https://doi.org/10.1007/978-3-319-02663-3 zbMATHGoogle Scholar
  46. 46.
    da Veiga, L.B., Lovadina, C., Vacca, G.: Divergence free virtual elements for the Stokes problem on polygonal meshes (2015). arXiv:1510.01655v1Google Scholar
  47. 47.
    da Veiga, L.B., Brezzi, F., Marini, L.D., Russo, A.: H(div) and H(curl) conforming virtual element methods. Numer. Math., 1–30 (2015). https://doi.org/10.1007/s00211-015-0746-1 MathSciNetzbMATHGoogle Scholar
  48. 48.
    da Veiga, L.B., Brezzi, F., Marini, L.D., Russo, A.: Virtual element method for general second-order elliptic problems on polygonal meshes. Math. Models Methods Appl. Sci. 26, 729–750 (2016). https://doi.org/10.1142/S0218202516500160 MathSciNetzbMATHGoogle Scholar
  49. 49.
    Desbrun, M., Hirani, A.N., Leok, M., Marsden, J.E.: Discrete exterior calculus (2005). arXiv:math/0508341v2Google Scholar
  50. 50.
    Di Pietro, D.A., Ern, A.: Hybrid high-order methods for variable-diffusion problems on general meshes. C.R. Math. 353, 31–34 (2015). https://doi.org/10.1016/j.crma.2014.10.013 MathSciNetzbMATHGoogle Scholar
  51. 51.
    Di Pietro, D.A., Ern, A., Lemaire, S.: An arbitrary-order and compact-stencil discretization of diffusion on general meshes based on local reconstruction operators. Comput. Methods Appl. Math. 14 (2014).  https://doi.org/10.1515/cmam-2014-0018
  52. 52.
    Dodziuk, J.: Finite difference approach to the Hodge theory of harmonic functions. Am. J. Math. 98, 79–104 (1976)zbMATHGoogle Scholar
  53. 53.
    Durlofsky, L.J.: A triangle based mixed finite element finite volume technique for modeling two phase flow through porous media. J. Comput. Phys. 105, 252–266 (1993).  https://doi.org/10.1006/jcph.1993.1072 zbMATHGoogle Scholar
  54. 54.
    Durlofsky, L.J.: Accuracy of mixed and control volume finite element approximations to Darcy velocity and related quantities. Water Resour. Res. 30, 965–973 (1994). https://doi.org/10.1029/94WR00061 Google Scholar
  55. 55.
    Dziubek, A., Guidoboni, G., Harris, A., Hirani, A.N., Rusjan, E., Thistleton, W.: Effect of ocular shape and vascular geometry on retinal hemodynamics: a computational model. Biomech. Model. Mechanobiol. 15, 893–907 (2016). https://doi.org/10.1007/s10237-015-0731-8 Google Scholar
  56. 56.
    Edwards, M.G.: Unstructured, control-volume distributed, full-tensor finite-volume schemes with flow based grids. Comput. Geosci. 6, 433–452 (2002). https://doi.org/10.1023/A:1021243231313 MathSciNetzbMATHGoogle Scholar
  57. 57.
    Edwards, M.G., Rogers, C.F.: Finite volume discretization with imposed flux continuity for the general tensor pressure equation. Comput. Geosci. 2, 259–290 (1998). https://doi.org/10.1023/A:1011510505406 MathSciNetzbMATHGoogle Scholar
  58. 58.
    Elcott, S., Tong, Y., Kanso, E., Schröder, P., Desbrun, M.: Stable, circulation-preserving, simplicial fluids. ACM Trans. Graph. 26(1), 4 (2007). https://doi.org/10.1145/1189762.1189766 Google Scholar
  59. 59.
    Evans, J.A., Hughes, T.J.: Isogeometric divergence-conforming B-splines for the unsteady Navier-Stokes equations. J. Comput. Phys. 241, 141–167 (2013). https://doi.org/10.1016/j.jcp.2013.01.006 MathSciNetzbMATHGoogle Scholar
  60. 60.
    Forsyth, P.A.: A control-volume, finite-element method for local mesh refinement in thermal reservoir simulation. SPE Reserv. Eng. 5, 561–566 (1990). https://doi.org/10.2118/18415-PA Google Scholar
  61. 61.
    Gerritsma, M.: Edge functions for spectral element methods. In: Spectral and High Order Methods for Partial Differential Equations. Lecture Notes in Computational Science and Engineering, vol. 76, pp. 199–207. Springer, Berlin (2011). https://doi.org/10.1007/978-3-642-15337-2_17 Google Scholar
  62. 62.
    Gerritsma, M., Bochev, P.B.: A spectral mimetic least-squares method for the Stokes equations with no-slip boundary condition. Comput. Math. Appl. 71, 2285–2300 (2016). https://doi.org/10.1016/j.camwa.2016.01.033 MathSciNetGoogle Scholar
  63. 63.
    Gerritsma, M., Bouman, M., Palha, A.: Least-squares spectral element method on a staggered grid. In: Large-Scale Scientific Computing. Lecture Notes in Computer Science, vol. 5910, pp. 653–661. Springer, Berlin (2010)Google Scholar
  64. 64.
    Gerritsma, M., Hiemstra, R., Kreeft, J., Palha, A., Rebelo, P.P., Toshniwal, D.: The geometric basis of numerical methods. In: Spectral and High Order Methods for Partial Differential Equations. Lecture Notes in Computational Science and Engineering, vol. 95, pp. 17–35. Springer, Cham (2013)Google Scholar
  65. 65.
    Gunasekera, D., Cox, J., Lindsey, P.: The generation and application of K-orthogonal grid systems. In: SPE Reservoir Simulation Symposium, pp. 199–214. Society of Petroleum Engineers, Richardson (1997). https://doi.org/10.2118/37998-MS
  66. 66.
    Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration. Springer, Berlin (2006)zbMATHGoogle Scholar
  67. 67.
    Heinemann, Z.E., Brand, C.W., Munka, M., Chen, Y.M.: Modeling reservoir geometry with irregular grids. SPE Reserv. Eng. 6, 225–232 (1991). https://doi.org/10.2118/18412-PA Google Scholar
  68. 68.
    Herbin, R., Hubert, F.: Benchmark on discretization schemes for anisotropic diffusion problems on general grids. In: Finite Volumes for Complex Applications V: Problems and Perspectives, pp. 659–692. Wiley, Hoboken (2008)Google Scholar
  69. 69.
    Hermeline, F.: A finite volume method for the approximation of diffusion operators on distorted meshes. J. Comput. Phys. 160, 481–499 (2000).  https://doi.org/10.1006/jcph.2000.6466 MathSciNetzbMATHGoogle Scholar
  70. 70.
    Hiemstra, R., Toshniwal, D., Huijsmans, R., Gerritsma, M.: High order geometric methods with exact conservation properties. J. Comput. Phys. 257, 1444–1471 (2014). https://doi.org/10.1016/j.jcp.2013.09.027 MathSciNetzbMATHGoogle Scholar
  71. 71.
    Hiptmair, R.: PIER. In: Geometric Methods for Computational Electromagnetics, vol. 42, pp. 271–299. EMW Publishing, Cambridge (2001)Google Scholar
  72. 72.
    Hirani, A.: Discrete exterior calculus. Ph.D. thesis, California Institute of Technology (2003)Google Scholar
  73. 73.
    Hirani, A.N., Nakshatrala, K.B., Chaudhry, J.H.: Numerical method for Darcy flow derived using discrete exterior calculus. Int. J. Comput. Methods Eng. Sci. Mech. 16, 151–169 (2015). https://doi.org/10.1080/15502287.2014.977500 MathSciNetGoogle Scholar
  74. 74.
    Hyman, J.M., Scovel, J.C.: Deriving mimetic difference approximations to differential operators using algebraic topology. Technical report, Los Alamos National Laboratory (1990)Google Scholar
  75. 75.
    Hyman, J.M., Steinberg, S.: The convergence of mimetic methods for rough grids. Comput. Math. Appl. 47, 1565–1610 (2004)MathSciNetzbMATHGoogle Scholar
  76. 76.
    Hyman, J.M., Shashkov, M., Steinberg, S.: The numerical solution of diffusion problems in strongly heterogeneous non-isotropic materials. J. Comput. Phys. 132, 130–148 (1997)MathSciNetzbMATHGoogle Scholar
  77. 77.
    Hyman, J.M., Morel, J., Shashkov, M., Steinberg, S.: Mimetic finite difference methods for diffusion equations. Comput. Geosci. 6, 333–352 (2002)MathSciNetzbMATHGoogle Scholar
  78. 78.
    Kikinzon, E., Kuznetsov, Y., Lipnikov, K., Shashkov, M.: Approximate static condensation algorithm for solving multi-material diffusion problems on meshes non-aligned with material interfaces. J. Comput. Phys. (2017). https://doi.org/10.1016/j.jcp.2017.06.048 MathSciNetzbMATHGoogle Scholar
  79. 79.
    Kouranbaeva, S., Shkoller, S.: A variational approach to second-order multisymplectic field theory. J. Geom. Phys. 35, 333–366 (2000). https://doi.org/10.1016/S0393-0440(00)00012-7 MathSciNetzbMATHGoogle Scholar
  80. 80.
    Kraus, M., Maj, O.: Variational integrators for nonvariational partial differential equations. Physica D 310, 37–71 (2015). https://doi.org/10.1016/j.physd.2015.08.002 MathSciNetzbMATHGoogle Scholar
  81. 81.
    Kreeft, J., Gerritsma, M.: Mixed mimetic spectral element method for Stokes flow: a pointwise divergence-free solution. J. Comput. Phys. 240, 284–309 (2013). https://doi.org/10.1016/j.jcp.2012.10.043 MathSciNetGoogle Scholar
  82. 82.
    Kreeft, J., Palha, A., Gerritsma, M.: Mimetic framework on curvilinear quadrilaterals of arbitrary order, p. 69. arXiv:1111.4304 (2011)Google Scholar
  83. 83.
    Lie, K., Krogstad, S., Ligaarden, I.S., Natvig, J.R., Nilsen, H.M., Skaflestad, B.: Open-source MATLAB implementation of consistent discretisations on complex grids. Comput. Geosci. 16, 297–322 (2012). https://doi.org/10.1007/s10596-011-9244-4 zbMATHGoogle Scholar
  84. 84.
    Manzini, G., Putti, M.: Mesh locking effects in the finite volume solution of 2-D anisotropic diffusion equations. J. Comput. Phys. 220, 751–771 (2007). https://doi.org/10.1016/j.jcp.2006.05.026 MathSciNetzbMATHGoogle Scholar
  85. 85.
    Marsden, J.E., West, M.: Discrete mechanics and variational integrators. Acta Numer. 10, 357–514 (2001). https://doi.org/10.1017/S096249290100006X, published online:2003MathSciNetzbMATHGoogle Scholar
  86. 86.
    Mullen, P., Crane, K., Pavlov, D., Tong, Y., Desbrun, M.: Energy-preserving integrators for fluid animation. ACM Trans. Graph. 28(3), 38 (2009). https://doi.org/10.1145/1531326.1531344 Google Scholar
  87. 87.
    Neuman, S.P.: Theoretical derivation of Darcy’s law. Acta Mech. 25, 153–170 (1977). https://doi.org/10.1007/BF01376989 zbMATHGoogle Scholar
  88. 88.
    Nicolaides, R.: Discrete discretization of planar div-curl problems. SIAM J. Numer. Anal. 29, 32–56 (1992)MathSciNetzbMATHGoogle Scholar
  89. 89.
    Nilsen, H.M., Natvig, J.R., Lie, K.A.: Accurate modeling of faults by multipoint, mimetic, and mixed methods. SPE J., 568–579 (2012). https://doi.org/10.2118/149690-pa Google Scholar
  90. 90.
    Palagi, C.L., Aziz, K.: Use of Voronoi grid in reservoir simulation. SPE Adv. Technol. Ser. 2, 69–77 (1994). https://doi.org/10.2118/22889-PA Google Scholar
  91. 91.
    Palha, A., Gerritsma, M.: Mimetic least-squares spectral/hp finite element method for the Poisson equation. In: Large-Scale Scientific Computing. Lecture Notes in Computer Science, vol. 5910, pp. 662–670. Springer, Berlin (2010)Google Scholar
  92. 92.
    Palha, A., Gerritsma, M.: Spectral element approximation of the Hodge-⋆  operator in curved elements. In: Spectral and High Order Methods for Partial Differential Equations. Lecture Notes in Computational Science and Engineering, vol. 76, pp. 283–291. Springer, Berlin (2010)zbMATHGoogle Scholar
  93. 93.
    Palha, A., Gerritsma, M.: A mass, energy, enstrophy and vorticity conserving (MEEVC) mimetic spectral element discretization for the 2D incompressible Navier-Stokes equations. J. Comput. Phys. 328, 200–220 (2017). https://doi.org/10.1016/j.jcp.2016.10.009 MathSciNetGoogle Scholar
  94. 94.
    Palha, A., Rebelo, P.P., Hiemstra, R., Kreeft, J., Gerritsma, M.: Physics-compatible discretization techniques on single and dual grids, with application to the Poisson equation of volume forms. J. Comput. Phys. 257, 1394–1422 (2014). https://doi.org/10.1016/j.jcp.2013.08.005 MathSciNetzbMATHGoogle Scholar
  95. 95.
    Palha, A., Rebelo, P.P., Gerritsma, M.: Mimetic spectral element advection. In: Spectral and High Order Methods for Partial Differential Equations - ICOSAHOM 2012. Lecture Notes in Computational Science and Engineering, vol. 95, pp. 325–335. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-01601-6 zbMATHGoogle Scholar
  96. 96.
    Palha, A., Koren, B., Felici, F.: A mimetic spectral element solver for the Grad–Shafranov equation. J. Comput. Phys. 316, 63–93 (2016). https://doi.org/10.1016/j.jcp.2016.04.002 MathSciNetzbMATHGoogle Scholar
  97. 97.
    Pavlov, D., Mullen, P., Tong, Y., Kanso, E., Marsden, J.E., Desbrun, M.: Structure preserving discretization of incompressible fluids. Physica D 240, 443–458 (2011)MathSciNetzbMATHGoogle Scholar
  98. 98.
    Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12, 629–639 (1990). https://doi.org/10.1109/34.56205 Google Scholar
  99. 99.
    Perot, J.B.: Conservation properties of unstructured staggered mesh schemes. J. Comput. Phys. 159, 58–89 (2000)MathSciNetzbMATHGoogle Scholar
  100. 100.
    Perot, J.B.: Discrete conservation properties of unstructured mesh schemes. Annu. Rev. Fluid Mech. 43, 299–318 (2011)MathSciNetzbMATHGoogle Scholar
  101. 101.
    Perot, J.B., Subramanian, V.: A discrete calculus analysis of the Keller Box scheme and a generalization of the method to arbitrary meshes. J. Comput. Phys. 226, 494–508 (2007)MathSciNetzbMATHGoogle Scholar
  102. 102.
    Perot, J.B., Subramanian, V.: Discrete calculus methods for diffusion. J. Comput. Phys. 224, 59–81 (2007)MathSciNetzbMATHGoogle Scholar
  103. 103.
    Perot, J.B., Vidovic, D., Wesseling, P.: Mimetic reconstruction of vectors. IMA Vol. Math. Appl. 142, 173 (2006)MathSciNetzbMATHGoogle Scholar
  104. 104.
    Rapetti, F.: High order edge elements on simplicial meshes. ESAIM Math. Model. Numer. Anal. 41, 1001–1020 (2007)MathSciNetzbMATHGoogle Scholar
  105. 105.
    Rapetti, F.: Whitney forms of higher order. SIAM J. Numer. Anal. 47, 2369–2386 (2009)MathSciNetzbMATHGoogle Scholar
  106. 106.
    Rebelo, P.P., Palha, A., Gerritsma, M.: Mixed mimetic spectral element method applied to Darcy’s problem. In: Spectral and High Order Methods for Partial Differential Equations - ICOSAHOM 2012. Lecture Notes in Computational Science and Engineering, vol. 95, pp. 373–382. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-01601-6__30
  107. 107.
    Robidoux, N.: A new method of construction of adjoint gradients and divergences on logically rectangular smooth grids. In: Finite Volumes for Complex Applications: Problems and Perspectives, pp. 261–272. Éditions Hermès, Rouen (1996)Google Scholar
  108. 108.
    Robidoux, N.: Numerical solution of the steady diffusion equation with discontinuous coefficients. Ph.D. thesis, University of New Mexico, Albuquerque (2002)Google Scholar
  109. 109.
    Robidoux, N.: Polynomial histopolation, superconvergent degrees of freedom, and pseudospectral discrete Hodge operators (2008). Unpublished: http://people.math.sfu.ca/~nrobidou/public_html/prints/histogram/histogram.pdf
  110. 110.
    Robidoux, N., Steinberg, S.: A discrete vector calculus in tensor grids. Comput. Methods Appl. Math. 11, 23–66 (2011).  https://doi.org/10.2478/cmam-2011-0002 MathSciNetzbMATHGoogle Scholar
  111. 111.
    Shashkov, M.: Conservative finite-difference methods on general grids. CRC Press, Boca Raton (1996)zbMATHGoogle Scholar
  112. 112.
    Sovinec, C., Glasser, A., Gianakon, T., Barnes, D., Nebel, R., Kruger, S., Schnack, D., Plimpton, S., Tarditi, A., Chu, M., Team, N.: Nonlinear magnetohydrodynamics simulation using high-order finite elements. J. Comput. Phys. 195, 355–386 (2004). https://doi.org/10.1016/j.jcp.2003.10.004 zbMATHGoogle Scholar
  113. 113.
    Steinberg, S.: A discrete calculus with applications of higher-order discretizations to boundary-value problems. Comput. Methods Appl. Math. 42, 228–261 (2004)zbMATHGoogle Scholar
  114. 114.
    Steinberg, S., Zingano, J.P.: Error estimates on arbitrary grids for 2nd-order mimetic discretization of Sturm-Liouville problems. Comput. Methods Appl. Math. 9, 192–202 (2009)MathSciNetzbMATHGoogle Scholar
  115. 115.
    Tarhasaari, T., Kettunen, L., Bossavit, A.: Some realizations of a discrete Hodge operator: a reinterpretation of finite element techniques. IEEE Trans. Magn. 35, 1494–1497 (1999)Google Scholar
  116. 116.
    Taylor, G.I.: Production and dissipation of vorticity in a turbulent fluid. Proc. R. Soc. A Math. Phys. Eng. Sci. 164, 15–23 (1938).  https://doi.org/10.1098/rspa.1938.0002 zbMATHGoogle Scholar
  117. 117.
    Tonti, E.: On the formal structure of physical theories. Technical report, Italian National Research Council (1975)Google Scholar
  118. 118.
    Wang, Y., Hajibeygi, H., Tchelepi, H.A.: Algebraic multiscale solver for flow in heterogeneous porous media. J. Comput. Phys. 259, 284–303 (2014). https://doi.org/10.1016/j.jcp.2013.11.024 MathSciNetzbMATHGoogle Scholar
  119. 119.
    Whitney, H.: Geometric Integration Theory. Dover Publications, Mineola (1957)zbMATHGoogle Scholar
  120. 120.
    Wu, X.H., Parashkevov, R.: Effect of grid deviation on flow solutions. SPE J. 14, 67–77 (2009). https://doi.org/10.2118/92868-PA Google Scholar
  121. 121.
    Younes, A., Ackerer, P., Delay, F.: Mixed finite elements for solving 2-D diffusion-type equations. Rev. Geophys. 48, RG1004 (2010). https://doi.org/10.1029/2008RG000277 Google Scholar
  122. 122.
    Young, L.C.: Rigorous treatment of distorted grids in 3D. In: SPE Reservoir Simulation Symposium. Society of Petroleum Engineers, Richardson (1999). https://doi.org/10.2118/51899-MS
  123. 123.
    Zhang, X., Schmidt, D., Perot, J.B.: Accuracy and conservation properties of a three-dimensional unstructured staggered mesh scheme for fluid dynamics. J. Comput. Phys. 175, 764–791 (2002)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Marc Gerritsma
    • 1
  • Artur Palha
    • 2
  • Varun Jain
    • 1
  • Yi Zhang
    • 1
  1. 1.Faculty of Aerospace EngineeringDelft University of TechnologyDelftThe Netherlands
  2. 2.Department of Mechanical EngineeringEindhoven University of TechnologyEindhovenThe Netherlands

Personalised recommendations