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Function Spaces and Finite Element Approximations

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Mixed Finite Element Methods and Applications

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Abstract

In this chapter we present function spaces and suitable finite element approximations of them, which we shall use in order to apply the abstract theory of the previous chapters to problems of practical interest.

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References

  1. D.A. Adams. Sobolev spaces. Academic Press, New York, 1975.

    MATH  Google Scholar 

  2. B. Ahmad, A. Alsedi, F. Brezzi, L.D. Marini, and A. Russo. Equivalent projectors for virtual element methods. Submitted.

    Google Scholar 

  3. A. Alonso and A. Valli. Some remarks on the characterization of the space of tangential traces of H(rot; Ω) and the construction of an extension operator. Manuscripta Math., 89(2):159–178, 1996.

    Article  MathSciNet  MATH  Google Scholar 

  4. C. Amrouche, C. Bernardi, M. Dauge, and V. Girault. Vector potentials in three-dimensional non-smooth domains. Math. Methods Appl. Sci., 21(9):823–864, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  5. D. N. Arnold. Spaces of finite element differential forms. In U. Gianazza, F. Brezzi, P. Colli Franzone, and G. Gilardi, editors, Analysis and Numerics of Partial Differential Equations. Springer, 2013. 19 pages. To appear. arXiv preprint 1204.1351.

    Google Scholar 

  6. D.N. Arnold and G. Awanou. Finite element differential forms on cubical meshes. submitted to Mathematics of Computation, 2012.

    Google Scholar 

  7. D.N. Arnold, D. Boffi, and F. Bonizzoni. Approximation by tensor product finite element differential forms. Submitted. arXiv preprint 1212.6559, 2012.

    Google Scholar 

  8. D.N. Arnold, D. Boffi, and R.S. Falk. Approximation by quadrilateral finite elements. Math. Comp., 71(239):909–922 (electronic), 2002.

    Google Scholar 

  9. D.N. Arnold, D. Boffi, and R.S. Falk. Quadrilateral H(div) finite elements. SIAM J. Numer. Anal., 42(6):2429–2451 (electronic), 2005.

    Google Scholar 

  10. D.N. Arnold, D. Boffi, R.S. Falk, and L. Gastaldi. Finite element approximation on quadrilateral meshes. Comm. Numer. Methods Engrg., 17(11):805–812, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  11. D.N. Arnold, R.S. Falk, and R. Winther. Differential complexes and stability of finite element methods. I. The de Rham complex. In Compatible spatial discretizations, volume 142 of IMA Vol. Math. Appl., pages 24–46. Springer, New York, 2006.

    Google Scholar 

  12. D.N. Arnold, R.S. Falk, and R. Winther. Differential complexes and stability of finite element methods. II. The elasticity complex. In Compatible spatial discretizations, volume 142 of IMA Vol. Math. Appl., pages 47–67. Springer, New York, 2006.

    Google Scholar 

  13. D.N. Arnold, R.S. Falk, and R. Winther. Finite element exterior calculus, homological techniques, and applications. Acta Numerica, 15:1–155, 2006.

    Article  MathSciNet  MATH  Google Scholar 

  14. D.N. Arnold, R.S. Falk, and R. Winther. Finite element exterior calculus: from Hodge theory to numerical stability. Bull. Amer. Math. Soc., 47:281–354, 2010.

    Article  MathSciNet  MATH  Google Scholar 

  15. I. Babuška and A.K. Aziz. Survey lectures on the mathematical foundations of the finite element method. In A.K. Aziz, editor, The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations. Academic Press, New-York, 1972.

    Google Scholar 

  16. L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L.D. Marini, and A. Russo. Basic principles of virtual element methods. Math. Models Methods Appl. Sci., 23 (2013), no. 1, 199–214.

    Article  MathSciNet  MATH  Google Scholar 

  17. A. Bermúdez, P. Gamallo, M. R. Nogueiras, and R. Rodríguez. Approximation of a structural acoustic vibration problem by hexahedral finite elements. IMA J. Numer. Anal., 26(2):391–421, 2006.

    Article  MathSciNet  MATH  Google Scholar 

  18. D. Boffi. A note on the de Rham complex and a discrete compactness property. Appl. Math. Lett., 14(1):33–38, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  19. D. Boffi, M. Conforti, and L. Gastaldi. Modified edge finite elements for photonic crystals. Numer. Math., 105 (2006), pp. 249–266.

    Article  MathSciNet  MATH  Google Scholar 

  20. D. Boffi, M. Costabel, M. Dauge, and L. Demkowicz. Discrete compactness for the h p version of rectangular edge finite elements. SIAM J. Numer. Anal., 44(3):979–1004 (electronic), 2006.

    Google Scholar 

  21. D. Boffi and L. Gastaldi. On the quadrilateral Q 2-P 1 element for the Stokes problem. Internat. J. Numer. Methods Fluids, 39(11):1001–1011, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  22. D. Boffi, F. Kikuchi, and J. Schöberl. Edge element computation of Maxwell’s eigenvalues on general quadrilateral meshes. Math. Models Methods Appl. Sci., 16(2):265–273, 2006.

    Article  MathSciNet  MATH  Google Scholar 

  23. A. Bossavit. Whitney forms: a class of finite elements for three-dimensional computations in electromagnetism. IEEE Proc. A, 135:493–500, 1988.

    Google Scholar 

  24. F. Brezzi, J. Douglas, R. Duran, and M. Fortin. Mixed finite elements for second order elliptic problems in three variables. Numer. Math., 51:237–250, 1987.

    Article  MathSciNet  MATH  Google Scholar 

  25. F. Brezzi, J. Douglas, M. Fortin, and L.D. Marini. Efficient rectangular mixed finite elements in two and three space variables. Math. Model. Numer. Anal., 21:581–604, 1987.

    MathSciNet  MATH  Google Scholar 

  26. F. Brezzi, J. Douglas, and L.D. Marini. Recent results on mixed finite element methods for second order elliptic problems. In Balakrishanan, Dorodnitsyn, and Lions, editors, Vistas in Applied Math., Numerical Analysis, Atmospheric Sciences, Immunology. Optimization Software Publications, New York, 1986.

    Google Scholar 

  27. F. Brezzi and L.D. Marini. On the numerical solution of plate bending problems by hybrid methods. R.A.I.R.O. Anal. Numer., pages 5–50, 1975.

    Google Scholar 

  28. A. Buffa. Hodge decomposition on the boundary of a polyhedron: the multi-connected case. Math. Mod. Meth. Appl. Sci., 11(9):1491–1504, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  29. A. Buffa and P. Ciarlet Jr. On traces for functional spaces related to Maxwell’s equations. I. An integration by parts formula in Lipschitz polyhedra. Math. Methods Appl. Sci., 24(1):9–30, 2001.

    Google Scholar 

  30. A. Buffa and P. Ciarlet Jr. On traces for functional spaces related to Maxwell’s equations. II. Hodge decompositions on the boundary of Lipschitz polyhedra and applications. Math. Methods Appl. Sci., 24(1):31–48, 2001.

    Google Scholar 

  31. A. Buffa, M. Costabel, and D. Sheen. On traces for H(c u r l, Ω) in Lipschitz domains. J. Math. Anal. Appl., 276(2):845–867, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  32. J. Céa. Approximation variationnelle et convergence des éléments finis; un test. Journées Eléments Finis, Université de Rennes, 1976.

    Google Scholar 

  33. M. Cessenat. Mathematical methods in electromagnetism, volume 41 of Series on Advances in Mathematics for Applied Sciences. World Scientific Publishing Co. Inc., River Edge, NJ, 1996.

    Google Scholar 

  34. P.G. Ciarlet. The finite element method for elliptic problems. North-Holland, Amsterdam, 1978.

    MATH  Google Scholar 

  35. P.G. Ciarlet. Mathematical elasticity. Vol. I. North-Holland Publishing Co., Amsterdam, 1988. Three-dimensional elasticity.

    Google Scholar 

  36. P.G. Ciarlet. Mathematical elasticity. Vol. II. North-Holland Publishing Co., Amsterdam, 1997. Theory of plates.

    Google Scholar 

  37. P.G. Ciarlet and P.A. Raviart. Interpolation theory over curved elements with applications to finite element methods. Comp. Meth. Appl. Mech. Eng., 1:217–249, 1972.

    Article  MathSciNet  MATH  Google Scholar 

  38. P.G. Ciarlet and P.A. Raviart. A mixed finite element method for the biharmonic equation. In C. de Boor, editor, Mathematical Aspects of Finite Element in Partial Differential Equations. Academic Press, New York, 1974.

    Google Scholar 

  39. P. Clément. Approximation by finite element functions using local regularization. R.A.I.R.O. Anal. Mumer., 9:77–84, 1975.

    Google Scholar 

  40. M. Costabel and M. Dauge. Computation of resonance frequencies for Maxwell equations in non-smooth domains. In Topics in computational wave propagation, volume 31 of Lect. Notes Comput. Sci. Eng., pages 125–161. Springer, Berlin, 2003.

    Google Scholar 

  41. M. Crouzeix and P.A. Raviart. Conforming and non-conforming finite element methods for solving the stationary Stokes equations. R.A.I.R.O. Anal. Numer., 7:33–76, 1973.

    Google Scholar 

  42. M.C. Delfour and Z.-P. Zolésio. Shapes and Geometries: Analysis, Differential Calculus, and Optimization. Advances in Design and Control. SIAM, Philadelphia, 2001.

    Google Scholar 

  43. L. Demkowicz, P. Monk, L. Vardapetyan, and W. Rachowicz. De Rham diagram for h p finite element spaces. Comput. Math. Appl., 39(7–8):29–38, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  44. J. Douglas and J.E. Roberts. Mixed finite element methods for second order elliptic problems. Math. Applic. Comp., 1:91–103, 1982.

    MathSciNet  MATH  Google Scholar 

  45. J. Douglas and J.E. Roberts. Global estimates for mixed methods for second order elliptic equations. Math. Comp., 44:39–52, 1985.

    Article  MathSciNet  MATH  Google Scholar 

  46. T. Dupont and L.R. Scott. Polynomial approximation of functions in Sobolev spaces. Math. of Comp., 34:441–463, 1980.

    Article  MathSciNet  MATH  Google Scholar 

  47. R.S. Falk, P. Gatto, and P. Monk. Hexahedral H(div) and H(curl) finite elements. ESAIM: Mathematical Modelling and Numerical Analysis, 45:115–143, 2011.

    Article  MathSciNet  Google Scholar 

  48. M. Fortin. Utilisation de la méthode des éléments finis en mécanique des fluides. Calcolo, 12:405–441, 1975.

    Article  MathSciNet  Google Scholar 

  49. M. Fortin and M. Soulie. A non-conforming piecewise quadratic finite element on triangles. Int. J. Num. Meth. Eng., 19:505–520, 1983.

    Article  MathSciNet  MATH  Google Scholar 

  50. B. Fraeijs de Veubeke. Variational principles and the patch test. Int. J. Numer. Meth. Eng., 8:783–801, 1974.

    Article  MathSciNet  MATH  Google Scholar 

  51. V. Girault and P.A. Raviart. Finite Element Approximation of Navier-Stokes Equations, volume 749 of Lectures Notes in Math. Springer-Verlag, Berlin, 1979.

    Google Scholar 

  52. V. Girault and P.A. Raviart. Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms. Springer-Verlag, Berlin, 1986.

    Book  MATH  Google Scholar 

  53. F.H. Harlow and R.E. Welsch. Numerical calculation of time dependent viscous incompressible flow. Phys. Fluids, 8:–2182, 1965.

    Google Scholar 

  54. J.P. Hennart, J. Jaffré, and J.E. Roberts. A constructive method for deriving finite elements of nodal type. Numer. Math., 55:701–738, 1988.

    Article  Google Scholar 

  55. R. Hiptmair. Canonical construction of finite elements. Math. Comp., 68(228):1325–1346, 1999.

    Article  MathSciNet  MATH  Google Scholar 

  56. R. Hiptmair. Finite elements in computational electromagnetism. Acta Numer., 11:237–339, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  57. B.M. Irons and A. Razzaque. Experience with the patch-test for convergence of finite elements. In A.K. Aziz, editor, Mathematics of Finite Element Method with Applications to Partial Differential Equations. Univ. of Maryland, Baltimore, 1972.

    Google Scholar 

  58. P. Jamet. Estimation d’erreur pour des éléments finis droits presque dégénérés. R.A.I.R.O. Anal. Numer., 10, 3:43–62, 1976.

    Google Scholar 

  59. F. Kikuchi, M. Okabe, and H. Fujio. Modification of the 8-node serendipity element. Comp. Methods Appl. Mech. Engrg., 179:91–109, 1999.

    Article  MATH  Google Scholar 

  60. P. Lascaux and P. Lesaint. Some non-conforming finite elements for the plate bending problem. R.A.I.R.O. Anal. Numer., 9:9–53, 1975.

    Google Scholar 

  61. P. Lesaint. Nodal methods for the transport equation. In The Mathematics of Finite Elements and Applications V , pages 562–569, Oxbridge, England, 1985. MAFELAP 984, Proc 5th Conf.

    Google Scholar 

  62. J.L. Lions and E. Magenes. Problèmes aux limites non-homogènes et applications. Dunod, Paris, 1968.

    MATH  Google Scholar 

  63. J.E. Marsden and T.J.R. Hughes. Mathematical Foundations of Elasticity. Prentice Hall, New York, 1983.

    MATH  Google Scholar 

  64. G. Matthies. Mapped finite elements on hexahedra. Necessary and sufficient conditions for optimal interpolation errors. Numer. Algorithms, 27(4):317–327, 2001.

    Google Scholar 

  65. R. H. McNeal and R. L. Harder. Eight nodes or nine? Int. J. Numer. Methods Engrg., 33:1049–1058, 1992.

    Article  Google Scholar 

  66. P. Monk. Finite element methods for Maxwell’s equations. Numerical Mathematics and Scientific Computation. Oxford University Press, New York, 2003.

    Book  MATH  Google Scholar 

  67. S.E. Mousavi and Sukumar N. Numerical integration of polynomials and discontinuous functions on irregularconvex polygons and polyhedrons. Comput.Mech., 47:535–554, 2011.

    Google Scholar 

  68. Russel T.F. Naff R.L. and Wilson J. D. Shape functions for velocity interpolation in general hexahedral cells. Comput. Geosci., 6:285–314, 2002.

    Article  MathSciNet  Google Scholar 

  69. J. Nečas. Les méthodes directes en théorie des équations elliptiques. Masson, Paris, 1967.

    MATH  Google Scholar 

  70. J.-C. Nédélec. Mixed finite elements in \({\mathbb{R}}^{3}\). Numer. Math., 35(3):315–341, 1980.

    Article  MathSciNet  MATH  Google Scholar 

  71. J.-C. Nédélec. A new family of mixed finite elements in \({\mathbb{R}}^{3}\). Numer. Math., 50(1):57–81, 1986.

    Article  MathSciNet  MATH  Google Scholar 

  72. L. Paquet. Problèmes mixtes pour le système de Maxwell. Ann. Fac. Sci. Toulouse Math. (5), 4(2):103–141, 1982.

    Google Scholar 

  73. A. Pechstein and J. Schöberl. Tangential-displacement and normal-normal-stress continuous mixed finite elements for elasticity. Mathematical Models and Methods in Applied Sciences, 21(8):1761–1782, 2011.

    Article  MathSciNet  MATH  Google Scholar 

  74. A. Pechstein and J. Schöberl. Anisotropic mixed finite elements for elasticity. Int. J. Numer. Meth. Engng, 90:196–217, 2012.

    Article  MATH  Google Scholar 

  75. R. Rannacher and S. Turek. Simple nonconforming quadrilateral Stokes element. Numer. Methods Partial Differential Equations, 8(2):97–111, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  76. P.A. Raviart and J.M. Thomas. A mixed finite element method for second order elliptic problems. In I. Galligani and E. Magenes, editors, Mathematical Aspects of the Finite Element Method, volume 606 of Lectures Notes in Math. Springer-Verlag, New York, 1977.

    Google Scholar 

  77. P.A. Raviart and J.M. Thomas. Introduction à l’analyse numérique des équations aux dérivées partielles. Masson, Paris, 1983.

    MATH  Google Scholar 

  78. G. Strang and G.J. Fix. An Analysis of the Finite Element Method. Prentice Hall, New York, 1973. now published by Wellesley-Cambridge Press.

    Google Scholar 

  79. J.M Thomas. Sur l’analyse numérique des méthodes d’éléments finis hybrides et mixtes. Master’s thesis, Université Pierre et Marie Curie, Paris, 1977.

    Google Scholar 

  80. T.P. Thomas-Peter Fries and Ted Belytschko T. The extended/generalized finite element method: An overview of the method and its applications. Int. J. Numer. Meth. Engng., 84(3):253–304, 2010.

    Google Scholar 

  81. E. L. Wachspress. A Rational Finite Element Basis. Academic Press, New York, 1975.

    MATH  Google Scholar 

  82. H. Whitney. Geometric integration theory. Princeton University Press, Princeton, N. J., 1957.

    MATH  Google Scholar 

  83. J. Zhang and F. Kikuchi. Interpolation error estimates of a modified 8-node serendipity finite element. Numer. Math., 85(3):503–524, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  84. O. C. Zienkiewicz and R. L. Taylor. The Finite Element Method, Fourth edition, Volume 1: Basic Formulation and Linear Problems. McGraw-Hill, London, 1989.

    Google Scholar 

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Boffi, D., Brezzi, F., Fortin, M. (2013). Function Spaces and Finite Element Approximations. In: Mixed Finite Element Methods and Applications. Springer Series in Computational Mathematics, vol 44. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36519-5_2

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