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Compatible Discretizations in Two Dimensions

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Book cover Numerical Mathematics and Advanced Applications

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Abstract

In this paper we recall the construction of the dual finite element complex introduced in [11] and we investigate some applications. More precisely, we propose and analyze fully compatible discretizations for the magnetostatics and the Darcy flow equations in two dimensions, and we introduce an optimal matching condition for domain decomposition methods for Maxwell equations in three dimensions.

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References

  1. Amrouche, C., Bernardi, C., Dauge, M., Girault, V.: Vector potentials in threedimensional non-smooth domains. Math. Meth. Appl. Sci. 21, 823–864 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  2. Douglas N. Arnold: Differential complexes and numerical stability. Proceedings of the International Congress of Mathematicians, Vol. I (2002) (Beijing), Higher Ed. Press, 2002, pp. 137–157.

    Google Scholar 

  3. Arnold, Douglas N., Falk, Richard S., Winther, R.: Differential complexes and stability of finite element methods. i. the de rham complex. Proceedings of the IMA workshop on Compatible Spatial Discretizations for PDE, (2006) (to appear)

    Google Scholar 

  4. Arnold, Douglas N., Falk, Richard S., Winther, R.: Differential complexes and stability of finite element methods. ii. the elasticity complex. Proceedings of the IMA workshop on Compatible Spatial Discretizations for PDE, (2006) (to appear)

    Google Scholar 

  5. Ben Belgacem, F.: The mortar finite element method with lagrange multipliers. Numer. Math. 84, Issue 2, 173–197 (2000)

    Article  MathSciNet  Google Scholar 

  6. Ben Belgacem, F., Buffa, A., Maday, Y.: The mortar element method for Maxwell equations: first results. SIAM J. Num. Anal. 39, no. 3, 880–901 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bernardi, C., Maday, Y., Patera, A. T.: A new nonconforming approach to domain decomposition: The mortar elements method. Nonlinear partial differential equations and their applications (H. Brezis and J.L. Lions, eds.), Pitman, pp. 13–51 (1994)

    Google Scholar 

  8. Bochev, P.B., Hyman, J.M.: Principles of mimetic discretizations of differential operators. Proceedings of the IMA “Hot Topic conference” on Compatible spatial discretizations, (2004)

    Google Scholar 

  9. Bossavit, A.: The mathematics of finite elements and applications, vol. VI (Uxbridge, 1987), ch. Mixed finite elements and the complex of Whitney forms, pp. 1377–144, Academic Press, London (1988)

    Google Scholar 

  10. Brezzi, F., Fortin, M.: Mixed and hybrid finite element methods, vol. 15, Springer-Verlag, Berlin (1991)

    Google Scholar 

  11. Buffa, A., Christiansen, S. H.: A dual finite element complex on the barycentric refinement. Tech. Report PV-18, IMATI-CNR (2005)

    Google Scholar 

  12. Buffa, A., Ciarlet, Jr.,P.: On traces for functional spaces related to Maxwell's equations. Part I: An integration by parts formula in Lipschitz polyhedra. Math. Meth. Appl. Sci. 21, no. 1, 9–30 (2001)

    Article  MathSciNet  Google Scholar 

  13. Buffa, A., Ciarlet, Jr.,P.: On traces for functional spaces related to Maxwell's equations. Part II: Hodge decompositions on the boundary of Lipschitz polyhedra and applications, Math. Meth. Appl. Sci. 21, no. 1, 31–48 (2001)

    Article  MathSciNet  Google Scholar 

  14. Clemens, M., Weiland, T.: Discrete Electromagnetism with the Finite Integration Technique. Progress in Electromagnetic research PIER 32, 65–87 (2001)

    Article  Google Scholar 

  15. Grisvard, P.: Elliptic problems in nonsmooth domains. Monographs and studies in Mathematics, vol. 24, Pitman, London (1985)

    Google Scholar 

  16. Hiptmair, R.: Discrete Hodge operators. Numer. Math. 90, no. 2, 265–289 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  17. Hiptmair, R.: Finite elements in computational electromagnetism. Acta Numerica, 237–339 (2002)

    Google Scholar 

  18. Monk, P.: Finite Element Methods for Maxwell's Equations. Numerical Mathematics and Scientific Computation, Oxford University Press, (2003)

    Google Scholar 

  19. Nédélec, J.C.: Mixed finite element in R3. Numer. Math. 35, 315–341 (1980)

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Istituto di Matematica Applicata e Tecnologie Informatiche del CNR, Via Ferrata 1, 27100, Pavia, ITALY

    Annalisa Buffa

Authors
  1. Annalisa Buffa
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Editor information

Editors and Affiliations

  1. Departamento de Matemática Aplicada Facultade de Matemáticas, Universidade de Santiago de Compostela, 15706, Santiago de Compostela, Spain

    Alfredo Bermúdez de Castro , Dolores Gómez  & Peregrina Quintela ,  & 

  2. Departamento de Matématica Aplicada, Escola Politécnica Superior, 27002, Lugo, Spain

    Pilar Salgado

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Buffa, A. (2006). Compatible Discretizations in Two Dimensions. In: de Castro, A.B., Gómez, D., Quintela, P., Salgado, P. (eds) Numerical Mathematics and Advanced Applications. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-34288-5_1

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