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Algebraic Dual Polynomials for the Equivalence of Curl-Curl Problems

  • Marc GerritsmaEmail author
  • Varun Jain
  • Yi Zhang
  • Artur Palha
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Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 132)

Abstract

In this paper we will consider two curl-curl equations in two dimensions. One curl-curl problem for a scalar quantity F and one problem for a vector field E. For Dirichlet boundary conditions \(\boldsymbol {n} \times \boldsymbol {E} = \hat {E}_{\dashv }\) on E and Neumann boundary conditions \(\boldsymbol {n} \times \mathbf {{curl}}\,F=\hat {E}_{\dashv }\), we expect the solutions to satisfy E = curl F. When we use algebraic dual polynomial representations, these identities continue to hold at the discrete level. Equivalence will be proved and illustrated with a computational example.

Keywords

Spectral element method Algebraic dual polynomials Curl-curl problems 

References

  1. 1.
    Palha, A., Gerritsma, M.I.: 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)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Carstensen, C., Demkowicz, L., Gopalakrishnan, J.: Breaking spaces and form for the DPG method and applications including the Maxwell equations. Comput. Math. Appl. 72, 494–522 (2016)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Jain, V., Zhang, Y., Palha, A., Gerritsma, M.I.: Construction and application of algebraic dual polynomial representations for finite element methods. Comput. Methods Appl. Math. (2018, submitted), arXiv: 1712.09472v2Google Scholar
  4. 4.
    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. Methods Appl. Sci. 24, 9–30 (2001)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chung, E.T., Lee, C.S.: A staggred discontinuous Galerkin method for the curl-curl operator. IMA J. Numer. Anal. 32(3), 1241–1265 (2012)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Degerfeldt, D., Rylander, T.: A brick-tetrahedron finite-element interface with stable hybrid explicit-implicit time-stepping for Maxwell’s equations. J. Comput. Phys. 220, 383–393 (2006)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Gerritsma, M.I.: Edge functions for spectral element methods. In: Hesthaven, J.S., Rønquist, E.M. (eds.) Spectral and High Order Methods for Partial Differential Equations, vol. 76, pp. 199–207. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  8. 8.
    Kreyszig, E.: Introductory functional analysis with applications, 2nd edn. Wiley, New York (1978)zbMATHGoogle Scholar
  9. 9.
    Oden, J.T., Demkowicz, L.: Applied Functional Analysis, 2nd edn. Chapman & Hall/CRC, Boca Raton (2010)Google Scholar
  10. 10.
    Zhang, Y., Jain, V., Palha, A., Gerritsma, M.I.: The discrete Steklov–Poincaré operator using algebraic dual polynomials. Comput. Methods Appl. Math. 19(3), 645–661 (2019)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Marc Gerritsma
    • 1
    Email author
  • Varun Jain
    • 1
  • Yi Zhang
    • 1
  • Artur Palha
    • 2
  1. 1.Faculty of Aerospace EngineeringTU DelftDelftThe Netherlands
  2. 2.Department of Mechanical EngineeringEindhoven University of TechnologyEindhovenThe Netherlands

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