Numerische Mathematik

, Volume 109, Issue 4, pp 509–533

A nonconforming finite element method for a two-dimensional curl–curl and grad-div problem

Article

Abstract

A numerical method for a two-dimensional curl–curl and grad-div problem is studied in this paper. It is based on a discretization using weakly continuous P1 vector fields and includes two consistency terms involving the jumps of the vector fields across element boundaries. Optimal convergence rates (up to an arbitrary positive \({\epsilon}\)) in both the energy norm and the L2 norm are established on graded meshes. The theoretical results are confirmed by numerical experiments.

Keywords

Curl–curl and grad-div problem Nonconforming finite element methods Maxwell equations 

Mathematics Subject Classification (2000)

65N30 65N15 35Q60 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Apel, Th.: Anisotropic Finite Elements. Teubner, Stuttgart (1999)Google Scholar
  2. 2.
    Apel, Th., Sändig, A.-M., Whiteman, J.R.: Graded mesh refinement and error estimates for finite element solutions of elliptic boundary value problems in non-smooth domains. Math. Methods Appl. Sci. 19, 63–85 (1996)CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    Assous, F., Ciarlet, P. Jr., Labrunie, S., Segré, J.: Numerical solution to the time-dependent Maxwell equations in axisymmetric singular domains: the singular complement method. J. Comput. Phys. 191, 147–176 (2003)CrossRefMathSciNetMATHGoogle Scholar
  4. 4.
    Assous, F., Ciarlet, P. Jr., Garcia, E., Segré, J.: Time-dependent Maxwell’s equations with charges in singular geometries. Comput. Methods Appl. Mech. Eng. 196, 665–681 (2006)CrossRefMATHGoogle Scholar
  5. 5.
    Assous, F., Ciarlet, P. Jr., Labrunie, S., Lohrengel, S.: The singular complement method. In: Debit, N., Garbey, M., Hoppe, R., Keyes, D., Kuznetsov, Y., Périaux, J.(eds) Domain Decomposition Methods in Science and Engineering, pp. 161–189. CIMNE, Barcelona (2002)Google Scholar
  6. 6.
    Assous, F., Ciarlet, P. Jr., Sonnendrücker, E.: Resolution of the Maxwell equation in a domain with reentrant corners. M 2(N Math. Model. Numer. Anal. 32), 359–389 (1998)Google Scholar
  7. 7.
    Babuška, I., Osborn, J.: Eigenvalue problems. In: Ciarlet, P.G., Lions, J.L.(eds) Handbook of Numerical Analysis II, pp. 641–787. North-Holland, Amsterdam (1991)Google Scholar
  8. 8.
    Băcuţă, C., Nistor, V., Zikatanov, L.T.: Improving the rate of convergence of ‘high order finite elements’ on polygons and domains with cusps. Numer. Math. 100, 165–184 (2005)CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    Bathe, K.J., Nitikitpaiboon, C., Wang, X.: A mixed displacement-based finite element formulation for acoustic fluid-structure interaction. Comput. Struct. 56, 225–237 (1995)CrossRefMathSciNetMATHGoogle Scholar
  10. 10.
    Bermúdez, A., Rodríguez, R.: Finite element computation of the vibration modes of a fluid-solid system. Comput. Methods Appl. Mech. Eng. 119, 355–370 (1994)CrossRefMATHGoogle Scholar
  11. 11.
    Birman, M., Solomyak, M.: L 2-theory of the Maxwell operator in arbitrary domains. Russ. Math. Surv. 42, 75–96 (1987)CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    Boffi, D., Gastaldi, L.: On the “-grad div s curl rot” operator. In: Computational fluid and solid mechanics, vol. 1, 2 (Cambridge, MA, 2001), pp. 1526–1529. Elsevier, Amsterdam (2001)Google Scholar
  13. 13.
    Bonnet-Ben Dhia, A.-S., Hazard, C., Lohrengel, S.: A singular field method for the solution of Maxwell’s equations in polyhedral domains. SIAM J. Appl. Math. 59, 2028–2044 (1999)CrossRefMathSciNetMATHGoogle Scholar
  14. 14.
    Bossavit, A.: Discretization of electromagnetic problems: the “generalized finite differences” approach. In: Ciarlet, P.G., Schilders, W.H.A., Ter Maten, E.J.W.(eds) Handbook of numerical analysis, vol XIII, Handb. Numer. Anal., XIII, pp. 105–197. North-Holland, Amsterdam (2005)Google Scholar
  15. 15.
    Brenner, S.C., Carstensen, C.: Finite element methods. In: Stein, E., de Borst, R., Hughes, T.J.R.(eds) Encyclopedia of Computational Mechanics, pp. 73–118. Wiley, Weinheim (2004)Google Scholar
  16. 16.
    Brenner, S.C., Li, F., Sung, L.-Y.: A locally divergence-free interior penalty method for two dimensional curl–curl problems. SIAM J. Numer. Anal. 46, 1190–1211 (2008)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Brenner, S.C., Li, F., Sung, L.-Y.: A locally divergence-free nonconforming finite element method for the time-harmonic Maxwell equations. Math. Comp. 76, 573–595 (2007)CrossRefMathSciNetMATHGoogle Scholar
  18. 18.
    Brenner, S.C., Li, F., Sung, L.-Y.: A nonconforming penalty method for a two dimensional curl–curl problem. preprint, (2007)Google Scholar
  19. 19.
    Brenner, S.C., Li, F., Sung, L.-Y.: Parameter free nonconforming Maxwell eigensolvers without spurious eigenmodes. preprint, (2008)Google Scholar
  20. 20.
    Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, 2nd edn. Springer, Heidelberg (2002)MATHGoogle Scholar
  21. 21.
    Ciarlet, P. Jr.: Augmented formulations for solving Maxwell equations. Comput. Methods Appl. Mech. Eng. 194, 559–586 (2005)CrossRefMathSciNetMATHGoogle Scholar
  22. 22.
    Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)CrossRefMATHGoogle Scholar
  23. 23.
    Costabel, M.: A remark on the regularity of solutions of Maxwell’s equations on Lipschitz domains. Math. Methods Appl. Sci. 12, 36–368 (1990)CrossRefMathSciNetGoogle Scholar
  24. 24.
    Costabel, M.: A coercive bilinear form for Maxwell’s equations. J. Math. Anal. Appl. 157, 527–541 (1991)CrossRefMathSciNetMATHGoogle Scholar
  25. 25.
    Costabel, M., Dauge, M.: Maxwell and Lamé eigenvalues on polyhedra. Math. Methods Appl. Sci. 22, 243–258 (1999)CrossRefMathSciNetMATHGoogle Scholar
  26. 26.
    Costabel, M., Dauge, M.: Singularities of electromagnetic fields in polyhedral domains. Arch. Ration. Mech. Anal. 151, 221–276 (2000)CrossRefMathSciNetMATHGoogle Scholar
  27. 27.
    Costabel, M., Dauge, M.: Weighted regularization of Maxwell equations in polyhedral domains. Numer. Math. 93, 239–277 (2002)CrossRefMathSciNetMATHGoogle Scholar
  28. 28.
    Costabel, M., Dauge, M., Schwab, C.: Exponential convergence of hp-FEM for Maxwell equations with weighted regularization in polygonal domains. Math. Models Methods Appl. Sci. 15, 575–622 (2005)CrossRefMathSciNetMATHGoogle Scholar
  29. 29.
    Coulomb, J.L.: Finite element three dimensional magnetic field computation. IEEE Trans. Magnetics 17, 3241–3246 (1981)CrossRefGoogle Scholar
  30. 30.
    Crouzeix, M., Raviart, P.-A.: Conforming and nonconforming finite element methods for solving the stationary Stokes equations I. RAIRO Anal. Numér. 7, 33–75 (1973)MathSciNetGoogle Scholar
  31. 31.
    Cui, J.: Nonconforming Multigrid Methods for Maxwell’s Equations. PhD thesis, Louisiana State University (in preparation)Google Scholar
  32. 32.
    Dauge, M.: Elliptic Boundary Value Problems on Corner Domains. Lecture Notes in Mathematics, vol. 1341. Springer, Heidelberg (1988)Google Scholar
  33. 33.
    Demkowicz, L.: Finite Element Methods for Maxwell Equations. In: Stein, E., de Borst, R., Hughes, T.J.R.(eds) Encyclopedia of Computational Mechanics, pp. 723–737. Wiley, Weinheim (2004)Google Scholar
  34. 34.
    Girault, V., Raviart, P.-A.: Finite Element Methods for Navier–Stokes Equations. Theory and Algorithms. Springer, Berlin (1986)MATHGoogle Scholar
  35. 35.
    Grisvard, P.: Elliptic Problems in Non Smooth Domains. Pitman, Boston (1985)Google Scholar
  36. 36.
    Grisvard, P.: Singularities in Boundary Value Problems. Masson, Paris (1992)MATHGoogle Scholar
  37. 37.
    Hamdi, M.A., Ousset, Y., Verchery, G.: A displacement method for the analysis of vibrations of coupled fluid–structure systems. Int. J. Numer. Methods Eng. 13, 139–150 (1978)CrossRefMATHGoogle Scholar
  38. 38.
    Hazard, C., Lohrengel, S.: A singular field method for Maxwell’s equations: numerical aspects for 2D magnetostatics. SIAM J. Numer. Anal. 40, 1021–1040 (2002)CrossRefMathSciNetMATHGoogle Scholar
  39. 39.
    Hiptmair, R.: Finite elements in computational electromagnetism. Acta Numer. 11, 237–339 (2002)CrossRefMathSciNetMATHGoogle Scholar
  40. 40.
    Kato, T.: Perturbation Theory of Linear Operators. Springer, Berlin (1966)Google Scholar
  41. 41.
    Leis, R.: Zur Theorie elektromagnetischer Schwingungen in anisotopen inhomgenen Medien. Math. Z. 106, 213–224 (1968)CrossRefMathSciNetGoogle Scholar
  42. 42.
    Monk, P.: Finite Element Methods for Maxwell’s Equations. Oxford University Press, New York (2003)MATHGoogle Scholar
  43. 43.
    Nazarov, S.A., Plamenevsky, B.A.: Elliptic Problems in Domains with Piecewise Smooth Boundaries. de Gruyter, Berlin (1994)MATHGoogle Scholar
  44. 44.
    Nédélec, J.-C.: Mixed finite elements in R3. Numer. Math. 35, 315–341 (1980)CrossRefMathSciNetMATHGoogle Scholar
  45. 45.
    Nédélec, J.-C.: A new family of mixed finite elements in R3. Numer. Math. 50, 57–81 (1986)CrossRefMathSciNetMATHGoogle Scholar
  46. 46.
    Neittaanmäki, P., Picard, R.: Error estimates for the finite element approximation to a Maxwell-type boundary value problem. Numer. Funct. Anal. Optimiz. 2, 267–285 (1980)CrossRefMATHGoogle Scholar
  47. 47.
    Rahman, B.M.A., Davies, J.B.: Finite element analysis of optical and microwave waveguide problems. IEEE Trans. Microwave Theory Tech. 32, 20–28 (1984)CrossRefGoogle Scholar
  48. 48.
    Schatz, A.: An observation concerning Ritz-Galerkin methods with indefinite bilinear forms. Math. Comp. 28, 959–962 (1974)CrossRefMathSciNetMATHGoogle Scholar
  49. 49.
    Weber, C.: A local compactness theorem for Maxwell’s equations. Math. Meth. Appl. Sci. 2, 12–25 (1980)CrossRefMATHGoogle Scholar
  50. 50.
    Weiland, T.: On the unique numerical solution of Maxwellian eigenvalue problems in three dimensions. Part. Accel. 17, 227–242 (1985)Google Scholar
  51. 51.
    Witsch, K.J.: A remark on a compactness result in electromagnetic theory. Math. Methods. Appl. Sci 16, 123–129 (1993)CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of Mathematics and Center for Computation and TechnologyLouisiana State UniversityBaton RougeUSA
  2. 2.Department of MathematicsLouisiana State UniversityBaton RougeUSA
  3. 3.Department of Mathematical SciencesRensselaer Polytechnic InstituteTroyUSA

Personalised recommendations