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Algebraic convergence for anisotropic edge elements in polyhedral domains

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Abstract

We study approximation errors for the h-version of Nédélec edge elements on anisotropically refined meshes in polyhedra. Both tetrahedral and hexahedral elements are considered, and the emphasis is on obtaining optimal convergence rates in the H(curl) norm for higher order elements. Two types of estimates are presented: First, interpolation error estimates for functions in anisotropic weighted Sobolev spaces. Here we consider not only the H(curl)-conforming Nédélec elements, but also the H(div)-conforming Raviart-Thomas elements which appear naturally in the discrete version of the de Rham complex. Our technique is to transport error estimates from the reference element to the physical element via highly anisotropic coordinate transformations. Second, Galerkin error estimates for the standard H(curl) approximation of time harmonic Maxwell equations. Here we use the anisotropic weighted Sobolev regularity of the solution on domains with three-dimensional edges and corners. We also prove the discrete compactness property needed for the convergence of the Maxwell eigenvalue problem. Our results generalize those of [40] to the case of polyhedral corners and higher order elements.

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References

  1. Acosta, G., Durán, R.G.: The maximum angle condition for mixed and nonconforming elements: application to the Stokes equations. SIAM J. Numer. Anal. 37, 18–36 (1999) (electronic)

    Google Scholar 

  2. Al-Shenk, N.: Uniform error estimates for certain narrow Lagrange finite elements. Math. Comp. 63, 105–119 (1994)

    Google Scholar 

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

    Google Scholar 

  4. Apel, T.: Anisotropic finite elements: Local estimates and applications. Advances in Numerical Mathematics, Teubner, Stuttgart, 1999

  5. Apel, T.: Personal communication. July 2003

  6. Apel, T., Dobrowolski, M.: Anisotropic interpolation with applications to the finite element method. Computing 47, 277–293 (1992)

    Google Scholar 

  7. Apel, T., Nicaise, S.: The finite element method with anisotropic mesh grading for elliptic problems in domains with corners and edges. Math. Meth. Appl. Sci. 21, 519–549 (1998)

    Google Scholar 

  8. Babuška, I., Kellogg, R.B., Pitkäranta, J.: Direct and inverse error estimates for finite elements with mesh refinements. Numer. Math. 33, 447–471 (1979)

    Google Scholar 

  9. Birman, M., Solomyak, M.: L2-theory of the Maxwell operator in arbitrary domains. Russ. Math. Surv. 42(6), 75–96 (1987)

    Google Scholar 

  10. Boffi, D.: Fortin operator and discrete compactness for edge elements. Numer. Math. 87, 229–246 (2000)

    Google Scholar 

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

    Google Scholar 

  12. Brezzi, F., Fortin, M.: Mixed and hybrid finite element methods. Vol. 15, Springer-Verlag, Berlin, 1991

  13. Buffa, A.: Remarks on the discretization of some non-positive operator with application to heterogeneous Maxwell problems. SIAM J. Num. Anal. 43(1), 1–18 (2005)

    Google Scholar 

  14. Buffa, A., Costabel, M., Dauge, M.: Anisotropic regularity results for Laplace and Maxwell operators in a polyhedron. C. R. Math. Acad. Sci. Paris 336(7), 565–570 (2003)

    Google Scholar 

  15. Buffa, A., Hiptmair, R., von Petersdorff, T., Schwab, C.: Boundary element methods for Maxwell transmission problems in Lipschitz domains. Numer. Mathem. 95(3), 459–485 (2003)

    Google Scholar 

  16. Caorsi, S., Fernandes, P., Raffetto, M.: On the convergence of Galerkin finite element approximations of electromagnetic eigenproblems. SIAM J. Numer. Anal. 38, 580–607 (2000) (electronic)

    Google Scholar 

  17. Caorsi, S., Fernandes, P., Raffetto, M.: Spurious-free approximations of electromagnetic eigenproblems by means of Nedelec-type elements. M2AN Math. Model. Numer. Anal. 35, 331–354 (2001)

    Google Scholar 

  18. Cessenat, M.: Mathematical methods in Electromagnetism. Linear Theory and Applications, vol. 41 of Series of advances in mathematics for applied sciences, Word Scientific publishing, 1996

  19. Ciarlet, P.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam, 1978

  20. Ciarlet, P., Jr, Zou, J.: Fully discrete finite element approaches for time-dependant Maxwell’s equations. Numer. Math. 82, 193–219 (1999)

    Google Scholar 

  21. Costabel, M., Dauge, M.: Singularities of electromagnetic fields in polyhedral domains. Arch. Rational Mech. Anal. 151, 221–276 (2000)

    Google Scholar 

  22. Costabel, M., Dauge, M.: Weighted regularization of Maxwell equations in polyhdreal domains. Numer. Mathem. 93, 239–277 (2002)

    Google Scholar 

  23. Costabel, M., Dauge, M.: Computation of resonance frequencies for Maxwell equations in non smooth domains. In: M. Ainsworth, P. Davies, D. Duncan, P. Martin, B. Rynne (eds.), Topics in Computational Wave Propagation, Lecture Notes in Computational Science and Engineering., Vol. 31, Springer, 2003, pp. 125–161

  24. Costabel, M., Dauge, M., Nicaise, S.: Singularities of Maxwell interface problems. M2AN Math. Model. Numer. Anal. 33, 627–649 (1999)

    Google Scholar 

  25. Costabel, M., Dauge, M., Schwab, C.: Exponential convergence of the hp-FEM for the weighted regularization of Maxwell equations in polygonal domains. To appear in Math. Models Methods Appl. Sci. 15(4), 2005

    Google Scholar 

  26. Dauge, M.: Elliptic boundary value problems on corner domains. Lecture Notes in Mathematics, Springer Verlag, Berlin, 1988

  27. Descloux, J., Nassif, N., Rappaz, J.: On spectral approximation. I. The problem of convergence. RAIRO Anal. Numér. 12, 97–112, iii (1978)

    Google Scholar 

  28. Descloux, J., Nassif, N., Rappaz, J.: On spectral approximation. II. Error estimates for the Galerkin method. RAIRO Anal. Numér. 12, 113–119, iii (1978)

    Google Scholar 

  29. Durán, R.G.: Error estimates for 3-d narrow finite elements. Math. Comp. 68, 187–199 (1999)

    Google Scholar 

  30. Farhloul, M., Nicaise, S., Paquet, L.: Some mixed finite element methods on anisotropic meshes. M2AN Math. Model. Numer. Anal. 35, 907–920 (2001)

    Google Scholar 

  31. Girault, V., Raviart, P.-A.: Finite element methods for Navier-Stokes equations. Sringer-Verlag, Berlin, 1986

  32. Guo, B.Q.: The h-p version of the finite element method for solving boundary value problems in polyhedral domains. In: M. Costabel, M. Dauge, S. Nicaise (eds.), Boundary value problems and integral equations in nonsmooth domains (Luminy, 1993), Dekker, New York, 1995, pp. 101–120

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

    Google Scholar 

  34. Hiptmair, R.: Finite elements in computational electromagnetism. Acta Numerica, 2002, pp. 237–339

  35. Kikuchi, F.: On a discrete compactness property for the Nédélec finite elements. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36, 479–490 (1989)

    Google Scholar 

  36. Krízek, M.: On the maximum angle condition for linear tetrahedral elements. SIAM J. Numer. Anal. 29, 513–520 (1992)

    Google Scholar 

  37. Leis, R.: Zur Theorie elektromagnetischer Schwingungen in anisotropen inhomogenen Medien. Math. Z. 106, 213–224 (1968)

    Google Scholar 

  38. Monk, P.: Analysis of a finite element method for Maxwell’s equations. SIAM J. Numer. Anal. 29, 714–729 (1992)

    Google Scholar 

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

    Google Scholar 

  40. Nicaise, S.: Edge elements on anisotropic meshes and approximation of the Maxwell equations. SIAM J. Numer. Anal. 39, 784–816 (2001) (electronic)

    Google Scholar 

  41. Raviart, P.A., Thomas, J.M.: Primal hybrid finite element methods for second order elliptic problems. Math. Comput. 31, 391–413 (1977)

    Google Scholar 

  42. Raviart, P.A., Thomas, J.M.: A Mixed Finite Element Method for Second Order Elliptic Problems, vol. 606 of Springer Lecture Notes in Mathematics, Springer, New York, 1977, pp. 292–315

  43. Schwab, C.: p- and hp-finite element methods. Theory and applications in solid and fluid mechanics. The Clarendon Press Oxford University Press, New York, 1998

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Authors and Affiliations

  1. IMATI-CNR, Via Ferrata 1, 27100, Pavia, Italy

    A. Buffa

  2. IRMAR - Université de Rennes 1, Campus Beaulieu, 35042, Rennes Cedex, France

    M. Costabel & M. Dauge

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  1. A. Buffa
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  2. M. Costabel
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  3. M. Dauge
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Buffa, A., Costabel, M. & Dauge, M. Algebraic convergence for anisotropic edge elements in polyhedral domains. Numer. Math. 101, 29–65 (2005). https://doi.org/10.1007/s00211-005-0607-4

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