Skip to main content Accessibility help

The discrete compactness property for anisotropic edge elements on polyhedral domains

  • Ariel Luis Lombardi (a1) (a2)


We prove the discrete compactness property of the edge elements of any order on a class of anisotropically refined meshes on polyhedral domains. The meshes, made up of tetrahedra, have been introduced in [Th. Apel and S. Nicaise, Math. Meth. Appl. Sci. 21 (1998) 519–549]. They are appropriately graded near singular corners and edges of the polyhedron.



Hide All
[1] Apel, T. and Nicaise, S., The finite element method with anisotropic mesh grading for elliptic problems in domains with corners and edges, Math. Meth. Appl. Sci. 21 (1998) 519549.
[2] Boffi, D., Fortin operator and discrete compactness for edge elements. Numer. Math. 87 (2000) 229246.
[3] Boffi, D., Finite element approximation of eigenvalue problems. Acta Numer. 19 (2010) 1120.
[4] Buffa, A., Costabel, M. and Dauge, M., Algebraic convergence for anisotropic edge elements in polyhedral domains. Numer. Math. 101 (2005) 2965.
[5] Caorsi, S., Fernandes, P. and Raffetto, M., On the convergence of Galerkin finite element approximations of electromagnetic eigenproblems, SIAM J. Numer. Anal. 38 (2000) 580607.
[6] Caorsi, S., Fernandes, P. and Raffetto, M., Spurious-free approximations of electromagnetic eigenproblems by means of Nedelec-type elements. Math. Model. Numer. Anal. 35 (2001) 331354.
[7] V. Girault and P.A. Raviart, Finite Element Methods for Navier-Stokes Equations, in Theory and Applications. Springer-Verlag, Berlin (1986).
[8] Hiptmair, R., Finite elements in computational electromagnetism. Acta Numer. 11 (2002) 237339.
[9] Kikuchi, F., On a discrete compactness property for the Nédélec finite elements. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36 (1989) 479490.
[10] Krízek, M., On the maximum angle condition for linear tetrahedral elements. SIAM J. Numer. Anal. 29 (1992) 513520.
[11] R. Leis, Initial Boundary Value Problems in Mathematical Physics. John Wiley, New York (1986).
[12] Lombardi, A.L., Interpolation error estimates for edge elements on anisotropic meshes. IMA J. Numer. Anal. 31 (2011) 16831712.
[13] P. Monk, Finite Element Methods for Maxwell’s Equations. Oxford University Press, New York (2003).
[14] Monk, P. and Demkowicz, L., Discrete compactness and the approximation of Maxwell’s equations in R3. Math. Comp. 70 (2001) 507523.
[15] Nédélec, J.C., Mixed finite elements in R3. Numer. Math. 35 (1980) 315341.
[16] Nicaise, S., Edge elements on anisotropic meshes and approximation of the Maxwell equations. SIAM J. Numer. Anal. 39 (2001) 784816.
[17] P. A. Raviart and J.-M. Thomas, A mixed finite element method for second order elliptic problems, in Mathematical Aspects of the Finite Element Method, edited by I. Galligani and E. Magenes. Lect. Notes Math. 606 (1977).
[18] Weber, Ch., A local compactness theorem for Maxwell’s equations. Math. Meth. Appl. Sci. 2 (1980) 1225.


Related content

Powered by UNSILO

The discrete compactness property for anisotropic edge elements on polyhedral domains

  • Ariel Luis Lombardi (a1) (a2)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.