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The discrete compactness property for anisotropic edge elements on polyhedral domains∗

Published online by Cambridge University Press:  31 August 2012

Ariel Luis Lombardi
Ariel Luis Lombardi*
Affiliation:
Instituto de Ciencias, Universidad Nacional de General Sarmiento, J.M. Gutierrez 1150, Los Polvorines, B1613 GSX Provincia de Buenos Airesn, Argentina. aldoc7@dm.uba.ar Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428 Buenos Aires, Member of CONICET, Argentina
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Abstract

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.

Keywords

Discrete compactness propertyedge elementsanisotropic finite elementsMaxwell equations
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 1 , January 2013 , pp. 169 - 181
Copyright
© EDP Sciences, SMAI, 2012

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References

Références

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) 519–549. Google Scholar
Boffi, D., Fortin operator and discrete compactness for edge elements. Numer. Math. 87 (2000) 229–246. Google Scholar
Boffi, D., Finite element approximation of eigenvalue problems. Acta Numer. 19 (2010) 1–120. Google Scholar
Buffa, A., Costabel, M. and Dauge, M., Algebraic convergence for anisotropic edge elements in polyhedral domains. Numer. Math. 101 (2005) 29–65. Google Scholar
Caorsi, S., Fernandes, P. and Raffetto, M., On the convergence of Galerkin finite element approximations of electromagnetic eigenproblems, SIAM J. Numer. Anal. 38 (2000) 580–607. Google Scholar
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) 331–354. Google Scholar
V. Girault and P.A. Raviart, Finite Element Methods for Navier-Stokes Equations, in Theory and Applications. Springer-Verlag, Berlin (1986).
Hiptmair, R., Finite elements in computational electromagnetism. Acta Numer. 11 (2002) 237–339. Google Scholar
Kikuchi, F., On a discrete compactness property for the Nédélec finite elements. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36 (1989) 479–490. Google Scholar
Krízek, M., On the maximum angle condition for linear tetrahedral elements. SIAM J. Numer. Anal. 29 (1992) 513–520. Google Scholar
R. Leis, Initial Boundary Value Problems in Mathematical Physics. John Wiley, New York (1986).
Lombardi, A.L., Interpolation error estimates for edge elements on anisotropic meshes. IMA J. Numer. Anal. 31 (2011) 1683–1712. Google Scholar
P. Monk, Finite Element Methods for Maxwell’s Equations. Oxford University Press, New York (2003).
Monk, P. and Demkowicz, L., Discrete compactness and the approximation of Maxwell’s equations in R3. Math. Comp. 70 (2001) 507–523. Google Scholar
Nédélec, J.C., Mixed finite elements in R3. Numer. Math. 35 (1980) 315–341. Google Scholar
Nicaise, S., Edge elements on anisotropic meshes and approximation of the Maxwell equations. SIAM J. Numer. Anal. 39 (2001) 784–816. Google Scholar
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).
Weber, Ch., A local compactness theorem for Maxwell’s equations. Math. Meth. Appl. Sci. 2 (1980) 12–25. Google Scholar