The discrete compactness property for anisotropic edge elements on polyhedral domains
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 1, pp. 169-181.

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.

DOI : 10.1051/m2an/2012024
Classification : 65N30
Mots clés : discrete compactness property, edge elements, anisotropic finite elements, Maxwell equations
@article{M2AN_2013__47_1_169_0,
     author = {Lombardi, Ariel Luis},
     title = {The discrete compactness property for anisotropic edge elements on polyhedral domains},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {169--181},
     publisher = {EDP-Sciences},
     volume = {47},
     number = {1},
     year = {2013},
     doi = {10.1051/m2an/2012024},
     mrnumber = {2979513},
     zbl = {1281.78014},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2012024/}
}
TY  - JOUR
AU  - Lombardi, Ariel Luis
TI  - The discrete compactness property for anisotropic edge elements on polyhedral domains
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2013
SP  - 169
EP  - 181
VL  - 47
IS  - 1
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2012024/
DO  - 10.1051/m2an/2012024
LA  - en
ID  - M2AN_2013__47_1_169_0
ER  - 
%0 Journal Article
%A Lombardi, Ariel Luis
%T The discrete compactness property for anisotropic edge elements on polyhedral domains
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2013
%P 169-181
%V 47
%N 1
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2012024/
%R 10.1051/m2an/2012024
%G en
%F M2AN_2013__47_1_169_0
Lombardi, Ariel Luis. The discrete compactness property for anisotropic edge elements on polyhedral domains. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 1, pp. 169-181. doi : 10.1051/m2an/2012024. http://www.numdam.org/articles/10.1051/m2an/2012024/

[1] T. Apel and S. Nicaise, 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. | MR | Zbl

[2] D. Boffi, Fortin operator and discrete compactness for edge elements. Numer. Math. 87 (2000) 229-246. | MR | Zbl

[3] D. Boffi, Finite element approximation of eigenvalue problems. Acta Numer. 19 (2010) 1-120. | MR | Zbl

[4] A. Buffa, M. Costabel and M. Dauge, Algebraic convergence for anisotropic edge elements in polyhedral domains. Numer. Math. 101 (2005) 29-65. | MR | Zbl

[5] S. Caorsi, P. Fernandes and M. Raffetto, On the convergence of Galerkin finite element approximations of electromagnetic eigenproblems, SIAM J. Numer. Anal. 38 (2000) 580-607. | MR | Zbl

[6] S. Caorsi, P. Fernandes and M. Raffetto, Spurious-free approximations of electromagnetic eigenproblems by means of Nedelec-type elements. Math. Model. Numer. Anal. 35 (2001) 331-354. | Numdam | MR | Zbl

[7] V. Girault and P.A. Raviart, Finite Element Methods for Navier-Stokes Equations, in Theory and Applications. Springer-Verlag, Berlin (1986). | MR | Zbl

[8] R. Hiptmair, Finite elements in computational electromagnetism. Acta Numer. 11 (2002) 237-339. | MR | Zbl

[9] F. Kikuchi, On a discrete compactness property for the Nédélec finite elements. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36 (1989) 479-490. | MR | Zbl

[10] M. Krízek, On the maximum angle condition for linear tetrahedral elements. SIAM J. Numer. Anal. 29 (1992) 513-520. | MR | Zbl

[11] R. Leis, Initial Boundary Value Problems in Mathematical Physics. John Wiley, New York (1986). | MR | Zbl

[12] A.L. Lombardi, Interpolation error estimates for edge elements on anisotropic meshes. IMA J. Numer. Anal. 31 (2011) 1683-1712. | MR | Zbl

[13] P. Monk, Finite Element Methods for Maxwell's Equations. Oxford University Press, New York (2003). | MR | Zbl

[14] P. Monk and L. Demkowicz, Discrete compactness and the approximation of Maxwell's equations in R3. Math. Comp. 70 (2001) 507-523. | MR | Zbl

[15] J.C. Nédélec, Mixed finite elements in R3. Numer. Math. 35 (1980) 315-341. | EuDML | Zbl

[16] S. Nicaise, Edge elements on anisotropic meshes and approximation of the Maxwell equations. SIAM J. Numer. Anal. 39 (2001) 784-816. | MR | Zbl

[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). | MR | Zbl

[18] Ch. Weber, A local compactness theorem for Maxwell's equations. Math. Meth. Appl. Sci. 2 (1980) 12-25. | MR | Zbl

Cité par Sources :