Algunos resultados para elementos finitos de Nédélec sobre ma-llas anisotrópicas de tetraedros

ARIEL L. LOMBARDI

Concepción

Simposio; 8th CI2MA Focus Seminar; 2014

Centro de Investigación en Ingeniería Matemática, Universidad de Concepción

The first family of N´ed´elec´s edge elements, introduced incite{N1}, is a conforming family of finite elements in$H(mbox{curl})$. It is broadly used in the approximation ofelliptic partial differential equations in mixed form, such asMaxwell equation and their associated eigenproblems. Anisotropicmeshes appear naturally in applications when the solution presentsboundary layers or edge singularities. This is the case whenconsidering the time-harmonic Maxwell equations in a Lipschitzpolyhedron with nonconvex edges or corners.  The possibility ofusing anisotropic elements can make the design of such mesheseasier, reduce the number of elements and take advantage of thebest regularity properties of the solution. In fact, in manyproblems, the solutions have more regularity in the direction ofthe edges than transversally to them.In this talk firstly we show that uniform interpolation errorestimates for edge elements can be obtained on tetrahedral meshesunder the maximum angle condition. This condition allows forarbitrarily anisotropic elements needed for the discretisation ofelliptic problems in general polyhedra.Secondly, for the tetrahedral meshes used on general polyhedra, wediscuss a proof of the discrete compactness property (DCP) foredge elements of any order. The DCP was introduced by Kikuchicite{kikuchi} for edge elements of lowest order on shape-regularmeshes. It is a useful tool for the analysis of the approximationof Maxwell´s equations, both for the source problem as well as forcomputing the eigenvalues (or resonant frequencies) on a boundedcavity. The numerical approximation of both problems, and so, thevalidity of the DCP, has been considered  in different situationsby several authors: Boffi, Buffa, Caorsi, Costabel, Dauge,Fernandez, Hiptmair, Monk, Nicaise, Raffetto and others.In particular, the aspects mentioned before extend some results ofNicaise cite{N} and Buffa, Costabel and Dauge cite{BCD}.Precisely, we consider edge elements of any order, and we allowedge and corner refinements: our meshes are proposed in order tobe able to adequately approximate the homogeneous Dirichletproblem for the Laplace operator with a right hand side in $L^p$for some $pge 2$.egin{thebibliography}{99}ibitem{kikuchi}  F. Kikuchi, On a discrete compactness property for the N´ed´elec finite elements. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36(1989) 479--490.ibitem{BCD}  A. Buffa, M. Costabel and M. Dauge, Algebraic convergence for anisotropic edge elements in polyhedral domains. Numer.Math. 101 (2005) 29--65.ibitem{N1} J.C. N´ed´elec, Mixed Finite Elements in $mathbb R^3$,Numer. Math. 35 (1980) 315--341.ibitem{N}  S. Nicaise, Edge elements on anisotropic meshes and approximation of the Maxwell equations. SIAM J. Numer. Anal. 39(2001) 784--816.end{thebibliography}