Functional Analysis and Applications Seminar 

14/12/2007, 15:00 — 16:00 — Room P3.10, Mathematics Building
Martin Costabel, Université de Rennes 1, França

Computing edge singularity coefficients

In many physical and engineering applications, corner singularities of the solutions of elliptic boundary value problems contain important information. One can therefore expect that these singularities have been approximated by the finite element method for a long time. Classical approaches include the augmentation of the finite element spaces by singular functions (Fix method) and the use of dual singular functions for extracting the coefficient of the singular function either directly from the given data or by a post-processing procedure from a computed approximation of the solution. In 2D problems, these methods work well, because the singular functions are all known more or less explicitly, even for the most general elliptic boundary value problems, and there remains only a finite number of coefficients to calculate. In 3D problems, these methods have been studied theoretically, too, and they work well for the case of conical corner singularities. For the case of edge singularities, however, one has the curious situation that several numerical methods have been described and analyzed in detail, up to precise stability and error estimates, but very few actual numerical codes have been implemented. This has two main causes: The unknown coefficients now are functions, living in an infinite-dimensional function space, and the singular and dual singular functions are often too complicated to be used in practice without simplification, whereas over-simplification leads to insufficient precision. A simplification that does work is the recent "quasi-dual singular function method", developed in collaboration with M. Dauge (Rennes) and Z. Yosibash and N. Omer (Beere-Sheva). Here the singular and dual singular functions are approximated by an asymptotic expansion. In this method, moments of the coefficient function are computed from extrapolation of integrals over cylindrical surfaces neighboring the edge involving a finite-element approximation of the solution. The method has been analyzed theoretically in a rather general setting, and it integrates well as a post-processing tool in higher order finite element codes. Numerical results have been obtained for general second order scalar elliptic boundary value problems and for the system of linear elasticity, even in the anisotropic case.