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.