Anna-Margarete Sändig

Graded mesh refinement and error estimates for finite element solutions of elliptic boundary value problems in non-smooth domains

This paper is concerned with the effective numerical treatment of elliptic boundary value problems when the solutions contain singularities. The paper deals first with the theory of problems of this type in the context of weighted Sobolev spaces and covers problems in domains with conical vertices and non-intersecting edges, as well as polyhedral domains with Lipschitz boundaries. Finite element schemes on graded meshes for second-order problems in polygonal/polyhedral domains are then proposed for problems with the above singularities. These schemes exhibit optimal convergence rates with decreasing mesh size. Finally, we describe numerical experiments which demonstrate the efficiency of our technique in terms of ‘actual’ errors for specific (finite) mesh sizes in addition to the asymptotic rates of convergence.

Quasilinear Elliptic Systems of Second Order in Domains with Corners and Edges: Nemytskij Operator, Local Existence and Asymptotic Behaviour

We consider systems of quasilinear partial differential equations of second order in twoand three-dimensional domains with corners and edges. The analysis is performed in weighted Sobolev spaces with attached asymptotics generated by the asymptotic behaviour of the solutions of the corresponding linearized problems near boundary singularities. Applying the Local Invertibility Theorem in these spaces we find conditions which guarantee existence of small solutions of the nonlinear problem having the same asymptotic behaviour as the solutions of the linearized problem. The main tools are multiplication theorems and properties of composition (Nemytskij) operators in weighted Sobolev spaces. As application of the general results a steady-state drift-diffusion system is explained.

Innovative Solution of a 2-D Elastic Transmission Problem

This article is concerned with a boundary-field equation approach to a class of boundary value problems exterior to a thin domain. A prototype of this kind of problems is the interaction problem with a thin elastic structure. We are interested in the asymptotic behavior of the solution when the thickness of the elastic structure approaches to zero. In particular, formal asymptotic expansions will be developed, and their rigorous justification will be considered. As will be seen, the construction of these formal expansions hinges on the solutions of a sequence of exterior Dirichlet problems, which can be treated by employing boundary element methods. On the other hand, the justification of the corresponding formal procedure requires an independence on the thickness of the thin domain for the constant in the Korn inequality. It is shown that in spite of the reduction of the dimensionality of the domain under consideration, this class of problems are, in general, not singular perturbation problems, because of appropriate interface conditions. ¶Dedicated to Wolfgang L. Wendland on the occasion of his 70th birthday.