A New Finite Element Approach for Problems Containing Small Geometric Details

Wolfgang Hackbusch and Stefan A. Sauter

Address. Lehrstuhl Praktische Mathematik, Universitaet Kiel, 24098 Kiel, Germany

E-mail: wh@numerik.uni-kiel.de


Abstract. In this paper a new finite element approach is presented which allows the discretization of PDEs on domains containing small micro-structures with extremely few degrees of freedom. The applications of these so-called {\it Composite Finite Elements }are two-fold. They allow the efficient use of multi-grid methods to problems on complicated domains where, otherwise, it is not possible to obtain very coarse discretizations with standard finite elements. Furthermore, they provide a tool for discrete homogenization of PDEs without requiring periodicity of the data.

AMS classification. 65D05, 65N12, 65N15, 65N30, 65N50, 65N55

Key words. Finite Elements, Shortley-Weller discretization, complicated boundary