Adaptive Boundary Element Methods (Invited contribution)

Abstract

Boundary element methods are now a well understood and widely used tool in computational mechanics. Although the mesh construction is due to the dimensional reduction, considerably easier than in finite element computations, there is still a need for reliable rules and tools to construct meshes and assign polynomial degrees in a reasonable and optimal way. This is where adaptive methods start. In so-called feedback procedures, the approximating space is ‘extended’ in order to optimize the accuracy of the boundary element solution for a certain number of degrees of freedom. Adaptive methods are already well established in the finite element field, with quite well developed mathematical theory and very promising numerical results. In boundary element methods, on the other hand, there is nearly no theory available and only few numerical results on adaptive procedures have been reported. So this paper will review the achievements in adaptive finite element methods, will try to ‘transform’ some of the basic results to the boundary element situation and will show the possibilities and advantages of an adaptive BIEM in some numerical examples.