Adaptive Galerkin boundary element methods with panel clustering

In this paper, we will propose a boundary element method for solving classical boundary integral equations on complicated surfaces which, possibly, contain a large number of geometric details or even uncertainties in the given data. The (small) size of such details is characterised by a small parameter and the regularity of the solution is expected to be low in such zones on the surface (which we call the wire-basket zones). We will propose the construction of an initial discretisation for such type of problems. Afterwards standard strategies for boundary element discretisations can be applied such as the h, p, and the adaptive hp-version in a straightforward way. For the classical boundary integral equations, we will prove the optimal approximation results of our so-called wire-basket boundary element method and discuss the stability aspects. Then, we construct the panel-clustering and -matrix approximations to the corresponding Galerkin BEM stiffness matrix. The method is shown to have an almost linear complexity with respect to the number of degrees of freedom located on the wire basket

Robust multilevel elliptic problem solvers for anisotropic unstructured geometries

In this paper we introduce a new class of robust multilevel interface solvers for two-dimensional finite element discrete elliptic problems with highly varying coefficients corresponding to strongly anisotropic multigrade geometric decompositions. The global iterations convergence rate q < 1 is shown to be of the order q = 1 - O(log"-"1"/"2N) with respect to the number N of degrees of freedom on the single subdomain boundaries, uniformly upon the coarse and fine mesh sizes, jumps in the coefficients related to the original decomposition and aspect ratios of substructures. First we adapt the frequency filtering techniques [28] to construct robust additive/multiplicative smoothers on the unstructured coarse grid. As an alternative, some multilevel everaging procedure for successive coarse grid correction is proposed and analyzed. The resultant multilevel coarse grid preconditioner is shown to have the condition number independent upon the multiscale hierarchy of coarse mesh grading and jumps in the coefficients related to the coarset refinement level. The frequency cutting approach is applied for sparse approximation of the factorized cross terms (with respect to adjacent edges) related to the Schur complement on the edge space with an appropriate assembling [18] of parallel neighbouring thin strips. The proposed technique exhibited high serial and parallel performance in the skin diffusion processes modelling. It may be also applied in magnetostatics problems as well as in some composite materials simulations. (orig.)Available from TIB Hannover: RR 1606(94-36) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman