Coarsening of Finite Element and Boundary Element Spaces

In this paper, we will present composite boundary elements (CBE) for classical Fredholm boundary integral equations. These new boundary elements allow the low-dimensional discretisation of boundary integral equation where the minimal number of degrees of freedom is independent of the, possibly, huge number of charts which are necessary to describe a complicated surface. The applications are three-fold: (a) The coarse grid discretisation by composite boundary elements allow the use of multi-grid algorithms for solving the fine grid discretisation independently of the number of patches which are necessary to describe the surface. (b) If the accuracy requirements are moderate the composite boundary elements allow the low-dimensional discretisation of the integral equation. (c) A posteriori error indicators can be applied already to a low dimensional discretisation, which do not resolve the domain, to obtain a problem-adapted discretisation.