The Hybrid Galerkin Boundary

In the implementation of boundary element methods for 3D elliptic PDEs in a given geometry there are two main numerical tasks: the stiiness matrix assembly and the solution of the resulting linear systems. The linear solve (performed directly) is O(n 3)-where n is the number of degrees of freedom-and is asymp-totically the most expensive component. Hence there are a number of modern techniques (multipole, panel-clustering, wavelets) for reducing this cost. However for practical discretisations it is the formation of the stiiness matrix which often dominates (even though asymptotically it is only O(n 2)). Saving in the matrix assembly process may sometimes be achieved by replacing the variational Galerkin method with the collocation method. However, this is not a panacea. In some cases it is not clear what an appropriate set of collocation points should be and in a great many more cases the collocation method has no theoretical justiication. The last point is not just a mathematical nicety-a relevant convergence theory often leads to better practical techniques, such as distinguishing the best norms on which to base an adaptive procedure. In this paper we describe the rst results on a new quadrature method for computing the Galerkin stii-ness matrix arising from piecewise linear nite elements on surfaces in R 3. This method in eeect approximates the true Galerkin matrix away from the diagonal by appropriately averaged elements of a related discrete collocation (or Nystrr om) matrix and hence uses relatively few kernel evaluations. This \hybrid" Nystrr om-Galerkin method can be shown to be stable and to enjoy the same convergence rate in the energy norm as the Galerkin method itself. Numerical tests show that the method produces solutions which are qualitatively as good as the Galerkin method, but at substantially reduced cost. This paper is the rst to describe this new method and will do so only in the context of a single model example. Gen-eralisations and proofs of the theoretical results described here will be contained in the later paper Graham, Hackbusch & Sauter (1997).