Optimal Panel-Clustering in the Presence of Anisotropic Mesh Refinement

In this paper we propose and analyse a new enhanced version of the panel-clustering algorithm for discrete boundary integral equations on polyhedral surfaces in 3D, which is designed to perform efficiently even when the meshes contain the highly stretched elements needed for efficient discretisation when the solution contains edge singularities. The key features of our algorithm are: (i) the employment of partial analytic integration in the direction of stretching, yielding a new kernel function on a one dimensional manifold where the influence of the high aspect ratios in the stretched elements is removed and (ii) the introduction of a generalised admissibility condition with respect to the partially integrated kernel which ensures that certain stretched clusters which are inadmissible in the classical sense now become admissible. In the context of a model problem, we prove that our algorithm yields an accurate (up to discretisation error) matrix-vector multiplication which requires O(N log N) operations, where N is the number of degrees of freedom and κ is small and independent of the aspect ratio of the elements. We also show that the classical admissibility condition leads to a sub-optimal clustering algorithm for these problems. A numerical experiment shows that the theoretical estimates can be realised in practice. The generalised admissibility condition can be viewed as a simple addition to the classical method which may be useful in general when stretched meshes are present.