| Abstract: | In this paper, we study a highly scalable communication-free parallel domain boundary decomposition algorithm for the Laplace equation based on a hybrid method combining boundary integral equations and walk-on-spheres (BIE-WOS)
method, which provides a numerical approximation of the Dirichlet-to-Neumann
(DtN) mapping for the Laplace equation. The BIE-WOS is a local method on the
boundary of the domain and does not require a structured mesh, and only needs a
covering of the domain boundary by patches and a local mesh for each patch for a local BIE. A new version of the BIE-WOS method with second kind integral equations is
introduced for better error controls. The effect of errors from the Feynman-Kac formula
based path integral WOS method on the overall accuracy of the BIE-WOS method is
analyzed for the BIEs, especially in the calculation of the right hand sides of the BIEs.
For the special case of flat patches, it is shown that the second kind integral equation
of BIE-WOS method can be simplified where the local BIE solutions can be given in
closed forms. A key advantage of the parallel BIE-WOS method is the absence of communications during the computation of the DtN mapping on individual patches of
the boundary, resulting in a complete independent computation using a large number
of cluster nodes. In addition, the BIE-WOS has an intrinsic capability of fault tolerance for exascale computations. The nearly linear scalability of the parallel BIE-WOS
method on a large-scale cluster with 6400 CPU cores is verified for computing the DtN
mapping of exterior Laplace problems with Dirichlet data for several domains. |
