A Highly Scalable Boundary Integral Equation and Walk-on-Spheres (BIE-WOS) Method for the Laplace Equation with Dirichlet Data (Communications in Computational Physics, 2021)
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 theboundary of the domain and does not require a structured mesh, and only needs acovering 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 isintroduced for better error controls. The effect of errors from the Feynman-Kac formulabased path integral WOS method on the overall accuracy of the BIE-WOS method isanalyzed 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 equationof BIE-WOS method can be simplified where the local BIE solutions can be given inclosed 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 ofthe boundary, resulting in a complete independent computation using a large numberof 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-WOSmethod on a large-scale cluster with 6400 CPU cores is verified for computing the DtNmapping of exterior Laplace problems with Dirichlet data for several domains.