New Optimized Robin--Robin Domain Decomposition Methods using Krylov Solvers for the Stokes--Darcy System

Liu, Yingzhi; Boubendir, Yassine; He, Xiaoming; He, Yinnian

doi:10.1137/21M1417223

Numerous scientific and engineering applications require solutions to boundary value problems (BVPs) involving elliptic partial differential equations, such as the Laplace or Poisson equations, on geometrically intricate domains. We develop a Monte Carlo method for solving such BVPs with arbitrary first-order linear boundary conditions---Dirichlet, Neumann, and Robin. Our method directly generalizes thewalk on stars (WoSt)algorithm, which previously tackled only the first two types of boundary conditions, with a few simple modifications. Unlike conventional numerical methods, WoSt does not need finite element meshing or global solves. Similar to Monte Carlo rendering, it instead computes pointwise solution estimates by simulating random walks along star-shaped regions inside the BVP domain, using efficient ray-intersection and distance queries. To ensure WoSt producesbounded-varianceestimates in the presence of Robin boundary conditions, we show that it is sufficient to modify how WoSt selects the size of these star-shaped regions. Our generalized WoSt algorithm reduces estimation error by orders of magnitude relative to alternative grid-free methods such as thewalk on boundaryalgorithm. We also developbidirectionalandboundary value cachingstrategies to further reduce estimation error. Our algorithm is trivial to parallelize, scales sublinearly with increasing geometric detail, and enables progressive and view-dependent evaluation.

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