Overlapping Domain Decomposition for Meshless Finite Difference Methods

Abstract

Schwarz type domain decomposition methods generally require a partition of unity to combine solutions on subdomains. However, in mesh-based methods it is common to organize subdomains with minimal overlap, if any, which is facilitated by the availability of a mesh. This study analyzes how the continuity of the partition of unity affects the algebraic Schwarz method for Poisson and Stokes equations from a meshless point of view, whereby the underlying differential operators are discretized using the radial basis function finite difference (RBF-FD) method. We demonstrate numerically that, in this setting, small overlaps improve the performance of the domain decomposition, leading to smaller iteration counts, and therefore no disjoint partitioning technique is required.