Partition of Unity Coarse Spaces and Schwarz Methods with Harmonic Overlap

Coarse spaces play a crucial role in making Schwarz type domain decomposition methods scalable with respect to the number of subdomains. In this paper, we consider coarse spaces based on a class of partition of unity (PU) for some domain decomposition methods including the classical overlapping Schwarz method, and the new Schwarz method with harmonic overlap. PU has been used as a very powerful tool in the theoretical analysis of Schwarz type domain decomposition methods and meshless discretization schemes. In this paper, we show that PU can also be used effectively in the numerical construction of coarse spaces. PU based coarse spaces are easy to construct and need less communication than the standard finite-element-basis-function-based coarse space in distributed memory parallel implementations. We prove the new result that the condition number of the algorithms grows only linearly with respect to the relative size of the overlap. We also introduce the additive Schwarz method (AS) with harmonic overlap (ASHO), where all functions are made harmonic in part of the overlapping regions. As a result, the communication cost and condition number of ASHO is smaller than that of AS. Numerical experiments and a conditioning theory are presented in the paper.