Partition of Unity Coarse Spaces

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 a hybrid Schwarz. 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. Numerical experiments and a conditioning theory are presented in the paper.