Parallel subspace correction methods for nearly singular systems

In this paper we consider the parallel subspace correction (PSC) methods for the nearly singular systems. We apply the PSC methods as the preconditioners when we solve the nearly singular systems by the conjugate gradient methods. Our focus is to estimate the condition number of the preconditioned systems. We deduce the parameter independent estimates on the PSC preconditioners for the nearly singular systems, under appropriate assumptions on subspace decomposition. The main assumption is that the kernel of the singular part of the system can be decomposed into a sum of local kernel subspaces. Our estimates can be applied into actual problems, and two examples are analyzed in this paper. One is the elliptic problem with large jumps in the coefficients, the other is the planar nearly incompressible elasticity problem with the Scott-Vogelius finite element discretization. We prove that the related parallel multilevel methods for both examples are convergent uniformly, with respect to the coefficients and the mesh size.