On deflation and singular symmetric positive semi-definite matrices

For various applications, it is well-known that the deflated ICCG is an efficient method for solving linear systems with invertible coefficient matrix. We propose two equivalent variants of this deflated ICCG which can also solve linear systems with singular coefficient matrix, arising from discretization of the discontinuous Poisson equation with Neumann boundary conditions. It is demonstrated both theoretically and numerically that the resulting methods accelerate the convergence of the iterative process.Moreover, in practice the singular coefficient matrix has often been made invertible by modifying the last element, since this can be advantageous for the solver. However, the drawback is that the condition number becomes worse-conditioned. We show that this problem can completely be remedied by applying the deflation technique with just one deflation vector.