A reciprocal preconditioner for structured matrices arising from elliptic problems with jumping coefficients

We present a new explicit preconditioner for a class of structured matrices Γ(a) defined by a function a. These matrices may arise from elliptic problems by a standard discretization scheme admitting jumps in the diffusion coefficient. When the grid is fixed, the matrix is determined by the diffusion coefficient a. Our preconditioner P is of the form P=Γ-1(1)Γ(1/a)Γ-1(1) and can be considered as an approximation to the inverse matrix to Γ(a). We prove that P and Γ-1(a) are spectrally equivalent. However, the most interesting observation is that Γ(a)P has a cluster at unity. In the one-dimensional case this matrix is found to be equal to the identity plus a matrix of rank one. In more dimensions, a rigorous proof is given in the two-dimensional stratified case. Moreover, in a stratified case with M constant values for the coefficient a, a hypothesis is proposed that a wider set of M+1 points including unity is a proper cluster. In such cases the number of iterations does not depend dramatically on jumps of the diffusion coefficient. In more general cases, fast convergence is still demonstrated by many numerical experiments.