Sine transform based preconditioners for elliptic problems

We consider applying the preconditioned conjugate gradient (PCG) method to solve linear systems Ax = b where the matrix A comes from the discretization of second-order elliptic operators. Let (L +)) ?1 (L t +) denote the block Cholesky factorization of A with lower block triangular matrix L and diagonal block matrix. We propose a preconditioner M = (^ L +)) ?1 (^ L t +) with block diagonal matrix and lower block triangular matrix ^ L. The diagonal blocks of and the subdiagonal blocks of ^ L are respectively the optimal sine transform approximations to the diagonal blocks of and the subdiagonal blocks of L. We show that for 2-dimensional domains, the construction cost of M and the cost for each iteration of the PCG algorithm are of order O(n 2 log n). Furthermore, for rectangular regions, we show that the condition number of the preconditioned system M ?1 A is of order O(1). In contrast, the system preconditioned by the MILU and MINV methods are of order O(n). We will also show that M can be obtained from A by taking the optimal sine transform approximations of each sub-block of A. Thus the construction of M is similar to that of Level-1 circulant preconditioners. Our numerical results on 2-dimensional square and L-shaped domains show that our method converges faster than the MILU and MINV methods. Extension to higher-dimensional domains will also be discussed.