On the Schur component preconditioners

摘要

Some representations of the H~1/2 norm are used as Schur complement preconditioners in PCG-based domain decomposition algorithms for elliptic problems. These norm representations are efficient preconditioners. Here we give a new matrix representation of the H~α (0 < α < 1) norms by a special sparse Toeplitz matrix. It contains O(log(N)) non-zero entries at each row, where N is the number of rows. The special properties of this matrix ensure that it can be used as preconditioner. This is proved by estimating spectral equivalence constants and this fact has also been verified by numericla tests.