On the non hierarchical matrix representation of the negative, non integer order sobolev norms

Abstract

In this paper a new cyclic matrix representation of the Sobolev norms Ha, a ∈ (-1, 0) are presented. The matrix-vector multiplication by these matrices requires only O(N · log(N)) arithmetic operations, where N is the number of unknowns, The application of the new H-1/2 norm representation as Schur complement preconditioning matrix requires only matrix-vector multiplication. The efficiency of the construction to elliptic problems has been verified by numerical tests. © Springer-Verlag Berlin Heidelberg 2006.