A sparse H-matrix arithmetic: general complexity estimates

In a preceding paper (Hackbusch, Computing 62 (1999) 89–108), a class of matrices (H-matrices) has been introduced which are data-sparse and allow an approximate matrix arithmetic of almost linear complexity. Several types of H-matrices have been analysed in Hackbusch (Computing 62 (1999) 89–108) and Hackbusch and Khoromskij (Preprint MPI, No. 22, Leipzig, 1999; Computing 64 (2000) 21–47) which are able to approximate integral (nonlocal) operators in FEM and BEM applications in the case of quasi-uniform unstructured meshes. In the present paper, the general construction of H-matrices on rectangular and triangular meshes is proposed and analysed. First, the reliability of H-matrices in BEM is discussed. Then, we prove the optimal complexity of storage and matrix–vector multiplication in the case of rather arbitrary admissibility parameters η<1 and for finite elements up to the order 1 defined on quasi-uniform rectangular/triangular meshes in Rd, d=1,2,3. The almost linear complexity of the matrix addition, multiplication and inversion of H-matrices is also verified.