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 ofH-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 ¡ 1 and for nite elements up to the order 1 de ned on quasi-uniform rectangular=triangular meshes in R; d=1; 2; 3. The almost linear complexity of the matrix addition, multiplication and inversion of H-matrices is also veri ed. c © 2000 Elsevier Science B.V. All rights reserved. MSC: 65F05; 65F30; 65F50