Blended kernel approximation in the ℋ︁‐matrix techniques

Several types of ℋ-matrices were shown to provide a data-sparse approximation of non-local (integral) operators in FEM and BEM applications. The general construction is applied to the operators with asymptotically smooth kernel function provided that the Galerkin ansatz space has a hierarchical structure. The new class of ℋ-matrices is based on the so-called blended FE and polynomial approximations of the kernel function and leads to matrix blocks with a tensor-product of block-Toeplitz (block-circulant) and rank-k matrices. This requires the translation (rotation) invariance of the kernel combined with the corresponding tensor-product grids. The approach allows for the fast evaluation of volume/boundary integral operators with possibly non-smooth kernels defined on canonical domains/manifolds in the FEM/BEM applications. (Here and in the following, we call domains canonical if they are obtained by translation or rotation of one of their parts, e.g. parallelepiped, cylinder, sphere, etc.) In particular, we provide the error and complexity analysis for blended expansions to the Helmholtz kernel. Copyright © 2002 John Wiley & Sons, Ltd.

[1]  Christian Lage,et al.  Advanced boundary element algorithms , 1999 .

[2]  Sergej Rjasanow,et al.  Effective algorithms with circulant-block matrices , 1994 .

[3]  W. Hackbusch,et al.  On the fast matrix multiplication in the boundary element method by panel clustering , 1989 .

[4]  E. P. ZHIDKOV,et al.  Boundary integral equations on special surfaces and their applications , 1987 .

[5]  W. Hackbusch,et al.  On H2-Matrices , 2000 .

[6]  Boris N. Khoromskij,et al.  A Sparse H-Matrix Arithmetic. Part II: Application to Multi-Dimensional Problems , 2000, Computing.

[7]  Wolfgang Hackbusch,et al.  A Sparse Matrix Arithmetic Based on H-Matrices. Part I: Introduction to H-Matrices , 1999, Computing.

[8]  E. Tyrtyshnikov.  Mosaic-Skeleton approximations , 1996 .

[9]  W. Hackbusch,et al.  A sparse H -matrix arithmetic: general complexity estimates , 2000 .

[10]  Boris N. Khoromskij,et al.  H-matrix approximation on graded meshes , 1999 .

[11]  Wolfgang Dahmen,et al.  Wavelet approximation methods for pseudodifferential equations II: Matrix compression and fast solution , 1993, Adv. Comput. Math..

[12]  D. F. Hays,et al.  Table of Integrals, Series, and Products , 1966 .

[13]  Boris N. Khoromskij,et al.  Towards H-matrix approximation of linear complexity , 2001 .

[14]  W. Hackbusch.  Elliptic Differential Equations , 1992 .

[15]  L. Greengard,et al.  A new version of the Fast Multipole Method for the Laplace equation in three dimensions , 1997, Acta Numerica.