Mathematik in den Naturwissenschaften Leipzig Existence of H-Matrix Approximants to the Inverse FE-Matrix of Elliptic Operators with L ∞-Coefficients

AbstractThis article deals with the existence of blockwise low-rank approximants — so-called ℋ-matrices — to inverses of FEM matrices in the case of uniformly elliptic operators with L∞-coefficients. Unlike operators arising from boundary element methods for which the ℋ-matrix theory has been extensively developed, the inverses of these operators do not benefit from the smoothness of the kernel function. However, it will be shown that the corresponding Green functions can be approximated by degenerate functions giving rise to the existence of blockwise low-rank approximants of FEM inverses. Numerical examples confirm the correctness of our estimates. As a side-product we analyse the ℋ-matrix property of the inverse of the FE mass matrix.

[1]  G. Meinardus.  Approximation of Functions: Theory and Numerical Methods , 1967 .

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

[3]  Wolfgang Hackbusch,et al.  Construction and Arithmetics of H-Matrices , 2003, Computing.

[4]  W. Hackbusch.  Iterative Lösung großer schwachbesetzter Gleichungssysteme , 1991 .

[5]  H. Weinberger,et al.  An optimal Poincaré inequality for convex domains , 1960 .

[6]  Wolfgang Hackbusch,et al.  Adaptive refinement and clustering of H-matrices , 2001 .

[7]  Ivan P. Gavrilyuk,et al.  fMathematik in den Naturwissenschaften Leipzig H-matrix approximation for the operator exponential with applications , 2000 .

[8]  W. Hackbusch,et al.  Introduction to Hierarchical Matrices with Applications , 2003 .

[9]  W. Hackbusch,et al.  Integralgleichungen : Theorie und Numerik , 1989 .

[10]  S. Müller,et al.  Estimates for Green's matrices of elliptic systems byLp theory , 1995 .

[11]  Michael Grüter,et al.  The Green function for uniformly elliptic equations , 1982 .

[12]  Wolfgang Hackbusch,et al.  Theorie und Numerik elliptischer Differentialgleichungen , 1986, Teubner Studienbücher.

[13]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[14]  Wolfgang Hackbusch,et al.  Multi-grid methods and applications , 1985, Springer series in computational mathematics.

[15]  Mario Bebendorf,et al.  Approximation of boundary element matrices , 2000, Numerische Mathematik.

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

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

[18]  W. Hackbusch.  Integral Equations: Theory and Numerical Treatment , 1995 .

[19]  William F. Moss,et al.  Decay rates for inverses of band matrices , 1984 .

[20]  Mariano Giaquinta,et al.  Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. (AM-105), Volume 105 , 1984 .

[21]  Mario Bebendorf,et al.  Effiziente numerische Lösung von Randintegralgleichungen unter Verwendung von Niedrigrang-Matrizen , 2001 .

[22]  W. Hackbusch.  A Sparse Matrix Arithmetic Based on $\Cal H$-Matrices. Part I: Introduction to ${\Cal H}$-Matrices , 1999, Computing.