H-matrix approximation for elliptic solution operators in cylinder domains

Journal of Numerical Mathematics

Volumn 9, Issue 1, 2001, Pages 25-58

  Gavrilyuk, I P ^a   Hackbusch, W ^b   Khoromski, B N ^b  

^a Am Wartenberg 2 ^*  (China)

^b Inselstr 22 26   (China)

Author keywords

[No Author keywords available]

Indexed keywords

ALGORITHMS; APPROXIMATION THEORY; BOUNDARY VALUE PROBLEMS; CONVERGENCE OF NUMERICAL METHODS; FINITE ELEMENT METHOD; INTEGRAL EQUATIONS; ITERATIVE METHODS; MATHEMATICAL OPERATORS; PARALLEL PROCESSING SYSTEMS; PARTIAL DIFFERENTIAL EQUATIONS;

DOMAIN DECOMPOSITION METHOD; ELLIPTIC SOLUTION OPERATOR; MATRIX APPROXIMATION;

MATRIX ALGEBRA;

EID: 0034936175     PISSN: 15702820     EISSN: None     Source Type: Journal    
DOI: 10.1515/JNMA.2001.25     Document Type: Article

References (29)

1. V. I. Agoshkov and V. I. Lebedev, Poincaré-Steklov operators and domain decomposition meth­ods in variational problems. In :Vychisl. Protsessy i Systemy 2. Nauka, Moscow, 1985, pp. 173-227 (in Russian).
2. Z. Chen, R. Ewing, R. Lazarov, S. Maliasov, and YU. Kuznetsov, Multilevel preconditioners for mixed methods for second-order elliptic problems. Numer. Lin. Algebra with Appl. (1996) 3, 427-453.
3. R.Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 5, Evolutions Problems I. Springer-Verlag, 1992.
4. I. P. Gavrilyuk, Strongly P-positive operators and explicit representation of the solutions of initial value problems for second order differential equations in Banach space. J. Math. Anal. Appl. (1999) 236,327-349.
5. I. P. Gavrilyuk, W. Hackbusch, and B. N. Khoromskij, H-matrix approximation for the operator exponential with applications. Preprint MPI, No. 42, 2000.
6. I. P. Gavrilyuk and V. L. Makarov, Exponentially convergent parallel discretization methods for the first order evolution equations. Preprint NTZ12, Universitat Leipzig, 2000.
7. I. P. Gavrilyuk and V. L. Makarov, Explicit and approximate solutions of second order elliptic differential equations in Hilbert and Banach spaces. Numer. Funct. Anal. Optimization (1999) 20, 695-717.
8. I. P. Gavrilyuk and V. L. Makarov, Exact and approximate solutions of some operator equations based on the Cayley transform. Lin. Algebra and Appl. (1998) 282, 97-121.
9. I. P. Gavrilyuk and V. L.Makarov, Representation and approximation of the solution of an initial value problem for a first order differential eqation in Banach space. Z. Anal. Anwend. (1996) 15, 495-527.
10. I. P. Gavrilyuk and V. L. Makarov, An explicit boundary integral representation of the solution of the two-dimensional heat equation and its discretization. J. Integral Eq. Appl. (2000) 12, 63-83.
11. I. P. Gavrilyuk, V. L. Makarov, and R. Chapko, On the numerical solution of linear evolution problems with an integral operator coefficient. J. Integral Eq. Appl. (1999) 11, 1-20.
12. W. Hackbusch, Integral Equations. Theory and Numerical Treatment. ISNM 128, Birkhauser, Basel, 1995.
13. W. Hackbusch, Elliptic Differential Equations :Theory and Numerical Treatment. Springer­Verlag, Berlin, 1992.
14. W. Hackbusch, A sparse matrix arithmetic based on H-matrices. Part I :Introduction to H- matrices. Computing (1999) 62, 89-108.
15. W.Hackbusch and B. N. Khoromskij, A sparse H-matrix arithmetic. Part II :Application to multi-dimensional problems. Computing (2000) 64,21-47.
16. W.Hackbusch and B. N. Khoromskij, A sparse H-matrix arithmetic :General complexity esti­mates. J. Comp. Appl. Math. (2000) 125, 479-501.
17. W. Hackbusch and B. N. Khoromskij, Towards H-matrix approximation of the linear complex­ity. Preprint MPI, No.15, Leipzig, 2000. Siegfried PrÖßdorf Memorial Volume, Birkhauser Ver­lag, 2000 (to appear).
18. W. Hackbusch, B. N. Khoromskij, and S. Sauter, On H2-matrices. In :Lectures on Applied Math­ematics (Eds. H.-J.Bungartz, R. Hoppe, and C.Zenger). Springer-Verlag, Berlin, 2000, pp. 9­30
19. B. N. Khoromskij, Lectures on multilevel schur complement methods for elliptic differential equations. Preprint No. 98/4, ICA3, University of Stuttgart, 1998.
20. B. N. Khoromskij and S. PrÖssdorf, Multilevel preconditioning on the refined interface and opti­mal boundary solvers for the Laplace equation. Advances in Comp. Math. (1995) 4, 331-355.
21. B. N. Khoromskij and G.Schmidt, A fast interface solver for the biharmonic Dirichlet problem on polygonal domains. Numer. Math.(1998) 78, 577-596.
22. B. N. Khoromskij and G.Wittum, Robust Schur complement method for strongly anisotropic elliptic equations. Numer. Lin. Algebra Appl. (1999) 6, 621-653.
23. R. Kress, Linear Integral Equations. Springer-Verlag, Berlin, 1999.
24. A.Pazy, Semigroups of Linear Operator and Applications to Partial Differential Equations. Springer-Verlag, 1983.
25. H. Poincaré, La méthode de Neumann et le probléme de Dirichlet. Acta Mathematica (1896) 20.
26. A. A. Samarskii, I. P. Gavrilyuk, and V. L. Makarov, Stability and regularization of three-level difference schemes with unbounded operator coefficients in Banach spaces. Preprint NTZ 39, University of Leipzig, 1998, pp. 1-14 (to appear in Dokl. Ros. Akad. Nauk).
27. D. Sheen, I. H. Sloan, and V. Thomée, A parallel method for time-discretization of parabolic problems based on contour integral representation and quadrature. Math. Comp. (2000) 69, 177-195.
28. V. A. Steklov, General Methods for the Solution of Principal Mathematical Physics Problems. Kharkov Mathematical Society Publ., Kharkov, 1901.
29. F. Stenger, Numerical Methods Based on Sinc and Analytic Functions. Springer-Verlag, 1993.