Two-scale Boolean Galerkin discretizations for Fredholm integral equations of the second kind

SIAM Journal on Numerical Analysis

Volumn 45, Issue 1, 2007, Pages 296-312

  Liu, Fang ^a,b   Zhou, Aihui ^a  

^a INSTITUTE OF COMPUTATIONAL MATHEMATICS AND SCIENTIFIC ENGINEERING COMPUTING   (China)

^b UNIVERSITY OF CHINESE ACADEMY OF SCIENCES   (China)

Author keywords

Boolean Galerkin discretization; Fredholm integral equation; Multidimension; Two scale

Indexed keywords

APPROXIMATION ALGORITHMS; BOOLEAN FUNCTIONS; DEGREES OF FREEDOM (MECHANICS); INTEGRAL EQUATIONS;

BOOLEAN GALERKIN DISCRETIZATION; DISCRETIZATION ALGORITHMS; FREDHOLM INTEGRAL EQUATION; MULTIDIMENSION;

GALERKIN METHODS;

EID: 34547961823     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/050633007     Document Type: Article

References (29)

1. R. A. ADAMS, Sobolev Spaces, Academic Press, New York, 1975.
2. K. BITTNER, Fast algorithms for periodic spline wavelets on sparse grids, SIAM J. Sci. Comput., 20 (1999), pp. 1192-1213.
3. H. BRUNNER, On the numerical solution of nonlinear Volterra-Fredholm integral equations by collocation methods, SIAM J. Numer. Anal., 27 (1990), pp. 987-1000.
4. H. J. BUNGARTZ AND M. GRIEBEL, Sparse grids, Acta Numer., 13 (2004), pp. 1-123.
5. H. BUNGARTZ, M. GRIEBEL, D. RÖSCHKE, AND C. ZENGER, A proof of convergence for the combination technique for the Laplace equation using tools of symbolic computation, Math. Comput. Simulation, 42 (1996), pp. 595-605.
6. H. BUNGARTZ, M. GRIEBEL, AND U. RÜDE, Extrapolation, combination, and sparse grid techniques for elliptic boundary value problems, Comput. Methods Appl. Mech. Engrg., 116 (1994), pp. 243-252.
7. Z. CHEN AND Y. XU, The Petrov-Galerkin and iterated Petrov-Galerkin methods for second-kind integral equations, SIAM J. Numer. Anal., 35 (1998), pp. 406-434.
8. P. G. CIARLET, The Finite Element Method for Elliptic Problems, SIAM, Philadelphia, 2002.
9. F. CUKER AND S. SMALE, On the mathematical foundations of learning, Bull. Amer. Math. Soc. (N.S.), 39 (2002), pp. 1-49.
10. F.-J. DELVOS, d-variate Boolean interpolation, J. Approx. Theory, 34 (1982), pp. 99-114.
11. M. GRIEBEL, P. OSWALD, AND T. SCHIEKOFER, Sparse grids for boundary integral equations, Numer. Math., 83 (1999), pp. 279-312.
12. M. GRIEBEL, M. SCHNEIDER, AND C. ZENGER, A combination technique for the solution of sparse grid problems, in Proceedings of the IMACS International Symposium on Iterative Methods in Linear Algebra, P. de Groen and R. Beauwens, eds., Elsevier, Amsterdam, 1992, pp. 263-281.
13. H. KANEKO AND Y. XU, Degenerate kernel method for Hemmerstein equations, Math. Comp., 56 (1991), pp. 141-148.
14. R. KRESS, Linear Integral Equations, Springer-Verlag, Berlin, 1989.
15. Q. LIN. N. YAN, AND A. ZHOU, A sparse finite element method with high accuracy: Part I, Numer. Math., 88 (2001), pp. 731-742.
16. F. LIU AND A. ZHOU, Two-scale finite element discretizations for partial differential equations, J. Comput. Math., 24 (2006), pp. 373-392.
17. F. LIU AND A. ZHOU, Localization and parallelizations for two-scale finite element discretizations, Comm. Pure Appl. Anal., to appear.
18. K. T. MYNBAEV, Approximate solution of a Fredholm integral equation of the second kind in a multidimensional domain, Soviet Math. Dokl., 43 (1991), pp. 543-546.
19. C.-H. PAO AND N. E. BICKERS, Renormalization-group acceleration of self-consistent field solutions: Two-dimensional Hubbard model, Phys. Rev. B, 49 (1994), pp. 1586-1599.
20. C. PFLAUM, Diskretisierung Elliptischer Differentialgleichungen mit Dünnen Gittern, Dissertation, Technische Universität München, München, Germany, 1995.
21. C. PFLAUM, Convergence of the combination technique for second-order elliptic differential equations, SIAM J. Numer. Anal., 34 (1997), pp. 2431-2455.
22. C. PFLAUM AND A. ZHOU, Error analysis of the combination technique, Numer. Math., 84 (1999), pp. 327-350.
23. U. RÜDE AND A. ZHOU, Multiparameter extrapolation methods for boundary integral equations, Adv. Comput. Math., 9 (1998), pp. 173-190.
24. I. H. SOLAN AND V. THOMÉE, Supercontiergence of the Galerkin iterates for integral equations of the second kind, J. Integral Equations, 9 (1985), pp. 1-23.
25. G. VAINIKKO, Multidimensional Weakly Singular Integral Equations, Lectures Notes Math. 1549, Springer-Verlag, Berlin, 1993.
26. Y. XU AND A. ZHOU, Fast Boolean approximations methods for solving integral equations in high dimensions, J. Integral Equations Appl., 16 (2004), pp. 83-110.
27. K. YOSIDA, Functional Analysis, 6th od., Springer-Verlag, Berlin, 1980.
28. C. ZENGER, Sparse grids, in Parallel Algorithms for Partial Differential Equations: Proceedings of the Sixth GAMM-Seminar (Kiel, 1990), Notes Numer. Fluid Mech. 31, W. Hackbusch, ed., Vieweg, Braunschweig, 1991, pp. 241-251.
29. A. ZHOU, Multi-parameter error resolution for the collocation method of Volteira integral equations, BIT, 37 (1997), pp. 978-987.