An energy-minimizing interpolation for robust multigrid methods

SIAM Journal on Scientific Computing

Volumn 21, Issue 4, 1999, Pages 1632-1649

  Wan, W L ^a,b   Chan, Tony F ^b   Smith, Barry ^c  

^a STANFORD UNIVERSITY   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

^c ARGONNE NATIONAL LABORATORY   (United States)

Author keywords

Elliptic differential equations; Energy minimization; Interpolation; Multigrid; Nonsmooth coefficient

Indexed keywords

ENERGY MINIMIZING INTERPOLATION; ROBUST MULTIGRID METHODS;

APPROXIMATION THEORY; INTEGRAL EQUATIONS; MATHEMATICAL OPERATORS; PARTIAL DIFFERENTIAL EQUATIONS; PROBLEM SOLVING;

INTERPOLATION;

EID: 0033293066     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1064827598334277     Document Type: Article

References (45)

1. R. E. ALCOUFFE, A. BRANDT, J. E. DENDY, JR., AND J. W. PAINTER, The multi-grid method for the diffusion equation with strongly discontinuous coefficients, SIAM J. Sci. Statist. Comput., 2 (1981), pp. 430-454.
2. D. BRAESS, Towards algebraic multigrid for elliptic problems of second order, Computing, 55 (1995), pp. 379-393.
3. J. BRAMBLE, Multigrid Methods, Longman Scientific and Technical, Harlow, UK, 1993.
4. J. BRAMBLE, J. PASCIAK, J. WANG, AND J. XU, Convergence estimates for multigrid algorithms without regularity assumptions, Math. Comp., 57 (1991), pp. 23-45.
5. J. BRAMBLE, J. PASCIAK, J. WANG, AND J. XU, Convergence estimates for product iterative methods with applications to domain decomposition and multigrid, Math. Comp., 57 (1991), pp. 1-21.
6. A. BRANDT, Multi-level adaptive solutions to boundary-value problems, Math. Comp., 31 (1977), pp. 333-390.
7. A. BRANDT, S. MCCORMICK, AND J. W. RUGE, Algebraic multigrid (AMG) for automatic multigrid solution with application to geodetic computations, in Sparsity and Its Applications, D. J. Evans, ed., Cambridge University Press, Cambridge, UK, 1984.
8. R. CHAN, T. CHAN, M. NG, W. TANG, AND C. WONG, Preconditioned iterative methods for high-resolution image reconstruction with multisensors, in Proceedings SPIE 3461, Advanced Signal Processing Algorithms, Architectures and Implementation VI, 1998.
9. R. H. CHAN, T. F. CHAN, AND W. L. WAN, Multigrid for Differential-Convolution Problems Arising from Image Processing, Technical Report CAM 97-20, Department of Mathematics, UCLA, Los Angeles, CA, 1997.
10. T. F. CHAN, S. Go, AND L. ZIKATANOV, Lecture notes on multilevel methods for elliptic problems on unstructured grids, in Lecture Course 28th Computational Fluid Dynamics, von Karman Institute for Fluid Dynamics, Belgium, 1997.
11. T. F. CHAN, J. XU, AND L. ZIKATANOV, An Agglomeration Multigrid Method for Unstructured Grids, Technical Report CAM 98-8, Department of Mathematics, UCLA, Los Angeles, CA, 1998.
12. J. E. DENDY, JR., Black box multigrid, J. Comput. Phys., 48 (1982), pp. 366-386.
13. J. E. DENDY, JR., Black box multigrid for nonsymmetric problems, Appl. Math. Comput., 13 (1983), pp. 261-283.
14. J. E. DENDY, JR., M. P. IDA, AND J. M. RUTLEDGE, A semi-coarsening multigrid algorithm for SIMD machines, SIAM J. Sci. Statist. Comput., 13 (1992), pp. 1460-1469.
15. J. E. DENDY, JR., S. F. MCCORMICK, J. W. RUGE, T. F. RUSSELL, AND S. SCHAFFER, Multigrid methods for three-dimensional petroleum reservoir simulation, in 10th Symposium on Reservoir Simulation, Houston, TX, Society of Petroleum Engineers (SPE), 1989, pp. 19-25.
16. J. E. DENDY, JR., AND C. C. TAZARTES, Grandchild of the frequency decomposition multigrid method, SIAM J. Sci. Comput., 16 (1994), pp. 307-319.
17. M. DRYJA, B. SMITH,. AND O. WIDLUND, Schwarz analysis of iterative sub structuring algorithms for problems in three dimensions, SIAM J. Numer. Anal., 31 (1994), pp. 1662-1694.
18. M. DRYJA AND O. B. WIDLUND, Towards a unified theory of domain decomposition algorithms for elliptic problems, in Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, T. F. Chan, R. Glowinski, J. Periaux, and O. B. Widlund, eds., SIAM, Philadelphia, PA, 1990, pp. 3-21.
19. B. ENGQUIST AND E. LUO, Convergence of a multigrid method for elliptic equations with highly oscillatory coefficients, SIAM J. Numer. Anal., 34 (1997), pp. 2254-2273.
20. J. FUHRMANN, A modular algebraic multilevel method, in Virtual Proceedings of the Ninth International GAMM-Workshop on Parallel Multigrid Methods, 1996.
21. GMD SOFTWARE, http://mgnet.ccs.uky.edu/mgnet/Codes/gmd.
22. W. HACKBUSCH, Ein iteratives Verfahren zur schnellen Auflosung elliptischer Randwert probleme, Technical Report 76-12, Universität zu Köln, Köln, Germany, 1976.
23. W. HACKBUSCH, Multi-grid Methods and Applications, Springer-Verlag, Berlin, 1985.
24. W. HACKBUSCH, The frequency decomposition multigrid method, part I: Application to anisotropic equations, Numer. Math., 56 (1989), pp. 229-245.
25. T. Y. HOU AND X. H. WU, A multiscale finite element method for elliptic problems in composite materials and porous media, J. Comput. Phys., 134 (1997), pp. 169-189.
26. T. Y. HOU, X. H. WU, AND Z. CAI, Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients, Math. Comp., 68 (1996), pp. 913-943.
27. R. KETTLER, Analysis and comparison of relaxation schemes in robust multigrid and preconditioned conjugate gradient methods, in Multigrid Methods, W. Hackbusch and U. Trottenberg, eds., Springer-Verlag, New York, 1981, pp. 502-534.
28. R. KETTLER AND J.A. MEIJERINK, A Multigrid Method and a Combined Multigrid-Conjugate Gradient Method for Elliptic Problems with Strongly Discontinuous Coefficients in General Domains, Shell Publ. 604, KSEPL, Rijswijk, The Netherlands, 1981.
29. S. KNAPEK, Matrix-dependent multigrid-homogenization for diffusion problems, SIAM J. Sci. Comput., 20 (1998), pp. 515-533.
30. E. Luo, Multigrid Method for Elliptic Equation with Oscillatory Coefficients, Ph.D. thesis, Department of Mathematics, UCLA, Los Angeles, 1993.
31. J. MANDEL, M. BREZINA, AND P. VANEK, Energy Optimization of Algebraic Multigrid Bases, Technical Report UCD/CCM 125, Center for Computational Mathematics, University of Colorado, Denver, CO, 1998.
32. S. MCCORMICK, Multigrid Methods, SIAM, Philadelphia, PA, 1988.
33. A. REUSKEN, Multigrid with matrix-dependent transfer operators for a singular perturbation problem, Computing, 50 (1993), pp. 199-211.
34. A. REUSKEN, Multigrid with matrix-dependent transfer operators for convection-diffusion problems, in Multigrid Methods, vol. IV, Internat. Ser. Numer. Math. 116, Birkhäuser-Verlag, Basel, 1994, pp. 269-280.
35. J. W. RUGE AND K. STÜBEN, Algebraic multigrid, in Multigrid Methods, S. McCormick, ed., SIAM, Philadelphia, PA, 1987, pp. 73-130.
36. B. SMITH, P. BJORSTAD, AND W. GROPP, Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations, Cambridge University Press, Cambridge, UK, 1996.
37. R. A. SMITH AND A. WEISER, Semicoarsening multigrid on a hypercube, SIAM J. Sci. Statist. Comput., 13 (1992), pp. 1314-1329.
38. K. STUBEN, Algebraic multigrid: Experience and comparisons, Appl. Math. Comput., 13 (1983), pp. 419-451.
39. S. TA'ASAN, Multigrid Methods for Highly Oscillatory Problems, Ph.D. thesis, Weizmann Institute of Science, Rehovat, Israel, 1984.
40. P. VANĚK, J. MANDEL, AND M. BREZINA, Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems, Computing, 56 (1996), pp. 179-196.
41. W. L. WAN, Scalable and Multilevel Iterative Methods, Ph.D. thesis, Department of Mathematics, UCLA, Los Angeles, CA, 1998.
42. P. WESSELING, An Introduction to Multigrid Methods, Wiley, Chichester, UK, 1992.
43. J. XU, Theory of Multilevel Methods, Ph.D. thesis, Cornell University, Ithaca, NY, 1989.
44. J. XU, Iterative methods by space decomposition and subspace correction, SIAM Rev., 34 (1992), pp. 581-613.
45. P. M. DE ZEEUW, Matrix-dependent prolongations and restrictions in a blackbox multigrid solver, J. Comput. Appl. Math., 33 (1990), pp. 1-27.