A fast solver for the orthogonal spline collocation solution of the biharmonic Dirichlet problem on rectangles

Journal of Computational Physics

Volumn 191, Issue 2, 2003, Pages 601-621

  Bialecki, Bernard ^a  

^a Department of Geology and Geological Engineering   (United States)

Author keywords

Biharmonic Dirichlet problem; Fast Fourier transforms; Orthogonal spline collocation; Preconditioned conjugate gradient method; Schur complement

Indexed keywords

CONJUGATE GRADIENT METHOD;

BIHARMONIC DIRICHLET PROBLEMS; COLLOCATION SOLUTION; FAST SOLVERS; HERMITE; ORTHOGONAL SPLINE COLLOCATION; ORTHOGONAL SPLINES; PIECE-WISE; PRECONDITIONED CONJUGATE GRADIENT METHOD; SCHUR COMPLEMENT; SPLINE COLLOCATION;

FAST FOURIER TRANSFORMS;

EID: 0242362304     PISSN: 00219991     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0021-9991(03)00342-5     Document Type: Article

References (21)

1. I. Altas, J. Dym, M.M. Gupta, R.P. Manohar, Multigrid solution of automatically generated high-order discretizations for the biharmonic equation, SIAM J. Sci. Comput. 19 (1998) 1575-1585.
2. C. Bernardi, Y. Maday, Spectral Methods, in: P.G. Ciarlet, J.L. Lions (Eds.), Handbook of Numerical Analysis, vol. V, Elsevier Science, 1997.
3. B. Bialecki, G. Fairweather, K.R. Bennett, Fast direct solvers for piecewise Hermite bicubic orthogonal spline collocation equations, SIAM J. Numer. Anal. 29 (1992) 156-173.
4. B. Bialecki, A. Karageorghis, A Legendre spectral collocation method for the biharmonic Dirichlet problem, Math. Mod. Numer. Anal. 34 (2000) 637-662.
5. C. de Boor, A Practical Guide to Splines, Applied Mathematical Sciences, vol. 27, Springer, New York, 1978.
6. P.G. Ciarlet, P.A. Raviart, A mixed finite element method for the biharmonic equation, in: C. de Boor (Ed.), Symposium on Mathematical Aspects of Finite Elements in Partial Differential Equations, Academic Press, New York, 1974, pp. 125-143.
7. J.C. Diaz, G. Fairweather, P. Keast, FORTRAN packages for solving certain almost block diagonal linear systems by modified alternate row and column elimination, ACM Trans. Math. Software 9 (1983) 358-375.
8. J.C. Diaz, G. Fairweather, P. Keast, Algorithm 603 COLROW and ARCECO: FORTRAN packages for solving certain almost block diagonal linear systems by modified alternate row and column elimination, ACM Trans. Math. Software 9 (1983) 376-380.
9. J. Douglas Jr., T. Dupont, Collocation Methods for Parabolic Equations in a Single Space Variable, Lecture Notes in Mathematics, vol. 358, Springer, New York, 1974.
10. M.R. Hanisch, Multigrid preconditioning for the biharmonic Dirichlet problem, SIAM J. Numer. Anal. 30 (1993) 184-214.
11. W. Heinrichs, A spectral multigrid method for the Stokes problem in streamfunction formulation, J. Comput. Phys. 102 (1992) 310-318.
12. M.A. Kelmanson, Modified integral equation solution of viscous flows near sharp corners, Comput. Fluids 11 (1983) 307-324.
13. Z.-M. Lou, B. Bialecki, G. Fairweather, Orthogonal spline collocation methods for biharmonic problems, Numer. Math. 80 (1998) 267-303.
14. V.V. Meleshko, Biharmonic problem in a rectangle, Appl. Scientific Res. 58 (1998) 217-249.
15. V.V. Meleshko, A.M. Gomilko, A.A. Gourjii, Normal reactions in a clamped elastic rectangle plate, J. Eng. Math. 40 (2001) 377-398.
16. 1 finite element collocation method for elliptic equations, SIAM J. Numer. Anal. 17 (1980) 605-622.
17. W. Sun, Orthogonal collocation solution of biharmonic equations, Int. J. Comput. Math. 49 (1993) 221-232.
18. S. Timoshenko, S. Woinowsky-Krieger, Theory of Plates and Shells, McGraw-Hill, New York, 1959.
19. C. Van Loan, Computational Frameworks for the Fast Fourier Transform, SIAM, Philadelphia, 1992.
20. R.W. DeVries, P.J. Zandbergen, The numerical solution of the biharmonic equation, using a spectral multigrid method, in: W.F. Ballhaus, M.Y. Hussaini (Eds.), Advances in Fluid Dynamics, Springer, New York, 1989, pp. 25-35.
21. X. Zhang, Multilevel Schwarz methods for the biharmonic Dirichlet problem, SIAM. J. Sci. Comput. 15 (1994) 621-644.