Abstract
Cyclic reduction and Fourier analysis-cyclic reduction (FACR) methods are presented for the solution of the linear systems which arise when orthogonal spline collocation with piecewise Hermite bicubics is applied to boundary value problems for certain separable partial differential equations on a rectangle. On anN×N uniform partition, the cyclic reduction and Fourier analysis-cyclic reduction methods requireO(N 2log2 N) andO(N 2log2log2 N) arithmetic operations, respectively.
Similar content being viewed by others
References
J.C. Adams, Recent enhancements in MUDPACK, a multigrid software package for elliptic partial differential equations, Appl. Math. Comp. 43 (1991) 79–93.
K.R. Bennett, Parallel collocation methods for boundary value problems, Ph.D. Thesis, University of Kentucky, Lexington, KY (1991).
B. Bialecki, An alternating direction implicit method for solving orthogonal spline collocation linear systems, Numer. Math. 59 (1991) 413–429.
B. Bialecki, A fast domain decomposition Poisson solver on a rectangle for Hermite bicubic orthogonal spline collocation, SIAM J. Numer. Anal. 30 (1993) 425–434.
B. Bialecki, Preconditioned Richardson and minimal residual iterative methods for piecewise Hermite bicubic orthogonal spline collocation equations, SIAM J. Sci. Comp. 15 (1994) 668–680.
B. Bialecki, G. Fairweather and K.R. Bennett, Fast direct solvers for piecewise Hermite bicubic orthogonal spline collocation equations, SIAM J. Numer. Anal. 29 (1992) 156–173.
B. Bialecki, G. Fairweather and K.R. Remington, Fourier methods for piecewise Hermite bicubic orthogonal spline collocation, East-West J. Numer. Math. 2 (1994) 1–20.
R.F. Boisvert, Algorithm 651, Algorithm HFFT-Higher-order fast-direct solution of the Helmholtz equation, ACM Trans. Math. Software 13 (1987) 235–249.
C. De Boor,A Practical Guide to Splines, Applied Mathematical Sciences 27 (Springer, New York, 1978).
B.L. Buzbee, G.H. Golub and C.W. Nielson, On direct methods for solving Poisson's equations, SIAM J. Number. Anal. 7 (1970) 627–656.
O. Buneman, A compact non-iterative Poisson solver, Rep. 294, Stanford University Institute for Plasma Research, Stanford, CA (1969).
J.C. Diaz, G. Fairweather and 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.
J.C. Diaz, G. Fairweather and 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.
J. Douglas, Jr. and T. Dupont,Collocation Methods for Parabolic Equations in a Single Space Variable, Lecture Notes in Mathematics 385 (Springer, New York, 1974).
G. Fairweather, A note on the efficient implementation of certain Padé methods for linear parabolic problems, BIT 18 (1978) 106–109.
G. Fairweather,Finite Element Galerkin Methods for Differential Equations, Lecture Notes in Pure and Applied Mathematics, Vol. 34 (Marcel Dekker, New York, 1978).
E. Gallopoulos and Y. Saad, A parallel block cyclic reduction algorithm for the fast solution of elliptic equations, Parallel Comp. 10 (1989) 143–159.
P. Percell and M.F. Wheeler, AC 1 finite element collocation method for elliptic equations, SIAM J. Numer. Anal. 17 (1980) 605–622.
L. Reichel, The ordering of tridiagonal matrices in the cyclic reduction method for Poisson's equation, Numer. Math. 56 (1989) 215–227.
A.A. Samarskii and E.S. Nikolaev,Numerical Methods for Grid Equations, Vol. 1:Direct Methods (Birkhäuser, Basel, 1989).
P.N. Swarztrauber, A direct method for the discrete solution of separable elliptic equations, SIAM J. Numer. Anal. 11 (1974) 1136–1150.
P.N. Swarztrauber, The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson's equation on a rectangle, SIAM Rev. 19 (1977) 490–501.
P.N. Swarztrauber, Fast Poisson solvers, in:Studies in Numerical Analysis, ed. G.H. Golub, MAA Studies in Mathematics, Vol. 24 (Mathematical Association of America, 1984) pp. 319–370.
P.N. Swarztrauber and R.A. Sweet, Algorithm 541, efficient Fortran subprograms for the solution of separable elliptic partial differential equations, ACM Trans. Math. Software 5 (1979) 352–364.
R.A. Sweet, A cyclic reduction algorithm for solving block tridiagonal systems of arbitrary dimension, SIAM J. Numer. Anal. 14 (1977) 706–720.
C. Temperton, Direct methods for the solution of the discrete Poisson equation: some comparisons, J. Comp. Phys. 31 (1979) 1–20.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bialecki, B. Cyclic reduction and FACR methods for piecewise hermite bicubic orthogonal spline collocation. Numer Algor 8, 167–184 (1994). https://doi.org/10.1007/BF02142689
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02142689
Keywords
- Poisson's equation
- Dirichlet
- Neumann
- periodic boundary conditions
- separable equations
- piecewise Hermite cubics
- Gauss points
- orthogonal spline collocation
- almost block diagonal matrices
- Fourier analysis and cyclic reduction methods