Cyclic reduction and FACR methods for piecewise hermite bicubic orthogonal spline collocation

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(N2log2N) andO(N2log2log2N) arithmetic operations, respectively.

[1]  Patrick Keast,et al.  Algorithm 603: COLROW and ARCECO: FORTRAN Packages for Solving Certain Almost Block Diagonal Linear Systems by Modified Alternate Row and Column Elimination , 1983, TOMS.

[2]  Roland A. Sweet,et al.  Algorithm 541: Efficient Fortran Subprograms for the Solution of Separable Elliptic Partial Differential Equations [D3] , 1979, TOMS.

[3]  Graeme Fairweather,et al.  Fast direct solvers for piecewise Hermite bicubic orthogonal spline collocation equations , 1992 .

[4]  L. Reichel.  The ordering of tridiagonal matrices in the cyclic reduction method for Poisson's equation , 1989 .

[5]  K. Remington,et al.  Fourier Methods for Piecewise Hermite Bicubic Or- Thogonal Spline Collocation , 1994 .

[6]  G. Stoyan.  de Boor, C., A Practical Guide to Splines. Applied Mathematical Sciences 27. Berlin‐Heidelberg‐New York, Springer‐Verlag 1978. XXIV, 392 S., DM 32,50. US $ 17.90 , 1980 .

[7]  Carl de Boor,et al.  A Practical Guide to Splines , 1978, Applied Mathematical Sciences.

[8]  P. Swarztrauber.  THE METHODS OF CYCLIC REDUCTION, FOURIER ANALYSIS AND THE FACR ALGORITHM FOR THE DISCRETE SOLUTION OF POISSON'S EQUATION ON A RECTANGLE* , 1977 .

[9]  Bernard Bialecki.  A fast domain decomposition Poisson solver on a rectangle for Hermite bicubic orthogonal spline collocation , 1993 .

[10]  Gene H. Golub,et al.  On direct methods for solving Poisson's equation , 1970, Milestones in Matrix Computation.

[11]  Clive Temperton,et al.  Direct methods for the solution of the discrete Poisson equation: Some comparisons , 1979 .

[12]  Shirley Dex,et al.  JR 旅客販売総合システム（マルス）における運用及び管理について , 1991 .

[13]  Jim Douglas,et al.  Collocation Methods for Parabolic Equations in a Single Space Variable , 1974 .

[14]  Bernard Bialecki,et al.  Preconditioned Richardson and Minimal Residual Iterative Methods for Piecewise Hermite Bicubic Orthogonal Spline Collocation Equations , 1994, SIAM J. Sci. Comput..

[15]  R. Sweet.  A Cyclic Reduction Algorithm for Solving Block Tridiagonal Systems of Arbitrary Dimension , 1977 .

[16]  Patrick Keast,et al.  FORTRAN Packages for Solving Certain Almost Block Diagonal Linear Systems by Modified Alternate Row and Column Elimination , 1983, TOMS.

[17]  G. Fairweather.  A note on the efficient implementation of certain Padé Methods for linear parabolic problems , 1978 .

[18]  John C. Adams.  Recent enhancements in MUDPACK, a multigrid software package for elliptic partial differential equations , 1991 .

[19]  Bernard Bialecki.  An alternating direction implicit method for orthogonal spline collocation linear systems , 1991 .

[20]  A. A. Samarskii,et al.  Numerical Methods for Grid Equations , 2018 .

[21]  G. Fairweather.  Finite Element Galerkin Methods for Differential Equations , 1978 .

[22]  Ronald F. Boisvert,et al.  Algorithm 651: Algorithm HFFT–high-order fast-direct solution of the Helmholtz equation , 1987, TOMS.

[23]  Mary F. Wheeler,et al.  A $C^1 $ Finite Element Collocation Method for Elliptic Equations , 1980 .

[24]  P. Swarztrauber.  A direct Method for the Discrete Solution of Separable Elliptic Equations , 1974 .

[25]  K. R. Bennett,et al.  Parallel collocation methods for boundary value problems , 1991 .

[26]  Yousef Saad,et al.  A Parallel Block Cyclic Reduction Algorithm for the Fast Solution of Elliptic Equations , 1987, ICS.