Fast Poisson solvers for problems with sparsity

Author:
Alexandra Banegas

Journal:
Math. Comp. **32** (1978), 441-446

MSC:
Primary 65F10; Secondary 65N20

DOI:
https://doi.org/10.1090/S0025-5718-1978-0483338-8

MathSciNet review:
0483338

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Abstract | References | Similar Articles | Additional Information

Abstract: Fast Poisson solvers, which provide the numerical solution of Poisson's equation on regions that permit the separation of variables, have proven very useful in many applications. In certain of these applications the data is sparse and the solution is only required at relatively few mesh points. For such problems this paper develops algorithms that allow considerable savings in computer storage as well as execution speed. Results of numerical experiments are given.

**[1]**James R. Bunch and Donald J. Rose (eds.),*Sparse matrix computations*, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. MR**0448792****[2]**O. BUNEMAN, A*Compact Noniterative Poisson Solver*, Rep. SUIPR-294, Inst. Plasma Research, Stanford Univ., 1969.**[3]**B. L. Buzbee, G. H. Golub, and C. W. Nielson,*On direct methods for solving Poisson’s equations*, SIAM J. Numer. Anal.**7**(1970), 627–656. MR**287717**, https://doi.org/10.1137/0707049**[4]**J. W. COOLEY, P. A. W. LEWIS & P. D. WELCH, "The fast Fourier transform algorithm: Programming consideration in the calculation of sine, cosine and Laplace transform,"*J. Sound Vib.*, v. 12, 1970, pp. 315-337.**[5]**Ȧke Björck and Germund Dahlquist,*Numerical methods*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1974. Translated from the Swedish by Ned Anderson; Prentice-Hall Series in Automatic Computation. MR**0368379****[6]**D. Fischer, G. Golub, O. Hald, C. Leiva, and O. Widlund,*On Fourier-Toeplitz methods for separable elliptic problems*, Math. Comp.**28**(1974), 349–368. MR**415995**, https://doi.org/10.1090/S0025-5718-1974-0415995-2**[7]**R. W. Hockney,*A fast direct solution of Poisson’s equation using Fourier analysis*, J. Assoc. Comput. Mach.**12**(1965), 95–113. MR**213048**, https://doi.org/10.1145/321250.321259**[8]**R. W. HOCKNEY, "The potential calculation and some applications,"*Methods in Computational Physics*, Vol. 9, Academic Press, New York, 1970.**[9]**D. P. O'LEARY & O. WIDLUND, ERDA-NYU report. (To appear.)**[10]**W. PROSKUROWSKI,*Numerical Solution of Helmholtz's Equation by Implicit Capacitance Matrix Methods*, Report 6402, Lawrence Berkeley Laboratory, February 1977.**[11]**Wlodzimierz Proskurowski and Olof Widlund,*On the numerical solution of Helmholtz’s equation by the capacitance matrix method*, Math. Comp.**30**(1976), no. 135, 433–468. MR**421102**, https://doi.org/10.1090/S0025-5718-1976-0421102-4**[12]**P. SWARZTRAUBER & R. SWEET,*Efficient FORTRAN Subprograms for the Solution of Elliptic Partial Differential Equations*, Report NCAR-1N/1A-109, National Center for Atmospheric Research, Boulder, Colorado, 1975.**[13]**O. WIDLUND, "On the use of fast methods for separable finite difference equations for the solution of general elliptic problems,"*Sparse Matrices and Their Applications*(D. J. Rose and R. A. Willoughby, Editors), Plenum Press, New York, 1972.**[14]**O. WIDLUND,*Capacitance Matrix Methods for Helmholtz' Equation on General Bounded Regions*, Proc. from a July 1976 Meeting in Oberwolfach. (To appear.)

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DOI:
https://doi.org/10.1090/S0025-5718-1978-0483338-8

Article copyright:
© Copyright 1978
American Mathematical Society