Capacitance matrix methods for the Helmholtz equation on general three-dimensional regions
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- by Dianne P. O’Leary and Olof Widlund PDF
- Math. Comp. 33 (1979), 849-879 Request permission
Abstract:
Capacitance matrix methods provide techniques for extending the use of fast Poisson solvers to arbitrary bounded regions. These techniques are further studied and developed with a focus on the three-dimensional case. A discrete analogue of classical potential theory is used as a guide in the design of rapidly convergent iterative methods. Algorithmic and programming aspects of the methods are also explored in detail. Several conjugate gradient methods are discussed for the solution of the capacitance matrix equation. A fast Poisson solver is developed which is numerically very stable even for indefinite Helmholtz equations. Variants thereof allow substantial savings in primary storage for problems on very fine meshes. Numerical results show that accurate solutions can be obtained at a cost which is proportional to that of the fast Helmholtz solver in use.References
- Alexandra Banegas, Fast Poisson solvers for problems with sparsity, Math. Comp. 32 (1978), no. 142, 441–446. MR 483338, DOI 10.1090/S0025-5718-1978-0483338-8
- Randolph E. Bank and Donald J. Rose, Marching algorithms for elliptic boundary value problems. I. The constant coefficient case, SIAM J. Numer. Anal. 14 (1977), no. 5, 792–829. MR 502000, DOI 10.1137/0714055
- Randolph E. Bank and Donald J. Rose, Marching algorithms for elliptic boundary value problems. I. The constant coefficient case, SIAM J. Numer. Anal. 14 (1977), no. 5, 792–829. MR 502000, DOI 10.1137/0714055 C. G. BROYDEN, "Quasi-Newton methods" in Numerical Methods for Unconstrained Optimization (W. Murray, Ed.), Academic Press, New York, 1972, pp. 87-106. O. BUNEMAN, A Compact Non-Iterative Poisson Solver, Report SUIPR-294, Inst. Plasma Research, Standford University, 1969.
- B. L. Buzbee and Fred W. Dorr, The direct solution of the biharmonic equation on rectangular regions and the Poisson equation on irregular regions, SIAM J. Numer. Anal. 11 (1974), 753–763. MR 362944, DOI 10.1137/0711061
- B. L. Buzbee, F. W. Dorr, J. A. George, and G. H. Golub, The direct solution of the discrete Poisson equation on irregular regions, SIAM J. Numer. Anal. 8 (1971), 722–736. MR 292316, DOI 10.1137/0708066
- 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, DOI 10.1137/0707049
- J. Albrecht and L. Collatz (eds.), Numerische Behandlung von Differentialgleichungen. Band 3, Internationale Schriftenreihe zur Numerischen Mathematik [International Series of Numerical Mathematics], vol. 56, Birkhäuser Verlag, Basel, 1981 (German). MR 784038, DOI 10.1007/978-3-0348-5454-2
- Paul Concus, Gene H. Golub, and Dianne P. O’Leary, A generalized conjugate gradient method for the numerical solution of elliptic partial differential equations, Sparse matrix computations (Proc. Sympos., Argonne Nat. Lab., Lemont, Ill., 1975) Academic Press, New York, 1976, pp. 309–332. MR 0501821 J. W. COOLEY, P. A. W. LEWIS & P. D. WELSH, "The fast Fourier transform algorithm: Programming considerations in the calculation of sine, cosine and Laplace transforms," J. Sound Vib., v. 12, 1970, pp. 315-337.
- James W. Daniel, The conjugate gradient method for linear and nonlinear operator equations, SIAM J. Numer. Anal. 4 (1967), 10–26. MR 217987, DOI 10.1137/0704002 S. C. EISENSTAT, "Complexity bounds for Gaussian elimination." (To appear.)
- S. C. Eisenstat, M. H. Schultz, and A. H. Sherman, Applications of an element model for Gaussian elimination, Sparse matrix computations (Proc. Sympos., Argonne Nat. Lab., Lemont, Ill., 1975) Academic Press, New York, 1976, pp. 85–96. MR 0458822
- M. Engeli, Th. Ginsburg, H. Rutishauser, and E. Stiefel, Refined iterative methods for computation of the solution and the eigenvalues of self-adjoint boundary value problems, Mitt. Inst. Angew. Math. Zürich 8 (1959), 107. MR 145689
- 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, DOI 10.1090/S0025-5718-1974-0415995-2
- George E. Forsythe and Wolfgang R. Wasow, Finite-difference methods for partial differential equations, Applied Mathematics Series, John Wiley & Sons, Inc., New York-London, 1960. MR 0130124
- P. R. Garabedian, Partial differential equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR 0162045 J. A. GEORGE, The Use of Direct Methods for the Solution of the Discrete Poisson Equation on Non-Rectangular Regions, Computer Science Department Report 159, Stanford University, 1970.
- Alan George, Nested dissection of a regular finite element mesh, SIAM J. Numer. Anal. 10 (1973), 345–363. MR 388756, DOI 10.1137/0710032
- R. M. Hayes, Iterative methods of solving linear problems on Hilbert space, Contributions to the solution of systems of linear equations and the determination of eigenvalues, National Bureau of Standards Applied Mathematics Series, No. 39, U.S. Government Printing Office, Washington, D.C., 1954, pp. 71–103. MR 0066563
- Magnus R. Hestenes, The conjugate-gradient method for solving linear systems, Proceedings of Symposia in Applied Mathematics. Vol. VI. Numerical analysis, McGraw-Hill Book Company, Inc., New York, for the American Mathematical Society, Providence, R.I., 1956, pp. 83–102. MR 0084178
- Magnus R. Hestenes and Eduard Stiefel, Methods of conjugate gradients for solving linear systems, J. Research Nat. Bur. Standards 49 (1952), 409–436 (1953). MR 0060307, DOI 10.6028/jres.049.044
- R. W. Hockney, A fast direct solution of Poisson’s equation using Fourier analysis, J. Assoc. Comput. Mach. 12 (1965), 95–113. MR 213048, DOI 10.1145/321250.321259 R. W. HOCKNEY, "Formation and stability of virtual electrodes in a cylinder," J. Appl. Phys., v. 39, 1968, pp. 4166-4170. R. W. HOCKNEY, "The potential calculation and some applications," Methods in Computational Physics, Vol. 9, Academic Press, New York, 1970. R. W. HOCKNEY, POT 4-A Fast Direct Poisson Solver for the Rectangle Allowing Some Mixed Boundary Conditions and Internal Electrodes, IBM Research, R. C. 2870, 1970.
- Alan J. Hoffman, Michael S. Martin, and Donald J. Rose, Complexity bounds for regular finite difference and finite element grids, SIAM J. Numer. Anal. 10 (1973), 364–369. MR 347065, DOI 10.1137/0710033
- Alston S. Householder, The theory of matrices in numerical analysis, Blaisdell Publishing Co. [Ginn and Co.], New York-Toronto-London, 1964. MR 0175290
- H. Y. Huang, Unified approach to quadratically convergent algorithms for function minimization, J. Optim. Theory Appl. 5 (1970), 405–423. MR 288939, DOI 10.1007/BF00927440
- Shmuel Kaniel, Estimates for some computational techniques in linear algebra, Math. Comp. 20 (1966), 369–378. MR 234618, DOI 10.1090/S0025-5718-1966-0234618-4
- Oliver Dimon Kellogg, Foundations of potential theory, Die Grundlehren der mathematischen Wissenschaften, Band 31, Springer-Verlag, Berlin-New York, 1967. Reprint from the first edition of 1929. MR 0222317, DOI 10.1007/978-3-642-86748-4
- Cornelius Lanczos, An iteration method for the solution of the eigenvalue problem of linear differential and integral operators, J. Research Nat. Bur. Standards 45 (1950), 255–282. MR 0042791, DOI 10.6028/jres.045.026 D. G. LUENBERGER, Introduction to Linear and Nonlinear Programming, Addison-Wesley, Reading, Mass., 1973.
- E. Dale Martin, A generalized-capacity-matrix technique for computing aerodynamic flows, Internat. J. Comput. & Fluids 2 (1974), no. 1, 79–97. MR 347233, DOI 10.1016/0045-7930(74)90006-1
- Geraldine E. Myers, Properties of the conjugate-gradient and Davidon methods, J. Optim. Theory Appl. 2 (1968), 209–219. MR 250451, DOI 10.1007/BF00937366
- J.-C. Nédélec and J. Planchard, Une méthode variationnelle d’éléments finis pour la résolution numérique d’un problème extérieur dans $R^{3}$, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 7 (1973), no. R-3, 105–129 (French, with English summary). MR 424022 C. C. PAIGE, The Computation of Eigenvalues and Eigenvectors of Very Large Sparse Matrices, Ph.D. thesis, London University, Institute of Computer Science, 1971.
- Victor Pereyra, Wlodzimierz Proskurowski, and Olof Widlund, High order fast Laplace solvers for the Dirichlet problem on general regions, Math. Comp. 31 (1977), no. 137, 1–16. MR 431736, DOI 10.1090/S0025-5718-1977-0431736-X
- G. N. Polozhii, The method of summary representation for numerical solution of problems of mathematical physics, International Series of Monographs in Pure and Applied Mathematics, Vol. 79, Pergamon Press, Oxford-New York-Paris, 1965. Authorized translation incorporating revisions and new material supplied by the author; Translated from the Russian by G. J. Tee; Translation edited by K. L. Stewart. MR 0230481
- W. Proskurowski, On the numerical solution of the eigenvalue problem of the Laplace operator by a capacitance matrix method, Computing 20 (1978), no. 2, 139–151 (English, with German summary). MR 619766, DOI 10.1007/BF02252343
- Włodzimierz Proskurowski, Numerical solution of Helmholtz’s equation by implicit capacitance matrix methods, ACM Trans. Math. Software 5 (1979), no. 1, 36–49. MR 520747, DOI 10.1145/355815.355818 W. PROSKUROWSKI, Four FORTRAN Programs for Numerically Solving Helmholtz’s Equation in an Arbitrary Bounded Planar Region, Report LBL-7516, Lawrence Berkeley Laboratory, 1978.
- 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, DOI 10.1090/S0025-5718-1976-0421102-4 W. PROSKUROWSKI & O. WIDLUND, "A finite element-capacitance method for the Neumann problem for Laplace’s equation." (To appear.) A. S. L. SHIEH, Fast Poisson Solver on Nonrectangular Domains, Ph.D. thesis, New York University, 1976.
- A. S. L. Shieh, On the convergence of the conjugate gradient method for singular capacitance matrix equations from the Neumann problem of the Poisson equation, Numer. Math. 29 (1977/78), no. 3, 307–327. MR 659094, DOI 10.1007/BF01389215
- A. S. L. Shieh, Fast Poisson solvers on general two-dimensional regions for the Dirichlet problem, Numer. Math. 31 (1978/79), no. 4, 405–429. MR 516582, DOI 10.1007/BF01404568
- Gilbert Strang and George J. Fix, An analysis of the finite element method, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1973. MR 0443377
- Paul N. Swarztrauber, A direct method for the discrete solution of separable elliptic equations, SIAM J. Numer. Anal. 11 (1974), 1136–1150. MR 368399, DOI 10.1137/0711086
- Paul 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), no. 3, 490–501. MR 438732, DOI 10.1137/1019071
- Paul N. Swarztrauber and Roland A. Sweet, The direct solution of the discrete Poisson equation on a disk, SIAM J. Numer. Anal. 10 (1973), 900–907. MR 368454, DOI 10.1137/0710075 P. SWARZTRAUBER & R. SWEET, Efficient FORTRAN Subprograms for the Solution of Elliptic Partial Differential Equations, Report NCAR-TN/IA-109, National Center for Atmospheric Research, Boulder, Colorado, 1975.
- Roland A. Sweet, A cyclic reduction algorithm for solving block tridiagonal systems of arbitrary dimension, SIAM J. Numer. Anal. 14 (1977), no. 4, 706–720. MR 445859, DOI 10.1137/0714048
- Vidar Thomée, On the convergence of difference quotients in elliptic problems, Numerical Solution of Field Problems in Continuum Physics (Proc. Sympos. Appl. Math., Durham, N.C., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 186–200. MR 0260206 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, Ed.), Plenum Press, New York, 1972, pp. 121-134.
- Olof Widlund, Capacitance matrix methods for Helmholtz’s equation on general bounded regions, Numerical treatment of differential equations (Proc. Conf., Math. Forschungsinst., Oberwolfach, 1976) Lecture Notes in Math., Vol. 631, Springer, Berlin, 1978, pp. 209–219. MR 0474873
- Robert B. Wilhelmson and James H. Ericksen, Direct solutions for Poisson’s equation in three dimensions, J. Comput. Phys. 25 (1977), no. 4, 319–331. MR 474874, DOI 10.1016/0021-9991(77)90001-8
Additional Information
- © Copyright 1979 American Mathematical Society
- Journal: Math. Comp. 33 (1979), 849-879
- MSC: Primary 65N99; Secondary 65F10
- DOI: https://doi.org/10.1090/S0025-5718-1979-0528044-7
- MathSciNet review: 528044