Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



High order fast Laplace solvers for the Dirichlet problem on general regions

Authors: Victor Pereyra, Wlodzimierz Proskurowski and Olof Widlund
Journal: Math. Comp. 31 (1977), 1-16
MSC: Primary 65N15; Secondary 65B05
MathSciNet review: 0431736
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Highly accurate finite difference schemes are developed for Laplace's equation with the Dirichlet boundary condition on general bounded regions in $ {R^n}$. A second order accurate scheme is combined with a deferred correction or Richardson extrapolation method to increase the accuracy. The Dirichlet condition is approximated by a method suggested by Heinz-Otto Kreiss. A convergence proof of his, previously not published, is given which shows that, for the interval size h, one of the methods has an accuracy of at least $ O({h^{5.5}})$ in $ {L_2}$. The linear systems of algebraic equations are solved by a capacitance matrix method. The results of our numerical experiments show that highly accurate solutions are obtained with only a slight additional use of computer time when compared to the results obtained by second order accurate methods.

References [Enhancements On Off] (What's this?)

  • [1] R. BARTELS & J. W. DANIEL, "A conjugate gradient approach to nonlinear elliptic boundary value problems in irregular regions," Conference on the Numerical Solution of Differential Equations (Univ. of Dundee, Scotland, 1973), Lecture Notes in Math., vol. 363, edited by G. A. Watson, Springer-Verlag, Berlin and New York, 1974, pp. 1-11. MR 48 #10032. MR 0440965 (55:13833)
  • [2] J. H. BRAMBLE & B. E. HUBBARD, "Approximation of derivatives by finite difference methods in elliptic boundary value problems," Contributions to Differential Equations, v. 3, 1964, pp. 399-410. MR 29 #4208. MR 0166935 (29:4208)
  • [3] R. BULIRSCH & J. STOER, "Fehlerabschätzungen und Extrapolation mit rationalen Funktionen bei Verfahren vom Richardson-Typus," Numer. Math., v. 6, 1964, pp. 413-427. MR 31 #861. MR 0176589 (31:861)
  • [4] P. CONCUS & G. H. GOLUB, "Use of fast direct methods for the efficient numerical solution of nonseparable elliptic equations," SIAM J. Numer. Anal., v. 10, 1973, pp. 1103-1120. MR 49 #6636. MR 0341890 (49:6636)
  • [5] P. CONCUS & G. H. GOLUB, "A generalized conjugate gradient method for nonsymmetric systems of linear equations," Proc. Second Internat. Sympos. on Computing Methods in Applied Sciences and Engineering, IRIA, Paris, 1975. (To appear.) Report CS-76-535, Computer Science Dept., Standford Univ., 1976. MR 0468130 (57:7968)
  • [6] D. FISCHER, G. GOLUB, O. HALD, C. LEIVA & O. WIDLUND, "On Fourier-Toeplitz methods for separable elliptic problems," Math. Comp., v. 28, 1974, pp. 349-368. MR 0415995 (54:4072)
  • [7] G. E. FORSYTHE & W. R. WASOW, Finite-Difference Methods for Partial Differential Equations, Wiley, New York, 1960. MR 23 #B3156. MR 0130124 (23:B3156)
  • [8] E. ISAACSON & H. B. KELLER, Analysis of Numerical Methods, Wiley, New York, 1966. MR 34 #924. MR 0201039 (34:924)
  • [9] A. JAMESON, Accelerated Iteration Schemes for Transonic Flow Calculations Using Fast Poisson Solvers, ERDA Report C00-3077-82, New York Univ.. 1975.
  • [10] D. P. O'LEARY, Hybrid Conjugate Gradient Algorithms for Elliptic Systems, Report CS-76-548, Computer Science Dept., Stanford Univ., 1976.
  • [11] E. D. MARTIN, "Progress in application of direct elliptic solvers for transonic flow computations," Aerodynamics Analyses Requiring Advanced Computers, NASA SP-347, 1975. (To appear.)
  • [12] E. D. MARTIN, "A fast semidirect method for computing transonic aerodynamic flows," Proc. AIAA Second Computational Fluid Dynamics Conference, 1975. (To appear.) MR 0459290 (56:17484)
  • [13] V. PEREYRA, "Accelerating the convergence of discretization algorithms," MRC Technical Report No. 687, Univ. of Wisconsin, 1966; SIAM J. Numer. Anal., v. 4, 1967, pp. 508-533. MR 36 #4778. MR 0221726 (36:4778)
  • [14] V. PEREYRA, "Iterated deferred corrections for nonlinear operator equations," Numer. Math., v. 10, 1967, pp. 316-323. MR 36 #4812. MR 0221760 (36:4812)
  • [15] V. PEREYRA, "Iterated deferred corrections for nonlinear boundary value problems," Numer. Math., v. 11, 1968, pp. 111-125. MR 37 #1091. MR 0225498 (37:1091)
  • [16] V. PEREYRA, "Highly accurate numerical solution of casilinear elliptic boundary-value problems in n dimensions," Math. Comp., v. 24, 1970, pp. 771-783. MR 44 #6165. MR 0288970 (44:6165)
  • [17] V. PEREYRA, High Order Finite Difference Solution of Differential Equations, Report CS-73-348, Computer Science Dept., Stanford Univ., 1973.
  • [18] W. PROSKUROWSKI & O. WIDLUND, "On the numerical solution of Helmholtz's equation by the capacitance matrix method," ERDA Report, New York Univ., 1975; Math. Comp., v. 30, 1976, pp. 433-468. MR 0421102 (54:9107)
  • [19] E. A. VOLKOV, "A method for improving the accuracy of grid solutions of the Poisson equation," Vyčisl. Mat., v. 1, 1957, pp. 62-80; English transl., Amer. Math. Soc. Transl. (2), v. 35, 1964, pp. 117-136. MR 22 #5131. MR 0114307 (22:5131)
  • [20] W. WASOW, "Discrete approximations to elliptic differential equations," Z. Angew. Math. Phys., v. 6, 1955, pp. 81-97. MR 18, 236. MR 0080369 (18:236e)
  • [21] O. WIDLUND, "A Lanczos method for a class of non-symmetric systems of linear equations." (Preprint.)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65N15, 65B05

Retrieve articles in all journals with MSC: 65N15, 65B05

Additional Information

Article copyright: © Copyright 1977 American Mathematical Society

American Mathematical Society