Abstract
Multi-grid methods are characterized by the simultaneous use of additional auxiliary grids corresponding to coarser step widths. Contrary to usual iterative methods the speed of convergence is very fast and does not tend to one if the step size approaches zero. The computational amount of one iteration is proportional toN, the number of grid points. Thus, a solution with accuracy ɛ requires 0 (|log ɛ|N) operations. In this paper we apply a multi-grid method to Helmholtz's equation (Dirichlet boundary data) in a general region and to a differential equation with variable coefficients subject to arbitrary boundary conditions.
Zusammenfassung
Mehrgittermethoden sind charakterisiert durch die gleichzeitige Verwendung zusätzlicher Hilfsgitter gröberer Schrittweite. Im Gegensatz zu üblichen Iterationsverfahren ist die Konvergenzgeschwindigkeit sehr schnell und geht bei kleiner werdender Schrittweite nicht gegen Eins. Der Rechenaufwand eines Iterationsschrittes ist proportinal zuN, der Anzahl der Gitterpunkte. Daher sind zur Berechnung einer Lösung mit der Genauigkeit ɛ 0 (|log ɛ|N) Rechenoperationen notwendig. In diesem Artikel verwenden wir die Mehrgittermethode zur Lösung der Helmholtz-Gleichung bei Dirichlet-Randbedingeungen in einem beliebigen Gebiet und zur Lösung einer Differentialgleichung mit variablen Koeffizienten bei beliebigen Radbedingungen.
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References
Bachvalov, N. S.: On the convergence of a relaxation method with natural constraints on the elliptic operator. Ž. vyčisl. Mat. mat. Fiz.6, 861–883 (1966).
Brandt, A.: Multi-level adaptive technique (MLAT) for fast numerical solution to boundary value problems. Proceedings of the Third International Conference on Numerical Methods in Fluid Mechanics, 1972 (Cabannes, H., Temam, R., eds.). (Lecture Notes in Physics, Vol. 18.), pp. 82–89. Berlin-Heidelberg-New York: Springer 1973.
Brandt, A.: Multi-level adaptive solutions to boundary-value problems. Math. Comput.31, 333–390 (1977).
Collatz, L.: The numerical treatment of differential equations, 3rd ed. Berlin-Heidelberg-New York: Springer 1966.
Fedorenko, R. P.: A relaxation method for solving elliptic difference equations. Ž. vyčisl. Mat. mat. Fiz.1, 922–927 (1961).
Fedorenko, R. P.: The speed of convergence of one iterative process. Ž. vyčisl. Mat. mat. Fiz.4, 559–564 (1964).
Hackbusch, W.: A fast iterative method for solving Poisson's equation in a general region. Numerical Treatment of Differential Equations, 1976 (Bulirsch, R., Grigorieff, R. D., Schröder, J., eds.). (Lecture Notes in Mathematics, Vol. 631.) Berlin-Heidelberg-New York: Springer 1970.
Hackbusch, W.: Ein iteratives Verfahren zur schnellen Auflösung elliptischer Randwertprobleme. Report 76-12, Mathematisches Institut (Angewandte Mathematik), Universität zu Köln, 1976.
Hackbusch, W.: On the convergence of a multi-grid iteration applied to finite element equations (to be published in Numerische Mathematik).
Hackbusch, W.: A fast numerical method for elliptic boundary value problems with variable coefficients. 2nd GAMM-Conference on Numerical Methods in Fluid Mechanics, DFVLR, Köln, 1977.
Meis, Th., Marcowitz, U.: Numerische Lösung partieller Diffenetialgleichungen. Berlin-Heidelberg-New York: Springer 1978.
Nicolaides, R. A.: On multiple grid and related techniques for solving discrete elliptic systems. J. Comp. Phys.19, 418–431 (1975).
Nicolaides, R. A.: On the observed rate of convergence of an iterative method applied to a model elliptic difference equation. Math. Comp.32, 127–133 (1978).
Nicolaides, R. A.: On thel 2 convergence of an algorithm for solving finite element equations. Math. Comp.31, 892–906 (1977).
Pereyra, V., Proskurowski, W., Widlund, O.: High order fast Laplace solvers for the Dirichlet problem on general regions. Math. Comp.31, 1–16 (1977).
Proskurowski, W., Widlund, O.: On the numerical solution of Helmholtz's equation by the capacitance matrix method. Math. Comp.30, 433–468 (1976).
Schröder, J., Trottenberg, U.: Reduktionsverfahren für Differentialgleichungen bei Randwertaufgaben I. Numerische Mathematik22, 37–68 (1973).
Shortley, G. H., Weller, R.: Numerical solution of Laplace's equation. J. Appl. Phys.9, 334–348 (1938).
South, J. C., Brandt, A.: Application of a multi-grid method to transonic flow calculations. ICASE Report 76-8, Nasa's Langley Research Center, 1976.
Wachspress, E. L.: A rational finite element basis. New York: Academic Press 1975.
Frederickson, P. O.: Fast approximate inversion of large sparse linear systems. Report. 7-75, Lakehead University, 1975.
Hackbusch, W.: On the computation of approximate eigenvalues and eigenfunctions of elliptic operators by means of a multi-grid method (to be published in SIAM J. Numer. Anal.).
Hackbusch, W.: Die schnelle Auflösung der Fredholmschen Integralgleichung zweiter Art (to be published in Beiträge zur Numerischen Mathematik9).
Hackbusch, W.: On the fast solving of nonlinear elliptic equations (to be submitted to Numerische Mathematik).
Wesseling, P.: Numerical solution of the stationary Navier-Stokes equations by means of a multiple grid method and Newton iteration. Report NA-18, Delft University, 1977.
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Hackbusch, W. On the multi-grid method applied to difference equations. Computing 20, 291–306 (1978). https://doi.org/10.1007/BF02252378
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DOI: https://doi.org/10.1007/BF02252378