Multi-level adaptive solutions to boundary-value problems. (English) Zbl 0373.65054


65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
35J40 Boundary value problems for higher-order elliptic equations
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
Full Text: DOI


[1] N. S. BAKHVALOV (BAHVALOV), ”Convergence of a relaxation method with natural constraints on an elliptic operator,” Ž. Vyčisl. Mat. i Mat. Fiz., v. 6, 1966, pp. 861-885. (Russian) MR 35 #6378. · Zbl 0154.41002
[2] A. BRANDT, ”Multi-level adaptive technique (MLAT) for fast numerical solution to boundary value problems,” Proc. 3rd Internat. Conf. on Numerical Methods in Fluid Mechanics (Paris, 1972), Lecture Notes in Physics, vol. 18, Springer-Verlag, Berlin and New York, 1973, pp. 82-89.
[3] A. BRANDT, Multi-Level Adaptive Techniques, IBM Research Report RC6026, 1976.
[4] A. BRANDT, ”Elliptic difference operators and smoothing rates.” (In preparation.)
[5] R. P. Fedorenko, A relaxation method of solution of elliptic difference equations, Ž. Vyčisl. Mat. i Mat. Fiz. 1 (1961), 922 – 927 (Russian).
[6] R. P. Fedorenko, On the speed of convergence of an iteration process, Ž. Vyčisl. Mat. i Mat. Fiz. 4 (1964), 559 – 564 (Russian).
[7] James M. Hyman, Mesh refinement and local inversion of elliptic partial differential equations, J. Computational Phys. 23 (1977), no. 2, 124 – 134. · Zbl 0346.65048
[8] Antony Jameson, Numerical solution of nonlinear partial differential equations of mixed type, Numerical solution of partial differential equations, III (Proc. Third Sympos. (SYNSPADE), Univ. Maryland, College Park, Md., 1975) Academic Press, New York, 1976, pp. 275 – 320.
[9] E. M. MURMAN, ”Analysis of embedded shock waves calculated by relaxation methods,” Proc. AIAA Conf. on Computational Fluid Dynamics (Palm Springs, Calif., 1973), AIAA, 1973, pp. 27-40.
[10] Carl E. Pearson, On non-linear ordinary differential equations of boundary layer type., J. Math. and Phys. 47 (1968), 351 – 358. · Zbl 0165.50503
[11] Y. SHIFTAN, Multi-Grid Method for Solving Elliptic Difference Equations, M. Sc. Thesis, Weizmann Institute of Science, Rehovot, Israel, 1972. (Hebrew)
[12] J. C. SOUTH, JR. & A. BRANDT, Application of a Multi-Level Grid Method to Transonic Flow Calculations, ICASE Report 76-8, NASA Langley Research Center, Hampton, Virginia, 1976.
[13] R. V. Southwell, Relaxation Methods in Engineering Science. A treatise on approximate computation, Oxford Engineering Science Series, Oxford University Press, New York, 1940.
[14] R. V. Southwell, Relaxation Methods in Theoretical Physics, Oxford, at the Clarendon Press, 1946. · Zbl 0061.27706
[15] Eduard Stiefel, Über einige Methoden der Relaxationsrechnung, Z. Angew. Math. Physik 3 (1952), 1 – 33 (German). · Zbl 0046.34104
[16] F. de la Vallee Poussin, An accelerated relaxation algorithm for iterative solution of elliptic equations, SIAM J. Numer. Anal. 5 (1968), 340 – 351. · Zbl 0165.50701 · doi:10.1137/0705029
[17] Eugene L. Wachspress, Iterative solution of elliptic systems, and applications to the neutron diffusion equations of reactor physics, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1966. · Zbl 0161.12203
[18] E. L. WACHSPRESS, ”Variational acceleration of linear iteration,” Proc. Army Workshop Watervliet Arsenal, Albany, New York, 1974.
[19] S. V. Ahamed, Accelerated convergence of numerical solution of linear and non-linear vector field problems, Comput. J. 8 (1965), 73 – 76. · Zbl 0135.38304 · doi:10.1093/comjnl/8.1.73
[20] I. BABUŠKA, W. RHEINBOLDT & C. MESZTENYI, Self-Adaptive Refinements in the Finite Element Method, Technical Report TR-375, Computer Science Department, University of Maryland, 1975.
[21] P. O. FREDERICKSON, Fast Approximate Inversion of Large Sparse Linear Systems, Math. Report 7-75, Lakehead University, Ontario, Canada, 1975.
[22] M. LENTINI & V. PEREYRA, An Adaptive Finite Difference Solver for Nonlinear Two Point Boundary Problems with Mild Boundary Layers, Report STAN-CS-75-530, Computer Science Department, Stanford University, Stanford, California, 1975. · Zbl 0358.65069
[23] R. A. Nicolaides, On multiple grid and related techniques for solving discrete elliptic systems, J. Computational Phys. 19 (1975), no. 4, 418 – 431. · Zbl 0363.65081
[24] A. Settari and K. Aziz, A generalization of the additive correction methods for the iterative solution of matrix equations, SIAM J. Numer. Anal. 10 (1973), 506 – 521. · Zbl 0256.65020 · doi:10.1137/0710046
[25] R. V. SOUTHWELL, ”Stress calculation in frameworks by the method of systematic relaxation of constraints. I, II,” Proc. Roy. Soc. London Ser. A, v. 151, 1935, pp. 56-95.
[26] Achi Brandt, Multi-level adaptive techniques (MLAT) for partial differential equations: ideas and software, Mathematical software, III (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1977) Academic Press, New York, 1977, pp. 277 – 318. Publ. Math. Res. Center, No. 39. · Zbl 0407.68037
[27] C. William Gear, Numerical initial value problems in ordinary differential equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. · Zbl 1145.65316
[28] W. HACKBUSH, Ein Iteratives Verfahren zur Schnellen Auflösung Elliptischer Randwertprobleme, Math. Inst., Universität zu Köln, Report 76-12 (November 1976). A short English version: ”A fast method for solving Poisson’s equation in a general region,” Numerische Behandlung von Differentialgleichungen (R. Bulirsch, R. D. Grigorieff & J. Schröder, Editors), Lecture Notes in Math., Springer-Verlag, Berlin and New York, 1977.
[29] R. A. Nicolaides, On the \?² convergence of an algorithm for solving finite element equations, Math. Comp. 31 (1977), no. 140, 892 – 906. · Zbl 0384.65052
[30] Robert D. Richtmyer and K. W. Morton, Difference methods for initial-value problems, Second edition. Interscience Tracts in Pure and Applied Mathematics, No. 4, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1967. · Zbl 0155.47502
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.