zbMATH — the first resource for mathematics

Numerical solution of nonlinear elliptic partial differential equations by a generalized conjugate gradient method. (English) Zbl 0385.65048

65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
35J60 Nonlinear elliptic equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
Algorithm 527
Full Text: DOI
[1] Axelsson, O.: On preconditioning and convergence acceleration in sparse matrix problems. Report 74-10, Data Handling Division, CERN, Geneva (1974). · Zbl 0354.65020
[2] Bank, R. E.: A FORTRAN implementation of the generalized marching algorithm. Trans. Math. Software. To appear.
[3] Bartels, R., Daniel, J. W.: A conjugate gradient approach to nonlinear elliptic boundary value problems in irregular regions. Proc. Conf. on the Numerical Solution of Differential Equations (Lecture Notes, Vol. 363), pp. 1–11. Berlin-Heidelberg-New York: Springer 1974. · Zbl 0287.65052
[4] Bertsekas, D.: Partial conjugate gradient methods for a class of optimal control problems. IEEE Trans. Automat Control AC-19 1974, 209–217. · Zbl 0296.49022 · doi:10.1109/TAC.1974.1100556
[5] Buzbee, B. L., Golub, G. H., Nielson, C. W.: On direct methods for solving Poisson’s equations. SIAM J. Numer. Anal.7, 627–656 (1970). · Zbl 0217.52902 · doi:10.1137/0707049
[6] Concus, P.: Numerical solution of the minimal surface equation. Math. Comp.21, 340–350 (1967). · Zbl 0189.16605 · doi:10.1090/S0025-5718-1967-0229394-6
[7] Concus, P.: Numerical solution of the minimal surface equation by block nonlinear successive overrelaxation. Information Processing 68, Proc. IFIP Congress 1968, pp. 153–158. Amsterdam: North-Holland 1969.
[8] Concus, P., Golub, G. H.: A generalized conjugate gradient method for nonsymmetric systems of linear equations. Proc. Second International Symposium on Computing Methods in Applied Sciences and Engineering IRIA, Paris, Dec. 1975 (Lecture Notes in Economics and Math. Systems, Vol. 134), pp. 56–65. Berlin-Heidelberg-New York: Springer 1976.
[9] Concus, P., Golub, G. H., O’Leary, D. P.: A generalized conjugate gradient method for the numerical solution of elliptic partial differential equations, in: Sparse Matrix Computations (Bunch, J. R., Rose, D. J., eds.), pp. 309–332. New York: Academic Press 1976.
[10] Daniel, J. W.: The conjugate gradient method for linear and nonlinear operator equations. Ph. D. Thesis, Stanford University, and SIAM J. Numer. Anal.4, 10–26 (1967). · Zbl 0154.40302
[11] Dixon, L. C. W.: Conjugate gradient algorithms: quadratic termination without linear searches. J. Inst. Maths. Applics.15, 9–18 (1975). · Zbl 0294.90076 · doi:10.1093/imamat/15.1.9
[12] Ehrlich, L. W.: On some experience using matrix splitting and conjugate gradient (abstract). SIAM Review18, 801 (1976).
[13] Fischer, D., Golub, G. H., Hald, O., Leiva, C., Widlund, O.: On Fourier-Toeplitz methods for separable elliptic problems. Math. Comp.28, 349–368 (1974). · Zbl 0277.65065 · doi:10.1090/S0025-5718-1974-0415995-2
[14] Fletcher, R., Reeves, C. M.: Function minimization by conjugate gradients. Computer J.7, 149–154 (1964). · Zbl 0132.11701 · doi:10.1093/comjnl/7.2.149
[15] Forsythe, G. E., Wasow, W. R.: Finite-difference Methods for Partial Differential Equations. New York: Wiley 1960. · Zbl 0099.11103
[16] Goldfarb, D.: A conjugate gradient method for nonlinear programming. Princeton University Press, Thesis, 1966.
[17] Hayes, L., Young, D. M., Schleicher, E.: The use of the accelerated SSOR method to solve large linear systems (abstract). SIAM Review18, 808 (1976).
[18] Hestenes, M., Stiefel, E.: Methods of conjugate gradients for solving linear systems. J. Res. Nat. Bur. Stand.49, 409–436 (1952). · Zbl 0048.09901
[19] Hockney, R. W.: The potential calculation and some applications. Methods in Computational Physics, 9. (Adler, B., Fernbach, S., Rotenberg, M., eds.), pp. 136–211. New York: Academic Press 1969.
[20] Lenard, M.: Practical convergence conditions for restarted conjugate gradient methods. MRC Report 1373, University of Wisconsin (December 1973).
[21] Meijerink, J. A., van der Vorst, H. A.: An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix. Math. Comp.31, 148–162 (1977). · Zbl 0349.65020
[22] Nazareth, L.: A conjugate direction algorithm without line searches. J. Opt. Th. Applic. (to appear). · Zbl 0348.65061
[23] O’Leary, D. P.: Hybrid conjugate gradient algorithms. Doctoral dissertation, Computer Science Department, Stanford University, Report No. STAN-CS-76-548 (1976).
[24] Ortega, J. M., Rheinboldt, W. C.: Iterative Solution of Nonlinear Equations in Several Variables. New York: Academic Press 1970. · Zbl 0241.65046
[25] Powell, M. J. D.: Restart procedures for the conjugate gradient method. Report C. S. S. 24, AERE, Harwell, England (1975). · Zbl 0396.90072
[26] Reid, J. K.: On the method of conjugate gradients for the solution of large sparse systems of linear equations, in: Large Sparse Sets of Linear Equations (Reid, J. K., ed.), pp. 231–254. New York: Academic Press 1971.
[27] Schecter, S.: Relaxation methods for convex problems. SIAM J. Numer. Anal.5, 601–612 (1968). · Zbl 0179.22701 · doi:10.1137/0705048
[28] Swartztrauber, P., Sweet, R.: Efficient FORTRAN subprograms for the solution of elliptic partial differential equations. Report No. NCAR-TN/IA-109, National Center for Atmospheric Research, Boulder, CO (1975).
[29] Underwood, R. R.: An approximate factorization procedure based on the block Cholesky decomposition and its use with the conjugate gradient method. Report No. NEDO-11386, General Electric Co., Nuclear Energy Systems Div., San Jose, CA (1976).
[30] Zangwill, W. I.: Nonlinear Programming, A Unified Approach. Englewood Cliffs, N. J.: Prentice-Hall 1969. · Zbl 0195.20804
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.