Abstract
A conjugate gradient and block iterative algorithm for element solution of penalty variational form of Navier-Stokes equations are presented. Because the algorithm of solving single variable minimizing problem is simplified, the computing time is greatly saved.
In this paper numerical examples are also provided.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Li Kai-tai, Huang Ai-xiang, Ma Yi-chen, Li Du and Liu Zhi-xiang, Optimal control finite element approximation for penalty variational formulatio of the Navier-Stokes problem, J. Xian Jiaotong University, Vol. 16, No. 1, 85–88, (1982). (in Chinese)
Li Du, Conjugate gradient algorithm and numerical experimentation for three-dimensional Navier-Stokes problem, J. Xian Jiaotong University, Vol. 16, No. 4, 81–90, (1982). (in Chinese)
Liu Zhi-xiang, The block iteration method and the program for penalty variational form of three-dimensional Navier-Stokes problems, J. Xian Jiaotong University, Vol. 16, No. 4, 91–102, (1982). (in Chinese)
Bristeau, M. O., O. Pironneau, R. Glowinski, J. Periaux and P. Perrier, On the numerical solution of nonlinear problems in fluid dynamics by least squares and finite element methods (I): Least square formulation and conjugate gradient solution of the continuous problems, Comp. Math. Appl. Mech. Eng. 17/18, (1979), 619–657.
Giraut, V., and P. A. Raviart, Finite Element Approximation for the Navier-Stokes Equations, Lecture Notes in Mathematics, Vol. 749, Springer-Verlag, Berlin, (1980).
Teman, R., Navier-Stokes Equations, North-Holland, Amsterdam, (1977).
Reddy, J. N., On the Mathematical Theory of the Penalty-Finite Elements for Navier-Stokes Equations, Proceedings of the Third International Conference on Finite Elements in Flow Problems, Vol. 2, (1980)
Zienkiewicz, O. C., Constrained Variational Principles and Penalty Function Methods in Finite Element Analysis, Lecture Notes in Mathematics, P. 363 (Edited by Dald and B. B. Eckman), Springer-Verlag, New York, (1974).
Falk, R. S., and J. T. King, A penalty and extrapolation method for the stationary Stokes equations, SIAM. J. Numer. Anal 13 (1979), 814–829.
Bercoviex, M., and M. Engelman, A finite element for the numerical solution of viscous incompressible flows, J. Comp. Phys. 30 (1979), 181–201.
Hughes, T. J. R., W. K. Liu and A. Brooks, Finite element analysis of incompressible viscous flows by the penalty function formulation, J. Comp. Phys., 30 (1979) 1–60.
Song, Y. J., J. T. Oden and N. Kikuchi, Discrete LBB-Conditions for RIP-Finite Element Methods, TICON Report, 80-7 (1980).
Oden, J. T., RIP-Methods for Stokesian Flows, Finite Elements in Fluids, Vol. 4, John Wiley Sons.
Oden, J. T., Penalty Methods and Selective Reduced Integration for Stokesian Flow, Proceedings of the Third International Conference on Finite Elements in Flow Problems, Banff. Alberta, Canada, (1980), 140–145.
Oden, J. T., Penalty Finite Element Methods for Constrained Problems in Elasticity, Symposium on Finite Element Methods, held in Hefei, Anhui Province, The People's Republic of China, (1981).
Author information
Authors and Affiliations
Additional information
Communicated by Chien Wei-zang.
Rights and permissions
About this article
Cite this article
Kai-tai, L., Ai-xiang, H., Du, L. et al. The conjugate gradient method and block iterative method for penalty finite element of three dimensional navier-stokes equation. Appl Math Mech 4, 927–941 (1983). https://doi.org/10.1007/BF01896178
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01896178