Skip to main content

Traditional Galerkin Methods

  • Chapter
Computational Galerkin Methods

Part of the book series: Springer Series in Computational Physics ((SCIENTCOMP))

Abstract

Galerkin methods have been used to solve problems in structural mechanics, dynamics, fluid flow, hydrodynamic stability, magnetohydrodynamics, heat and mass transfer, acoustics, microwave theory, neutron transport, etc. Problems governed by ordinary differential equations, partial differential equations, and integral equations have been investigated via a Galerkin formulation. Steady, unsteady, and eigenvalue problems have proved to be equally amenable to a Galerkin treatment. Essentially, any problem for which governing equations can be written down is a candidate for a Galerkin method.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

eBook
USD 16.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 16.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Ahlberg, J. H., Nilson, E. N., and Walsh, J. L. The Theory of Splines and Their Applications, Academic Press, New York (1967).

    MATH  Google Scholar 

  • Alexopoulos, N. G., and Uzunoglu, N. K. IEEE Trans. Mic. Th. Tech. MTT-26, 455–456 (1978).

    Article  Google Scholar 

  • Ames, W. F. Nonlinear Partial Differential Equations in Engineering, Academic Press, New York (1965).

    MATH  Google Scholar 

  • Ames, W. F. Nonlinear Partial Differential Equations in Engineering, Vol. II, Academic Press, New York (1972).

    Google Scholar 

  • Anderssen, R. S., and Mitchell, A. R. Math. Mech. Appl. Sci., 1, 3–15 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  • Anon. Prik. Mat. Mekh. 5, 337–341 (1941).

    Google Scholar 

  • Bartlett, E. P., and Kendall, R. M. NASA CR-1062 (1968).

    Google Scholar 

  • Belotserkovskii, O. M., and Chushkin, P. I. In Basic Developments in Fluid Dynamics (ed. M. Holt), Vol. 1, pp. 1–126 Academic Press, New York (1965).

    Google Scholar 

  • Belyaev, N. M., Kordyuk, O. L., and Ryadno, A. A. Inz. Fiz. Zh. 30, 512–518 (1976).

    Google Scholar 

  • Biezeno, C. B., and Koch, J. J. Ingenieur 38, 25–36 (1923).

    Google Scholar 

  • Bickley, W. G. Phil. Mag. (7) 32, 50–66 (1941).

    MathSciNet  Google Scholar 

  • Bourke, W., McAveney, B., Puri, K., and Thurling, R. Meth. in Comp. Phys. 17 267–325 (1977).

    Google Scholar 

  • Brebbia, C. A. The Boundary Element Method for Engineers, Pentech Press, London (1978).

    Google Scholar 

  • Catton, I., Ayyaswamy, P. S., and R. M. Clever Int. J. Heat Mass Trans. 17, 173–184 (1974).

    Article  MATH  Google Scholar 

  • Chandrasekhar, S. Hydrodynamic and Hydromagnetic Stability, Oxford U.P. (1961).

    Google Scholar 

  • Chattot, J. J., Guiu-Roux, J., and Laminie, J. Seventh International Conference on Numerical Methods in Fluid Dynamics, Proceedings, Stanford 1980 Lecture Notes in Physics, Vol. 141, pp. 107–112, Springer-Verlag, Berlin (1981).

    Google Scholar 

  • Chopra, I., and Durvasula, S. J. Sound Vib., 19 379–392 (1971).

    Article  ADS  MATH  Google Scholar 

  • Citron, S. J. J. Aero. Sci. 27, 317–318 (1980).

    Google Scholar 

  • Coen, S., and Gladwell, G. M. L. IEEE Trans. Mie. Th. Tech. MTT-25, 1–6 (1977).

    Google Scholar 

  • Collatz, L. The Numerical Treatment of Differential Equations, Springer-Verlag, Berlin (1960)

    MATH  Google Scholar 

  • Crandall, S. H. Engineering Analysis, McGraw-Hill, New York (1956).

    MATH  Google Scholar 

  • Dahlquist, G., Björck, A., and Anderson, N. Numerical Methods, Prentice-Hall, Englewood Cliffs, NJ (1974).

    Google Scholar 

  • Davis, S. S., and Johnson, M. L. 87th Meeting Acoustical Soc. of America, Paper KK-2 (1974).

    Google Scholar 

  • Demirdzic, I., Gosman, A. D., and Issa, R. Lecture Notes in Physics, Vol. 141, pp. 144–150, Springer-Verlag (1981).

    Google Scholar 

  • De Vahl Davis, G. Int. J. Heat Mass Trans. 11, 1675–1693 (1968).

    Article  Google Scholar 

  • DiPrima, R. C., and Sani, R. L. Quart. Appl. Math. 23, 183–187 (1965).

    MathSciNet  MATH  Google Scholar 

  • Dorodnitsyn, A. A. Advances in Aeronautical Sciences, Vol. 3, Pergamon, New York (1960).

    Google Scholar 

  • Dryden, H. L., Murnaghan, F. P., and Bateman, H. Hydrodynamics, p. 197, Dover, New York (1956).

    MATH  Google Scholar 

  • Duncan, W. J. ARC R and M 1798 (1937).

    Google Scholar 

  • Duncan, W. J. ARC R and M 1848 (1938).

    Google Scholar 

  • Durban, D. AIAA J. 15, 360–365 (1977).

    Article  ADS  MATH  Google Scholar 

  • Durvasula, S. AIAA J. 7, 461–466 (1971).

    Google Scholar 

  • Eversman, W., Cook, E. L., and Beckemeyer, R. J. J. Sound Vib. 38, 105–123 (1975).

    Article  ADS  MATH  Google Scholar 

  • Eversman, W., and Astley, R. J. J. Sound Vib. 74, 89–101 (1981).

    Article  ADS  MATH  Google Scholar 

  • Finlayson, B. A., and Scriven, L. E. App. Mech. Rev. 19, 735–748 (1966).

    Google Scholar 

  • Finlayson, B. A. Brit. Chem. Eng. 14, 179–182 (1969).

    Google Scholar 

  • Finlayson, B. A. The Method of Weighted Residuals and Variational Principles, Academic Press, New York (1972).

    MATH  Google Scholar 

  • Fletcher, C. A. J. In Numerical Simulation of Fluid Motion (ed. J. Noye ), pp. 537–550, North-Holland (1978).

    Google Scholar 

  • Fletcher, C. A. J. J. Comp. Phys. 33, 301–312 (1979).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Fletcher, C. A. J. “Burgers’ Equation: A Model for All Reasons”, in Numerical Solution of Partial Differential Equations (ed. J. Noye ), North-Holland (1982).

    Google Scholar 

  • Fletcher, C. A. J., and Holt, M. J. Comp. Phys. 18, 154–164 (1975).

    Article  ADS  MATH  Google Scholar 

  • Forsythe, G. E., Malcolm, M. A., and Moler, C. B. Computer Methods for Mathematical Computations, Prentice-Hall, Englewood Cliffs, NJ (1977).

    MATH  Google Scholar 

  • Frazer, R. A., Jones, W. P., and Skan, S. W. ARC R and M 1799 (1937).

    Google Scholar 

  • Fuller, E. L., Meneley, D. A., and Hetrick, D. L. Nucl. Sci. Eng. 40, 206–233 (1970).

    Google Scholar 

  • Galerkin, B. G. Vestnik Inzhenerov, Tech. 19, 897–908 (1915).

    Google Scholar 

  • Gear, C. W. SIAM Review 23, 10–24 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  • Green, J. W. J. Res. Nat. Bur. Std. 51, 127–132 (1953).

    MATH  Google Scholar 

  • Harrington, R. F. Field Computation by Moment Methods, Macmillan, New York (1968).

    Google Scholar 

  • Hamming, R. W. Numerical Methods for Scientists and Engineers, McGraw-Hill, New York, 2nd Edn. (1973).

    Google Scholar 

  • Hess, J. L., and Smith, A. M. O. J. Ship Research 8, 22–42 (1964).

    Google Scholar 

  • Heywood, J. G. Arch. Rat. Mech. Anal. 37, 48–60 (1970).

    Article  MathSciNet  MATH  Google Scholar 

  • Holt, M. Numerical Methods in Fluid Dynamics, Springer-Verlag (1977).

    Google Scholar 

  • Itoh, T., and Mittra, R. Arch. Elek. Übertragung 27, 456–458 (1973).

    Google Scholar 

  • Jain, M. K. Appl. Sci. Res. All, 177–188 (1962).

    Google Scholar 

  • Jameson, A., and Caughey, D. A. A Finite Volume Method for Transonic Potential Flow Calculations. AIAA Paper 77–635 (1977).

    Google Scholar 

  • Javeri, V. Int. J. Heat Mass Trans. 21, 1035–1040 (1978).

    Article  MATH  Google Scholar 

  • Kantorovich, L. V., and Krylov, V. I., Approximate Methods in Higher Analysis, Wiley, New York (1958).

    Google Scholar 

  • Ladyzhenskaya, O. A. The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York (1963).

    MATH  Google Scholar 

  • Ladyzhenskaya, O. A. Annual Review of Fluid Mechanics 7, 249–272 (1975).

    Article  ADS  Google Scholar 

  • Lee, Y., and Reynolds, W. C. Tech. Report FM-1, Dept. of Mech. Eng., Stanford Univ. (1964).

    Google Scholar 

  • Lynn, P. P. Int. J. Num. Meth. Eng. 8, 865–876 (1974).

    Article  MATH  Google Scholar 

  • Mikhlin, S. G. Variational Methods in Mathematical Physics, Pergamon, Oxford (1964).

    MATH  Google Scholar 

  • Mikhlin, S. G. The Numerical Performance of Variational Methods, Noordhoff, Groningen (1971).

    MATH  Google Scholar 

  • Milthorpe, J., and Steven, G. P. Finite Elements in Fluids, Vol. 3, pp. 89–110, Wiley, New York (1978).

    Google Scholar 

  • Murphy, J. D. “Application of the Generalised Galerkin Method to the Computation of Fluid Flows”, Proceedings 1st AIAA Computational Fluid Dynamics Conference, Palm Springs, pp. 63–68 (1973).

    Google Scholar 

  • Murphy, J. D. AIAA J. 15, 1307–1314 (1977).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Murti, P. R. K. Wear 26, 95–104 (1973).

    Article  Google Scholar 

  • Narasimha, R., and Deshpande, S. M. J. Fluid Mech. 36, 555–570 (1969).

    Article  ADS  MATH  Google Scholar 

  • Neuman, C. P. “Recent Developments in Discrete Weighted Residual Methods”, in Computational Methods in Nonlinear Engineering (ed. J. T. Oden) North-Holland, Amsterdam (1974).

    Google Scholar 

  • Nield, D. A. J. Fluid Mechanics 71, 441–454 (1975).

    Article  ADS  MATH  Google Scholar 

  • Nugmanov, Z. Kh. Izv. Vuz. Aria. Tekh. 18, 78–83 (1975).

    Google Scholar 

  • Oden, J. T. Finite Elements of Nonlinear Continua, McGraw-Hill. New York (1972)

    MATH  Google Scholar 

  • Orszag, S. A. J. Fluid Mechanics 50, 689–703 (1971).

    Article  ADS  MATH  Google Scholar 

  • Orszag, S. A. “Numerical Simulation of Turbulent Flows”, in Handbook of Turbulence (eds. W. Frost and T. H. Moulden), pp. 281 313, Plenum Press, New York (1977).

    Google Scholar 

  • Orszag, S. A. J. Comp. Phys. 37, 70 92 (1980).

    Google Scholar 

  • Pallone, A. J. Aero. Sci. 28, 449–456 (1961).

    MathSciNet  MATH  Google Scholar 

  • Panton, R. Z., and Sallee, H. B. Computers and Fluids 3, 257–269 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  • Platten, J. K., Flandroy, P., and Vanderborck, G. Int. J. Eng. Sci. 12, 995–1006 (1974).

    Article  MATH  Google Scholar 

  • Pomraning, G. C. Nucl. Sci. Eng. 24, 291–301 (1966).

    Google Scholar 

  • Poots, G. Quart. J. Mech. Appl. Math. 11, 257–267 (1958).

    Article  MathSciNet  MATH  Google Scholar 

  • Prabhu, M. S. S., and Durvasula, S. Comp. Structures 6, 177–185 (1976).

    Article  MATH  Google Scholar 

  • Quang, V. Z. Magnetohydrodynamics (Mg. Gidr.) 1, 83–88 (1974).

    Google Scholar 

  • Rizzi, A. W., and Inouye, M. AIAA J. 11, 1478–1485 (1973).

    Article  ADS  MATH  Google Scholar 

  • Rubbert, P. E., and Saaris, G. R. Review and Evaluation of a Three-Dimensional Lifting Potential Flow Analysis Method for Arbitrary Configurations, AIAA Paper 72–188 (1972).

    Google Scholar 

  • Schetz, J. A., J. Appl. Mech. 30, 263–268 (1963).

    Article  ADS  MATH  Google Scholar 

  • Shuleshko, P. Aust. J. Appl. Sci. 10, 1–16 (1959).

    MathSciNet  Google Scholar 

  • Sigilloto, V. G. J. Assoc. Comp. Mech. 14, 732–741 (1967).

    Article  Google Scholar 

  • Stephens, A. B. SIAM J. Num. Anal. 13, 607–614 (1976).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Straus, J. M. J. Fluid Mechanics 64, 51–63 (1974).

    Article  ADS  MATH  Google Scholar 

  • Tam, C. K. W. J. Sound Vib. 45, 91–104 (1976).

    Article  ADS  MATH  Google Scholar 

  • Temam, R. Numerical Analysis, Reidel (1973).

    Google Scholar 

  • Thomson, A. F., and Gopinath, A. IEEE Trans. Mic. Th. Tech. MTT-23, 648–655 (1975).

    Google Scholar 

  • Thorsen, R., and Landis, F. Int. J. Heat Mass Transfer 8, 189–192 (1965).

    Article  Google Scholar 

  • Truckenbrodt, E. J. Aero. Sci. 19, 428–429 (1952).

    MATH  Google Scholar 

  • Vichnevetsky, R. IEEE Trans. Comp. C-18, 499–512 (1969).

    Google Scholar 

  • Villadsen, J. V., and Stewart, W. E. Chem. Eng. Sci. 22, 1483–1501 (1967).

    Article  Google Scholar 

  • Viviand, H., and Ghazzi, W. La Recherche Aerospatiale 1974–5, 5, 247–260 (1974).

    Google Scholar 

  • Wadia, A. R., and Payne, F. R. In Advances in Computational Methods for Partial Differential Equations (eds. R. Vichnevetsky and R. S. Stepleman ), pp. 205–219, IMACS (1979).

    Google Scholar 

  • Wagner, H., and Pattabiraman, J. Z. Flugwiss. 21, 131–140 (1973).

    Google Scholar 

  • Williams, H. E. J. Hydronautics 9, 107–118 (1975).

    Article  Google Scholar 

  • Yamada, H. Rept. Res. Inst. Fluid Eng. Kyushu Univ. 3, 29 (1947).

    Google Scholar 

  • Yamada, H. Rept. Res. Inst. Fluid Eng. Kyushu Univ., 4, 27–42 (1948).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 1984 Springer-Verlag New York Inc.

About this chapter

Cite this chapter

Fletcher, C.A.J. (1984). Traditional Galerkin Methods. In: Computational Galerkin Methods. Springer Series in Computational Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-85949-6_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-85949-6_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-85951-9

  • Online ISBN: 978-3-642-85949-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics