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Galerkin Finite-Element Methods

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

Abstract

The Galerkin finite-element method has been the most popular method of weighted residuals, used with piecewise polynomials of low degree, since the early 1970s. The rise in the popularity of the Galerkin formulation and the concurrent decline in popularity of the variational finite-element formulation has coincided with the diversification of the finite-element method into areas remote from the structural birthplace of the method.

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References

  • Adam, Y. J. Comp. Phys. 24, 10–22 (1977).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Arakawa, A. J. Comp. Phys. 1, 119–143 (1960).

    Article  ADS  Google Scholar 

  • Astley, R. J., and Eversman, W. J. Sound Vib. 74, 103–121 (1981).

    Article  ADS  MATH  Google Scholar 

  • Astley, R. J., and Eversman, W. “Acoustic Transmission in Lined Ducts”, in Finite Elements in Fluids (ed. R. H. Gallagher), Vol. 4, Wiley, London (1982).

    Google Scholar 

  • Baker, A. J., and Manhardt, P. D. AIAA J. 16, 807–814 (1978)

    Article  ADS  MATH  Google Scholar 

  • Baker, A. J., and Soliman, M. O. J. Comp. Phys. 32, 289–324 (1979).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Bathe, K.-J., and Wilson, E. L. Numerical Methods in Finite Element Analysis, Prentice-Hall, Englewood Cliffs, NJ (1976).

    MATH  Google Scholar 

  • Birkhoff, G., and de Boor, C. R. In Approximations of Functions (ed. H. L. Garabedian ), pp. 164–190 Elsevier (1965).

    Google Scholar 

  • Bramble, J. H., and Schatz, A. H. Math. Comp. 31, 94–111 (1977).

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  • Ciment, M., Leventhal, S. H., and Weinberg, B. C. J. Comp. Phys. 28, 135–166 (1978)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Culham, W. E., and Varga, R. S. Soc. Pet. Eng. J. 11, 374–388 (1971).

    Google Scholar 

  • Cullen, M. J. P., and Hall, C. D. Quart, J. R. Met. Soc. 105, 571–592 (1979).

    Article  ADS  Google Scholar 

  • Cullen, M. J. P., and Morton, K. W. J. Comp. Phys. 34, 245–267 (1980).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Cushman, J. H. I. J. Num. Meth. Eng. 14, 1643–1651 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  • Cushman, J. H. I. J. Num. Meth. Eng. 17, 975–989 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  • Davis, R. T., and Rubin, S. G. Comp. and Fluids 8, 101–132 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  • Dennis, S. C. R., and Walker, J. D. A. J. Fluid Mech. 48, 771–789 (1971).

    Article  ADS  MATH  Google Scholar 

  • Dougalis, V. A., and Serbin, S. M. SIAM J. Num. Anal. 17, 431–446 (1980).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Douglas, J., and Dupont, T. Numer. Math. 22, 99–109 (1974).

    Article  MathSciNet  MATH  Google Scholar 

  • Dupont, T. SIAM J. Num. Anal. 10, 880–889 (1973).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Ergatoudis, J. G., Irons, B. M., and Zienkiewicz, O. C. Int. J. Solids Structures 4, 31–42 (1968)

    Article  MATH  Google Scholar 

  • Fairweather, G. Finite Element Galerkin Methods for Differential Equations, Dekker, New York (1978).

    MATH  Google Scholar 

  • Fletcher, C. A. J. The Application of the Finite Element Method to Two-Dimensional Inviscid Flow. WRE-TN-1606, Salisbury, South Australia (1976).

    Google Scholar 

  • Fletcher, C. A. J. Improved Integration Techniques for Fluid Flow Finite Element Formulations. WRE-TR-1810, Salisbury, South Australia (1977).

    Google Scholar 

  • Fletcher, C. A. J. “Burgers’ Equation: A Model for All Reasons”, in Numerical Solution of Partial Differential Equations (ed. J. Noye ), pp. 139–225, North-Holland, Amsterdam (1982a).

    Google Scholar 

  • Fletcher, C. A. J. “A Comparison of the Finite Element and Finite Difference Methods for Computational Fluid Dynamics”, in Finite Element Flow Analysis (ed. T. Kawai ), pp. 1003–1010, Univ. of Tokyo Press, (1982b).

    Google Scholar 

  • Gartling, D. K. Comp. Meth. App. Mech. Eng. 12, 365–382 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  • Gartling, D. K. “A Finite Element Analysis of Volumetrically Heated Fluids in an Axisymmetric Enclosure”. 3rd Finite Element in Flow Problems Conference, Banff, Canada, pp. 174–182 (1980).

    Google Scholar 

  • Gray, W. G., and Pinder, G. F. I. J. Num. Meth. Eng. 10, 893–923 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  • Grotkop, G. Comp. Meth. Appl. Mech. Eng. 2, 147–157 (1973).

    Article  MATH  Google Scholar 

  • Heubner, K. H. The Finite Element Method for Engineers, Wiley, New York (1975).

    Google Scholar 

  • Hughes, T. J. R., Taylor, R. L., and Levy, J. F. In Finite Elements in Fluids (ed. R. H. Gallagher et al.), Vol. 3, pp. 55–72, Wiley, London (1978).

    Google Scholar 

  • Hughes, T. J. R., Liu, W. K., and Brooks, A. J. Comp. Phys. 30, 1–60 (1979)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Irons, B. M. Int. J. Num. Meth. Eng. 2, 5–32 (1970).

    Article  ADS  MATH  Google Scholar 

  • Irons, B. M., and Razzaque, A. In The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (ed. A. K. Aziz ), pp. 557–587, Academic Press, (1972).

    Google Scholar 

  • Irons, B., and Ahmad, S. Techniques of Finite Elements, Wiley, Chichester (1980).

    Google Scholar 

  • Isaacson, E., and Keller, W. B. An Analysis of Numerical Methods, Wiley, New York (1966)

    Google Scholar 

  • Jennings, A. Matrix Computation for Engineers and Scientists, Wiley, London (1977)

    MATH  Google Scholar 

  • Jespersen, D. C. J. Comp. Phys. 16, 383–390 (1974).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Kawahara, M., Takeuchi, N., and Yoshida, T. Int. J. Num. Meth. Eng. 12, 331–351 (1978)

    Article  MATH  Google Scholar 

  • Khosla, P. K., and Rubin, S. G. J. Eng. Math. 13, 127–141 (1979).

    Article  MATH  Google Scholar 

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

    MATH  Google Scholar 

  • Mitchell, A. R., and Wait, R. The Finite Element Method in Partial Differential Equations, Wiley, London (1977).

    MATH  Google Scholar 

  • Norrie, D., and de Vries, G. Finite Element Bibliography, Plenum, New York (1976)

    Book  MATH  Google Scholar 

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

    MATH  Google Scholar 

  • Oden, J. T., and Reddy, J. N. An Introduction to the Mathematical Theory of Finite Elements, Wiley, New York (1976a).

    MATH  Google Scholar 

  • Oden, J. T., and Reddy, J. N. SIAM J. Num. Anal. 13, 393–404 (1976b).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Oden, J. T. “Penalty Methods and Selective Reduced Integration for Stokesian Flows”, in 3rd Finite Element in Flow Problems Conference, Banff, Canada, pp. 140–145 (1980).

    Google Scholar 

  • Phuoc, H. B., and Tanner, R. I. J. Fluid Mech. 98, 253–271 (1980).

    Article  ADS  MATH  Google Scholar 

  • Pinder, G. F., and Gray, W. G. Finite Element Simulation in Surface and Subsurface Hydrology, Academic Press, New York (1977).

    Google Scholar 

  • Richtmyer, R., and Morton, K. W. Difference Methods for Initial-Value Problems, Wiley, New York, 2nd Edn (1967).

    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 

  • Schreker, G. O., and Maus, J. R. Noise Characteristics of Jet-flap Type Exhaust Flows. NASA CR-2342 (1974).

    Google Scholar 

  • Schultz, M. Spline Analysis, Prentice-Hall, Englewood Cliffs, NJ (1973)

    MATH  Google Scholar 

  • Showalter, R. E. SIAM J. Num. Anal. 12, 456–463 (1975).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Staniforth, A. N. “A Review of the Applications of the Finite Element Method to Meteorological Flows,” in Finite Element Flow Analysis (ed. T. Kawai) Univ. of Tokyo Press, Tokyo (1982), pp. 835–842.

    Google Scholar 

  • Staniforth, A. N., and Daley, R. W. Mon. Weath. Rev. 107, 107–121 (1979).

    Article  ADS  Google Scholar 

  • Strang, G., and Fix, G. J. An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, NJ (1973).

    MATH  Google Scholar 

  • Swartz, B., and Wendroff, B. Math. Comp. 23, 37–50 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  • Taylor, C., and Davis, J. Comp. and Fluids 3, 125–148 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  • Taylor, C. “The Utilisation of the F. E. M. in the Solution of Some Free Surface Problems”, in 3rd Finite Element in Flow Problems Conference, Banff, Canada, pp. 54–81 (1980).

    Google Scholar 

  • Temam, R., and Thomasset, F. “Numerical Solution of the Navier-Stokes Equations by a Finite Element Method”, in 2nd Finite Element in Flow Problems Conference, St. Margharita Ligure, Italy (1976).

    Google Scholar 

  • Temam, R. “Some Finite Element Methods in Fluid Flow”, in Lecture Notes in Physics, No. 90, pp. 34–55 Springer-Verlag, Berlin, (1979a).

    Google Scholar 

  • Temam, R. Navier-Stokes Equations, North-Holland, Amsterdam (1979b).

    MATH  Google Scholar 

  • Thomée, V. Math. Comp. 31, 652–660 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  • Tuann, S.-Y., and Olson, M. D. J. Comp. Phys. 29, 1–19 (1978).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Wellford, L. C., and Oden, J. T. Comp. Meth. App. Mech. Eng. 5, 83–96 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  • Wheeler, M. F. SIAM J. Num. Anal. 10, 723–759 (1973).

    Article  ADS  Google Scholar 

  • Whiteman, J. R. A Bibliography for Finite Elements, Academic Press (1975)

    MATH  Google Scholar 

  • Zienkiewicz, O. C. The Finite Element Method, McGraw-Hill, London, 3rd Edn. (1977).

    MATH  Google Scholar 

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Fletcher, C.A.J. (1984). Galerkin Finite-Element Methods. In: Computational Galerkin Methods. Springer Series in Computational Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-85949-6_3

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  • DOI: https://doi.org/10.1007/978-3-642-85949-6_3

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  • Print ISBN: 978-3-642-85951-9

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