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Finite element methods for non-self-adjoint problems

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Topics in Numerical Analysis

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 965))

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5. References

  1. Agmon, S., Douglis, A., and Nirenberg, L, 1964. Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II. Comm. Pure Appl. Math, 17, 35–92.

    Article  MathSciNet  MATH  Google Scholar 

  2. Allen, D., and Southwell, R., 1955. Relaxation methods applied to determine the the motion, in two dimensions, of a viscous fluid past a fixed cylinder. Quart. J. Mech. and Appl. Math., VIII, 129–145.

    Article  MathSciNet  MATH  Google Scholar 

  3. Axelsson, O., 1981. Stability and error estimates of Galerkin finite element approximations for convection-diffusion equations. I.M.A. J. Numer. Anal. 1, 329–345.

    MathSciNet  MATH  Google Scholar 

  4. Babuŝka, I., and Aziz, A.K., 1972. Survey lectures on the mathematical foundations of the finite element method. The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (ed. A.K. Aziz), New York: Academic Press, 3–363.

    Chapter  Google Scholar 

  5. Barrett, J.W., 1980. Optimal Petrov-Galerkin methods. Ph.D. Thesis, University of Reading.

    Google Scholar 

  6. Barrett, J.W. and Morton, K.W., 1980. Optimal finite element solutions to diffusion-convection problems in one dimension. Int. J. Num. Meth. Engng. 15, 1457–1474.

    Article  MathSciNet  MATH  Google Scholar 

  7. Barrett, J.W., and Morton, K.W., 1981. Optimal Petrov-Galerkin methods through approximate symmetrization. I.M.A. J. Numer. Anal. 1, 439–468.

    MathSciNet  MATH  Google Scholar 

  8. Barrett, J.W., and Morton, K.W., 1981. Optimal finite element approximation for diffusion-convection problems. Proc. MAFELAP 1981 Conf. (ed. J.R. Whiteman).

    Google Scholar 

  9. Barrett, K.E., 1974. The numerical solution of singular-perturbation boundary-value problems. J. Mech. Appl. Math., 27, 57–68.

    Article  MathSciNet  MATH  Google Scholar 

  10. Barrett, K.E., 1977. Finite element analysis for flow between rotating discs using exponentially weighted basis functions. Int. J. Num. Meth. Engng., 11, 1809–1817.

    Article  MATH  Google Scholar 

  11. de Boor, C., and Swartz, B., 1973. Collocation at Gaussian points. SIAM J. Numer. Anal., 10, 582–606.

    Article  MathSciNet  MATH  Google Scholar 

  12. Bristeau, M.O., Pirronneau, O., Glowinski, R., Periaux, J., Perrier, P., and Poirier, G., 1980. Application of optimal control and finite element methods to the calculation of transonic flows and incompressible viscous flows. I.M.A. Conf. Numerical Methods in Applied Fluid Dynamics (ed. B. Hunt), Academic Press, 203–312.

    Google Scholar 

  13. Christie, I., Griffiths, D.F., Mitchell, A.R., and Zienkiewicz, O.C., 1976. Finite element methods for second order differential equations with significant first derivatives. Int. J. Num. Meth. Engng., 10, 1389–1396.

    Article  MathSciNet  MATH  Google Scholar 

  14. Ciarlet, P.G., 1978. The Finite Element Method for Elliptic Problems. North-Holland (Amsterdam).

    MATH  Google Scholar 

  15. Doolan, E.P., Miller, J.J.H., and Schilders, W.H.A., 1980. Uniform Numerical Methods for Problems with Initial and Boundary Layers. Dublin: Boole Press.

    MATH  Google Scholar 

  16. Douglas, J., Jr., and Dupont, T., 1973. Superconvergence for Galerkin methods for the two point boundary problem via local projections. Numer. Math., 21, 270–278.

    Article  MathSciNet  MATH  Google Scholar 

  17. Gresho, P.M., and Lee, R.L., 1979. Don't suppress the wiggles — they're telling you something. Finite Element Methods for Convection Dominated Flows (ed. T.J.R. Hughes) AMD Vol. 34, Am. Soc. Mech. Eng., 37–61.

    Google Scholar 

  18. Griffiths, D., and Lorenz, J, 1978. An analysis of the Petrov-Galerkin finite element method. Comp. Meth. Appl. Mech. Engng., 14, 39–64.

    Article  MathSciNet  MATH  Google Scholar 

  19. Guymon, G.L., Scott, V.H., and Herrmann, L.R., 1970. A general numerical solution of the two-dimensional diffusion-convection equation by the finite element method. Water Resources 6, 1611–1617.

    Article  Google Scholar 

  20. Guymon, G.L., 1970. A finite element solution of a one-dimensional diffusion-convection equation. Water Resources 6, 204–210.

    Article  Google Scholar 

  21. Heinrich, J.C., Huyakorn, P.S., Mitchell, A.R., and Zienkiewicz, O.C., 1977. An upwind finite element scheme for two-dimensional convective transport equations. Int. J. Num. Meth. Engng., 11, 131–143.

    Article  MATH  Google Scholar 

  22. Heinrich, J.C., and Zienkiewicz, O.C., 1979. The finite element method and ‘upwinding’ techniques in the numerical solution of convection dominated flow problems. Finite Element Methods for Convection Dominated Flows (ed. T.J.R. Hughes) AMD Vol. 34, Am. Soc. Mech. Eng., 105–136.

    Google Scholar 

  23. Hemker, P.W., 1977. A numerical study of stiff two-point boundary problems. Thesis, Amsterdam: Math. Cent.

    MATH  Google Scholar 

  24. Hughes, T.J.R., and Brooks, A., 1979. A multidimensional upwind scheme with no crosswind diffusion. Finite Element Methods for Convection Dominated Flows (ed. T.J.R. Hughes) AMD Vol. 34, Am. Soc. Mech. Eng., 19–35.

    Google Scholar 

  25. Hughes, T.J.R., and Brooks, A., 1981. A theoretical framework for Petrov-Galerkin methods with discontinuous weighting functions: application to the streamline-upwind procedure. To appear in Finite Elements in Fluids Vol. 4 (ed. R.H. Gallagher), J. Wiley & Sons: New York.

    Google Scholar 

  26. Johnson, C., and Nävert, U., 1981. An analysis of some finite element methods for advection-diffusion problems. Conf. on Analytical and Numerical Approaches to Asymptotic Problems in Analysis (eds. O. Axelsson, L.S. Frank and A. van der Sluis), North-Holland.

    Google Scholar 

  27. Lesaint, P., and Zlamal, M., 1979. Superconvergence of the gradient of finite element solutions. R.A.I.R.O. Numer. Amal., 13, 139–166.

    MathSciNet  MATH  Google Scholar 

  28. Micchelli, C.A., and Rivlin, T.J., 1976. A survey of optimal recovery. Optimal Estimation in Approximation Theory (ed. C.A. Micchelli & T.J. Rivlin), Plenam Press: New York.

    Google Scholar 

  29. Morton, K.W., and Barrett, J.W., 1980. Optimal finite element methods for diffusion-convection problems. Proc. Conf. Boundary and Interior Layers-Computational and Asymptotic Methods (ed. J.J.H. Miller), Boole Press: Dublin, 134–148.

    Google Scholar 

  30. Oden, J.T., and Reddy, J.N., 1976. An Introduction to the Mathematical Theory of Finite Elements, Wiley-Interscience: New York.

    MATH  Google Scholar 

  31. Strang, G., and Fix, G.J., 1973. An Analysis of the Finite Element Method, Prentice Hall: New York.

    MATH  Google Scholar 

  32. Thomée, V, and Westergren, B., 1968. Elliptic difference equations and interior regularity. Numer. Math. II, 196–210.

    Article  MathSciNet  MATH  Google Scholar 

  33. Zienkiewicz, O.C., Gallagher, R.H. and Hood, P., 1975. Newtonian and non-Newtonian viscous incompressible flow, temperature induced flows: finite element solution. 2nd. Conf. Mathematics of Finite Elements and Applications (ed. J.R. Whiteman), London: Academic Press.

    Google Scholar 

  34. Zienkiewicz, O.C., 1977. The Finite Element Method, London: McGraw Hill.

    MATH  Google Scholar 

  35. Zlamal, M., 1977. Some superconvergence results in the finite element method. Mathematical Aspects of Finite Element Methods. Springer-Verlag.

    Google Scholar 

  36. Zlamal, M., 1978. Superconvergence and reduced integration in the finite element method. Math. Comp. 32, 663–685.

    Article  MathSciNet  MATH  Google Scholar 

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Authors
  1. K. W. Morton
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Peter R. Turner

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© 1982 Spring-Verlag

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Morton, K.W. (1982). Finite element methods for non-self-adjoint problems. In: Turner, P.R. (eds) Topics in Numerical Analysis. Lecture Notes in Mathematics, vol 965. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0063202

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  • DOI: https://doi.org/10.1007/BFb0063202

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-11967-8

  • Online ISBN: 978-3-540-39558-4

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