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Finite Element Variational Crimes in the Solution of Nonlinear Stationary Problems

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Problems and Methods in Mathematical Physics

Part of the book series: TEUBNER-TEXTE zur Mathematik ((TTZM))

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Abstract

Let us consider the boundary value problem

$$ - \sum\limits_{i = 1}^2 {\frac{\partial }{{\partial {x_i}}}\left[ {{a_i}\left( {x,u\left( x \right),\nabla u\left( x \right)} \right)} \right]} + {a_0}\left( {x,u\left( x \right),\nabla u\left( x \right)} \right) = f\left( x \right)\;in\;\Omega ,$$
(1)

equipped with the mixed Dirichlet-Neumann boundary conditions

$$u\left| {{\Gamma _D} = {u_D},\;\sum\limits_{i = 1}^2 {{a_i}\left( { \cdot ,u,\nabla u} \right){n_i} = {\varphi _N}\;on\;{\Gamma _N}} } \right..$$
(2)

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References

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

    Google Scholar 

  2. Berger, H.: Finite-Element-Approximationen für transonische Strömungen. Dr. rer. nat. Dissertation, Universität Stuttgart, 1989.

    Google Scholar 

  3. Berger, H.: A convergent finite element formulation for transonic flow. Numer. Math. 56, 425–447, 1989.

    Article  MathSciNet  MATH  Google Scholar 

  4. Berger, H., Feistauer, M.: Analysis of the finite element variational crimes in the numerical approximation of transonic flow. Bericht Nr. 27, Nov. 90, Seminar Analysis und Anwendungen, Universität Stuttgart (to appear in Math. Comp. 1993).

    Google Scholar 

  5. Berger, H., Feistauer, M.: Mathematical theory of the finite element solution of transonic flow. In: Methoden und Verfahren der mathematischen Physik 37: Direct and Inverse Boundary Value Problems (R. Kleinman, R. Kress, E. Martensen, eds.), P. Lang, Frankfurt am Main — Bern — New York — Paris, 1991, 15–25.

    Google Scholar 

  6. Berger, H., Warnecke, G., Wendland, W.: Finite elements for transonic flows. Numer. Methods for P.D.E 6, (1990), 17–42.

    Article  MathSciNet  MATH  Google Scholar 

  7. Berger, H., Warnecke, G., Wendland, W.: Analysis of a FEM/BEM Coupling Method for Transonic Flow Computations. Preprint, Universität Stuttgart, 1993.

    Google Scholar 

  8. Chow, S.S.: Finite element error estimates for non-linear elliptic equations of monotone type. Numer. Math. 54, (1989), 373–393.

    Article  MathSciNet  MATH  Google Scholar 

  9. Ciariet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam, 1979.

    Google Scholar 

  10. Ciarlet, P.G., Raviart, P.A.: The combined effect of curved boundaries and numerical integration in isoparametric finite element method. In: The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (A.K. Aziz, ed.), Academic Press, New York, 1972, 409–474.

    Google Scholar 

  11. Douglas, J.: H 1-Galerkin methods for a nonlinear Dirichlet problem. In: Proc. Conf. on Mathematical Aspects of Finite Element Methods, Lecture Notes in Math., Vol. 606, Springer-Verlag, Berlin 1977, 64–86.

    Chapter  Google Scholar 

  12. Douglas, J., Dupont, T.: A Galerkin method for a nonlinear Dirichlet problem. Math. Comp. 29, (1975), 689–696.

    Article  MathSciNet  MATH  Google Scholar 

  13. Feistauer, M.: On the finite element approximation of a cascade flow problem. Numer. Math. 50, (1987), 655–684.

    Article  MathSciNet  MATH  Google Scholar 

  14. Feistauer, M.: Mathematical Methods in Fluid Dynamics. Pitman Monographs and Surveys in Pure and Applied Mathematics 67, Longman Scientific & Technical, Harlow, 1993.

    Google Scholar 

  15. Feistauer, M.: Finite element approximation of a problem with nonlinear Newton condition (in preparation).

    Google Scholar 

  16. Feistauer, M., Kalis, H., Rokyta, M.: Mathematical modelling of an electrolysis process. Comment. Math. Univ. Carolinae 30, (1989), 465–477.

    MathSciNet  MATH  Google Scholar 

  17. Feistauer, M., Křížek, M., Sobotíková, V.: An analysis of finite element variational crimes for a nonlinear elliptic problem of a nonmonotone type. Submitted to East-West J. Numer. Anal.

    Google Scholar 

  18. Feistauer, M., Mandel, J., Nečas, J.: Entropy regularization of the transonic potential flow problems. Comment. Math. Univ. Carolinae 25, (1984), 431–443.

    MATH  Google Scholar 

  19. Feistauer, M., Nečas, J.: On the solvability of transonic potential flow problems. Z. Anal. Anw. 4, (1985), 305–329.

    MATH  Google Scholar 

  20. Feistauer, M., Sobotíková, V.: Finite element approximation of nonlinear elliptic problems with discontinuous coefficients. RAIRO Model. Math. Anal. Numer. (M2AN) 24, (1990), 457–500.

    MATH  Google Scholar 

  21. Feistauer, M., Ženíšek, A.: Finite element solution of nonlinear elliptic problems. Numer. Math. 50, (1987), 451–475.

    Article  MathSciNet  MATH  Google Scholar 

  22. Feistauer, M., Ženíšek, A.: Compactness method in the finite element theory of nonlinear elliptic problems. Numer. Math. 52, (1988), 147–163.

    Article  MathSciNet  MATH  Google Scholar 

  23. Feistauer, M., Ženíšek, A.: Finite element variational crimes in nonlinear elliptic problems. In: Proc. of the ISNA 87, Teubner Texte zur Mathematik, Band 107, Lepzig, 1988, 28-35.

    Google Scholar 

  24. Frehse, J., Rannacher, R.: Optimal uniform convergence for the finite element approximation of a quasilinear elliptic boundary value problem. In: Proc. of the U. S.-German Symposium “Formulations and Computational Algorithms in Finite Element Analysis”, M.I.T, 1976.

    Google Scholar 

  25. Frehse, J., Rannacher, R.: Asymptotic L ∞-error estimates for linear finite element approximations of quasilinear boundary value problems. SIAM J. Numer. Anal. 15, (1978), 418–431.

    Article  MathSciNet  MATH  Google Scholar 

  26. Glowinski, R.: Numerical Methods for Nonlinear Variational Problems. Springer-Verlag, New York — Berlin — Heidelberg — Tokyo, 1984.

    Book  MATH  Google Scholar 

  27. Hlaváček, I., Křížek, M., Malý, J.: On Galerkin approximations of a quasilinear nonpotential elliptic problem of a nonmonotone type. J. Math. Anal. Appl. (to appear).

    Google Scholar 

  28. Mandel, J., Nečas, J.: Convergence of finite elements for transonic potential flow. SIAM J. Numer. Anal. 24, (1987), 985–996.

    Google Scholar 

  29. Strang, G.: Variational crimes in the finite element method. In: The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (A.K. Aziz, ed.), Academic Press, New York, 1972, 689–710.

    Google Scholar 

  30. Ženíšek, A.: The finite element method for nonlinear elliptic equations with discontinuous coefficients. Numer. Math. 58, (1990), 51–77.

    Article  MathSciNet  MATH  Google Scholar 

  31. Zlámal, M.: Curved elements in the finite element method I. SIAM J. Numer. Anal. 10, (1973), 229–240.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Faculty of Mathematics and Physics, Charles University Prague, Sokolovská 83, 186 00, Praha 8, Czech Republic

    Miloslav Feistauer

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  1. Miloslav Feistauer
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Editors and Affiliations

  1. Technical University, Chemnitz-Zwickau, Deutschland

    Lothar Jentsch  & Fredi Tröltzsch  & 

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© 1994 Springer Fachmedien Wiesbaden

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Feistauer, M. (1994). Finite Element Variational Crimes in the Solution of Nonlinear Stationary Problems. In: Jentsch, L., Tröltzsch, F. (eds) Problems and Methods in Mathematical Physics. TEUBNER-TEXTE zur Mathematik. Vieweg+Teubner Verlag, Wiesbaden. https://doi.org/10.1007/978-3-322-85161-1_3

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  • DOI: https://doi.org/10.1007/978-3-322-85161-1_3

  • Publisher Name: Vieweg+Teubner Verlag, Wiesbaden

  • Print ISBN: 978-3-322-85162-8

  • Online ISBN: 978-3-322-85161-1

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