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A multilevel algorithm for the biharmonic problem

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Summary

A finite element discretization of the mixed variable formulation of the biharmonic problem is considered. A multilevel algorithm for the numerical solution of the discrete equations is described. Convergence is proved under the assumption ofH 3-regularity.

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References

  1. Babuška, L., Aziz, K.: Survey lectures on the mathematical foundations of the finite element method. In: The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (A.K. Aziz, ed.), pp. 3–359. New York: Academic Press 1972

    Google Scholar 

  2. Babuška, I., Osborn, J., Pitkäranta, J.: Analysis of mixed methods using mesh dependent norms. Math. Comput.35, 1039–1062 (1980)

    Google Scholar 

  3. Bank, R., Dupont, T.: An optimal order process for solving finite element equations. Math. Comput.36, 35–51 (1981)

    Google Scholar 

  4. Bank, R.: A comparison of two multilevel iterative methods for nonsymmetric and indefinite finite element equations. Siam J. Numer. Anal.18, 724–744 (1981)

    Article  Google Scholar 

  5. Bjørstad, P.: Fast numerical solution of the biharmonic Dirichlet problem on rectangles. Siam J. Numer. Anal.20, 59–71 (1983)

    Article  Google Scholar 

  6. Blum, H., Rannacher, R.: On the boundary value problem of the biharmonic operator on domains with angular corners. Math. Methods Appl. Sci.2, 556–581 (1980)

    Google Scholar 

  7. Braess, D.: The convergence rate of a multigrid method with Gauss-Seidel relaxation for the Poisson equation. In: Multigrid Methods. Proceedings (W. Hackbusch, U. Trottenberg, eds.), pp. 368–386. Berlin: Springer 1981

    Google Scholar 

  8. Braess, D., Hackbusch, W.: A new convergence proof for the multigrid method including theV-cycle. Siam J. Numer. Anal.20, 967–975 (1983)

    Google Scholar 

  9. Brexzi, F.: On the existence, uniqueness and approximation of saddle point problems arising from Lagrangian multipliers. RAIRO8, 129–151 (1974)

    Google Scholar 

  10. Brezzi, F., Raviart, P.A.: Mixed finite element methods for 4th order elliptic problems. In: Topics in Numerical Analysis, III (J. Miller, ed.), pp. 33–59. New York: Academic Press 1978

    Google Scholar 

  11. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. Amsterdam: North Holland 1978

    Google Scholar 

  12. Ciarlet, P.G., Glowinski, R.: Dual iterative techniques for solving a finite element approximation of the biharmonic equation. Comput. Methods Appl. Mech.5, 277–295 (1975)

    Article  Google Scholar 

  13. Ciarlet, P.G., Raviart, P.A.: A mixed finite element method for the biharmonic equation. In: Mathematical Aspects of Finite Elements in Partial Differential Equations (C. de Boor, ed.), pp. 125–145. New York: Academic Press 1974

    Google Scholar 

  14. Glowinski, R., Pirroneau, O.: Numerical methods for the first biharmonic equation and for the two-dimensional Stokes problem. Siam Rev.21, 167–212 (1979)

    Google Scholar 

  15. Hackbusch, W.: Survey of convergence proofs for multigrid iterations. In: Special Topics of Applied Mathematics. (J. Frehse, D. Pallaschke, U. Trottenberg, eds.), pp. 151–164. Amsterdam: North-Holland 1980

    Google Scholar 

  16. Hackbusch, W.: Analysis and multigrid solutions of mixed finite element and mixed finite difference equations. Ruhr-Universität Bochum (Preprint 1980)

  17. Lions, J.L., Magenes, E.: Non-Homogeneous Boundary Value Problems with Applications, I. Berlin-Heidelberg-New York: Springer 1972

    Google Scholar 

  18. Verfürth, R.: A multilevel algorithm for mixed problems. Siam J. Numer. Anal.21, 264–271 (1984)

    Google Scholar 

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Peisker, P. A multilevel algorithm for the biharmonic problem. Numer. Math. 46, 623–634 (1985). https://doi.org/10.1007/BF01389663

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