Advances in Computational Mathematics

, Volume 19, Issue 4, pp 339–354 | Cite as

A New Boundary Element Method for the Biharmonic Equation with Dirichlet Boundary Conditions

  • Youngmok Jeon
  • William McLean
Article

Abstract

We consider a scalar boundary integral formulation for the biharmonic equation based on the Almansi representation. This formulation was derived by the first author in an earlier paper. Our aim here is to prove the ellipticity of the integral operator and hence establish convergence of and error bounds for Galerkin boundary element methods. The theory applies both in two and three dimensions, but only for star-shaped domains. Numerical results in two dimensions confirm our analysis.

biharmonic equations Almansi representation layer potential boundary integral equation 

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References

  1. [1]
    M. Ainsworth, W. McLean and T. Tran, The conditioning of boundary element equations on locally refined meshes and preconditioning by diagonal scaling, SIAMJ. Numer. Anal. 36 (1999) 1901–1932.Google Scholar
  2. [2]
    J. Chazarain and A. Piriou, Introduction to the Theory of Linear Partial Differential Equations (North-Holland, Amsterdam, 1982).Google Scholar
  3. [3]
    M. Costabel, Boundary integral operators on Lipschitz domains: Elementary results, SIAM J. Math. Anal. 12 (1988) 613–626.Google Scholar
  4. [4]
    M. Costabel and E.P. Stephan, Duality estimates for the numerical solution of integral equations, Numer. Math. 54 (1988) 339–353.Google Scholar
  5. [5]
    P. Grisvard, Elliptic Problems in Nonsmooth Domains (Pitman, London, 1985).Google Scholar
  6. [6]
    G.C. Hsiao and W.L. Wendland, The Aubin–Nitsche lemma for integral equations, J. Integral Equations 3 (1981) 299–315.Google Scholar
  7. [7]
    M. Jaswon and G. Symm, Integral EquationMethods in Potential Theory and Elastostatics (Academic Press, New York, 1977).Google Scholar
  8. [8]
    Y. Jeon, Scalar boundary integral equation formulas for the biharmonic equations – Numerical experiments, J. Comput. Appl. Math. 115 (2000) 269–282.Google Scholar
  9. [9]
    Y. Jeon, New indirect scalar boundary integral equation formulas for the biharmonic equation, J. Comput. Appl. Math. 135 (2001) 313–324.Google Scholar
  10. [10]
    W. McLean, Strongly Elliptic Systems and Boundary Integral Equations (Cambridge Univ. Press, Cambridge, 2000).Google Scholar
  11. [11]
    F. Paŕis and J. Cañas, Boundary Element Method, Fundamentals and Applications (Oxford Univ. Press, Oxford, 1997).Google Scholar
  12. [12]
    H. Power and L.C. Wrobel, Boundary Integral Methods in Fluid Mechanics (CMP, Souththampton, 1995).Google Scholar
  13. [13]
    H. Schulz, C. Schwab and W.L. Wendland, The computation of potentials near and on the boundary by an extraction technique for boundary element methods, Comput. Methods Appl. Mech. Engrg. 157 (1998) 225–238.Google Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Youngmok Jeon
    • 1
  • William McLean
    • 2
  1. 1.Department of MathematicsAjou UniversitySuwonKorea
  2. 2.School of MathematicsThe University of New South WalesSydneyAustralia

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