Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
권호 표지

A NOTE ON A FINITE ELEMENT METHOD DEALING WITH CORNER SINGULARITIES

  • Kim, Seok-Chan (Department of Applied Mathematics, Changwon National University) ;
  • Woo, Gyung-Soo (Department of Applied Mathematics, Changwon National University) ;
  • Park, Tae-Hoon (Department of Mathematics, Kookmin University)
  • Published : 2000.05.01

⟨ Previous Next ⟩

Abstract

Recently the first author and his coworker report a new finite element method for the Poisson equations with homogeneous Dirichlet boundary conditions on a polygonal domain with one re-entrant angle [7], They use the well-known fact that the solution of such problem has a singular representation, deduced a well-posed new variational problem for a regular part of solution and an extraction formula for the so-called stress intensity factor using tow cut-off functions. They use Fredholm alternative an Garding's inequality to establish the well-posedness of the variational problem and finite element approximation, so there is a maximum bound for mesh h theoretically. although the numerical experiments shows the convergence for every reasonable h with reasonable size y imposing a restriction to the support of the extra cut-off function without using Garding's inequality. We also give error analysis with similar results.

Keywords

References

  1. Numer. Math. v.33 Direct and inverse error estimates for finite elements with mesh refinements I. BABUSKA;R.B. KELLOGG;J. PITKARANTA
  2. Int. J. Numer. Methods Engrg. v.20 The post-processing approach in the finite element method - part 2: The calculation of stress intensity factors I. BABUSKA;A. MILLER
  3. Numer. Methods PDEs v.6 no.4 The p-version of the finite element method for domains with corners and for infinite domains I. BABUSKA;H. S. OH
  4. Computing v.28 On finite element methods for elliptic equations on domains with corners H. BLUM;M. DOBROWOLSKI
  5. SIAM Numer. Anal. v.29 Coefficients of the singularities for elliptic boundary value problems on domains with conical point Ⅲ. Finite element methods on polygonal domains M. BOURLARD;M. DAUGE;M.-S. LUBUMA;S. NICAISE
  6. Math. Comp. v.68 no.226 Multigrid methods for the computation of singular solutions and stress intensity factor Ⅰ: Corner singularities S. BRENNER
  7. Submitted to SIAM Numer. Anal. A Finite Element Method Using Singular Functions for The Poisson Equations: Corner Singularities Z. CAI;S. KIM
  8. Lecture Notes in Mathematics Elliptic Boundary Value Problems on Corner Domains M. DAUGE
  9. These de Troisieme Cycle Equations Integrales pour un Probleme Singulier dans le Plan M. DJAOUA
  10. Numerical Approximation of Elliptic Interface and Corner Problems M. DOBROWOLSKI
  11. J. Comput. Phy. v.13 On the use of singular functions with finite elements approximations G. J. FIX;S. GULATI;G. I. WAKOFF
  12. Elliptic Problems in Nonsmooth Domains P. GRISVARD
  13. C.R.A.S. Paris v.286 Resolution numerique par une methode d'elements finis du probleme de Dirichlet pour le Laplacien dans un polygone G. RAUGEL
  14. Math. Comp. v.32 no.141 Maximum norm estimates in the finite element method on plane polygonal domains A. SCHATZ;L. WAHLBIN
  15. Math. Comp. v.33 no.146 (refinements) A. SCHATZ;L. WAHLBIN
  16. J. Inst. Maths. Applics. v.16 Singularities in the solution of Laplace's equation in two dimensions R. W. THATCHER
  17. Math. Methods in Applied Sci. On the integral equation method for the plane mixed boundary value problem of Laplacian W. L. WENDLAND;E. STEPHAN;G. C. HSIAO