East Asian mathematical journal

  • 제38권5호
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  • Pages.603-610
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  • 2022
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  • 1226-6973(pISSN)
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  • 2287-2833(eISSN)
영남수학회 (The Youngnam Mathematical Society)
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SINGULAR AND DUAL SINGULAR FUNCTIONS FOR PARTIAL DIFFERENTIAL EQUATION WITH AN INPUT FUNCTION IN H1(Ω)

  • Woo, Gyungsoo (Department of Mathematics, Changwon National University) ;
  • Kim, Seokchan (Department of Mathematics, Changwon National University)
  • 투고 : 2022.08.12
  • 심사 : 2022.08.26
  • 발행 : 2022.09.30
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초록

In [6, 7] they introduced a new finite element method for accurate numerical solutions of Poisson equations with corner singularities. They consider the Poisson equations with homogeneous boundary conditions, compute the finite element solutions using standard FEM and use the extraction formula to compute the stress intensity factor(s), then they posed new PDE with a regular solution by imposing the nonhomogeneous boundary condition using the computed stress intensity factor(s), which converges with optimal speed. From the solution they could get an accurate solution just by adding the singular part. They considered a partial differential equation with the input function f ∈ L2(Ω). In this paper we consider a PDE with the input function f ∈ H1(Ω) and find the corresponding singular and dual singular functions. We also induce the corresponding extraction formula which are the basic element for the approach.

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과제정보

This author was financially supported by Changwon National University in 2021-2022.

참고문헌

  1. H. Blum and M. Dobrowolski, On finite element methods for elliptic equations on domains with corners, Computing, 28 (1982), 53-63. https://doi.org/10.1007/BF02237995
  2. Z. Cai and S.C. Kim, A finite element method using singular functions for the poisson equation: Corner singularities, SIAM J. Numer. Anal., 39:(2001), 286-299. https://doi.org/10.1137/S0036142999355945
  3. Z. Cai , S.C. Kim, S.D. Kim, S. Kong, A finite element method using singular functions for Poisson equations: Mixed boundary conditions, Comput. Methods Appl. Mech. Engrg. 195 (2006) 2635-2648 https://doi.org/10.1016/j.cma.2005.06.004
  4. P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, MA, 1985.
  5. S. Kim and S. Kong, Remarks on finite element methods for corner singularities using SIF, Honam Mathematical J., 38(2016), No.3, 661-674. https://doi.org/10.5831/HMJ.2016.38.3.661
  6. S. Kim and H.C. Lee, A finite element method for computing accurate solutions for Poisson equations with corner singularities using the stress intensity factor, Computers and Mathematics with Applications, 71(2016) 2330-2337. https://doi.org/10.1016/j.camwa.2015.12.023
  7. S. Kim and H.-C. Lee, Finite element method to control the domain singularities of Poisson equation using the stress intensity factor : mixed boundary condition, Int. J. Numer. Anal. Model, 14:4-5 (2017), 500-510.
  8. S. Kim B. Palta, and H.-S. Oh, Extraction formulas of stress intensity factors for the biharmonic equations containing crack singularities, Computers and Mathematics with Applications, 80(2020) 1142-1163. https://doi.org/10.1016/j.camwa.2020.05.026
  9. S. Kim B. Palta, J. Jeong and H.-S. Oh, Extraction of stress intensity factors of biharmonic equations with corner singularities corresponding to mixed boundary conditions of clamped, simply supported, and free (II), Computers and Mathematics with Applications, 109(2022) 235-259. https://doi.org/10.1016/j.camwa.2022.01.028