호남수학학술지 (Honam Mathematical Journal)

  • 제40권4호
  • /
  • Pages.785-794
  • /
  • 2018
  • /
  • 1225-293X(pISSN)
  • /
  • 2288-6176(eISSN)
호남수학회 (The Honam Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

FINITE ELEMENT SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATION WITH MULTIPLE CONCAVE CORNERS

  • Kim, Seokchan (Department of Mathematics, Changwon National University) ;
  • Woo, Gyungsoo (Department of Mathematics, Changwon National University)
  • 투고 : 2018.08.12
  • 심사 : 2018.08.31
  • 발행 : 2018.12.25
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

In [8] they introduced a new finite element method for accurate numerical solutions of Poisson equations with corner singularities. They consider the Poisson equations with homogeneous Dirichlet boundary condition with one corner singularity at the origin, and compute the finite element solution using standard FEM and use the extraction formula to compute the stress intensity factor, then pose a PDE with a regular solution by imposing the nonhomogeneous boundary condition using the computed stress intensity factor, which converges with optimal speed. From the solution they could get an accurate solution just by adding the singular part. This approach uses the polar coordinate and the cut-off function to control the singularity and the boundary condition. In this paper we consider Poisson equations with multiple singular points, which involves different cut-off functions which might overlaps together and shows the way of cording in FreeFEM++ to control the singular functions and cut-off functions with numerical experiments.

키워드

HNSHCY_2018_v40n4_785_f0001.png 이미지

FIGURE 1. A domain with multiple concave corners and corresponding polar coordinates

HNSHCY_2018_v40n4_785_f0002.png 이미지

FIGURE 2. Two polar coordinates on a T-shaped domain

TABLE 1. Errors and convergence rates of the λ1,h and λ2,h

HNSHCY_2018_v40n4_785_t0001.png 이미지

TABLE 2. Errors and convergence rates for uh with the Standard FEM

HNSHCY_2018_v40n4_785_t0002.png 이미지

TABLE 3. Errors and convergence rates for uh with our algorithmA

HNSHCY_2018_v40n4_785_t0003.png 이미지

참고문헌

  1. I. BABUSKA, R.B. KELLOGG, AND J. PITKARANTA, Direct and inverse error estimates for finite elements with mesh refinements, Numer. Math., 33 (1979), 447-471. https://doi.org/10.1007/BF01399326
  2. 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
  3. 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
  4. 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) 26352648
  5. G. J. FIX, S. GULATI, AND G. I. WAKOFF, On the use of singular functions with finite elements approximations, J. Comput. Phy., 13 (1973), 209-228. https://doi.org/10.1016/0021-9991(73)90023-5
  6. P. GRISVARD, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, MA, 1985.
  7. F. HECHT, New development in FreeFem++, J. Numer. Math. 20 (2012), no. 3-4, 251265.
  8. 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
  9. 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.