Journal of the Korean Society for Industrial and Applied Mathematics

  • 제22권2호
  • /
  • Pages.101-113
  • /
  • 2018
  • /
  • 1226-9433(pISSN)
  • /
  • 1229-0645(eISSN)
한국산업응용수학회 (Korean Society for Industrial and Applied Mathematics)
권호 표지
DOI QR코드

DOI QR Code

FINITE ELEMENT DUAL SINGULAR FUNCTION METHODS FOR HELMHOLTZ AND HEAT EQUATIONS

  • JANG, DEOK-KYU (DEPARTMENT OF MATHEMATICS, KANGWON NATIONAL UNIVERSITY) ;
  • PYO, JAE-HONG (DEPARTMENT OF MATHEMATICS, KANGWON NATIONAL UNIVERSITY)
  • 투고 : 2018.03.16
  • 심사 : 2018.06.11
  • 발행 : 2018.06.25
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

The dual singular function method(DSFM) is a numerical algorithm to get optimal solution including corner singularities for Poisson and Helmholtz equations. In this paper, we apply DSFM to solve heat equation which is a time dependent problem. Since the DSFM for heat equation is based on DSFM for Helmholtz equation, it also need to use Sherman-Morrison formula. This formula requires linear solver n + 1 times for elliptic problems on a domain including n reentrant corners. However, the DSFM for heat equation needs to pay only linear solver once per each time iteration to standard numerical method and perform optimal numerical accuracy for corner singularity problems. Because the Sherman-Morrison formula is rather complicated to apply computation, we introduce a simplified formula by reanalyzing the Sherman-Morrison method.

키워드

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. M. Bourlard, M. Dauge, M.-S. Lubuma, and S. Nicaise, Coeficients of the singularities for elliptic boundary value problems on domains with conical points III. Finite element methods on polygonal domains, SIAM Numer. Anal. 29 (1992), 136-155. https://doi.org/10.1137/0729009
  3. Z. Cai and S. Kim, A finite element method using singular functions for the Poisson equation: Corner singu- larities, SIAM J. Numer. Anal. 39 (2001), 286-299. https://doi.org/10.1137/S0036142999355945
  4. Z. Cai, S. C. Kim and B. C. Shin, Solution methods for the Poisson equation: corner singularities, SIAM J. SCI. COMPut. 23 (2001), 672-682. https://doi.org/10.1137/S1064827500372778
  5. H.J. Choi and J.R. Kweon The fourier finite element method for the corner singularity expansion of the heat equation Computers Math. Applic. 69 (2015), 13-30. https://doi.org/10.1016/j.camwa.2014.11.004
  6. Gene H. Colub and Charles F. Van Loan Matrix Computations, The John Hopkins University Press (1996).
  7. M. Djaoua, Equations Integrales pour un Probleme Singulier dans le Plan, These de Troisieme Cycle, Universite Pierre et Marie Curie, Paris, 1977.
  8. M. Dobrowolski, Numerical approximation of elliptic interface and corner problems, Habilitation-Schrift, Bonn, 1981.
  9. 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
  10. S.C. Kim, J.S. Lee and J.H. Pyo A finite element method using singular functions for Helmholtz equations: Part I, J. KSIAM 12 (2008), 13-23.

피인용 문헌

  1. An Optimal Finite Element Method with Uzawa Iteration for Stokes Equations including Corner Singularities vol.2021, pp.None, 2018, https://doi.org/10.1155/2021/5592982