NOTES ON NEW SINGULAR FUNCTION METHOD FOR DOMAIN SINGULARITIES

• Journal title : Honam Mathematical Journal
• Volume 29, Issue 4,  2007, pp.701-721
• Publisher : The Honam Mathematical Society
• DOI : 10.5831/HMJ.2007.29.4.701
Title & Authors
NOTES ON NEW SINGULAR FUNCTION METHOD FOR DOMAIN SINGULARITIES
Kim, Seok-Chan; Pyo, Jae-Hong; Xie, Shu-Sen; Yi, Su-Cheol;

Abstract
Recently, a new singular function(NSF) method was posed to get accurate numerical solution on quasi-uniform grids for two-dimensional Poisson and interface problems with domain singularities by the first author and his coworkers. Using the singular function representation of the solution, dual singular functions, and an extraction formula for stress intensity factors, the method poses a weak problem whose solution is in $\small{H^2({\Omega})}$ or $\small{H^2({\Omega}_i)}$. In this paper, we show that the singular functions, which are not in $\small{H^2({\Omega})}$, also satisfy the integration by parts and note that this fact suggests the possibility of different choice of the weak formulations. We show that the original choice of weak formulation of NSF method is critical.
Keywords
Finite element;Singular function method;
Language
English
Cited by
1.
CLASSIFICATION OF SINGULAR SOLUTIONS FOR THE POISSON PROBLEM WITH VARIOUS BOUNDARY CONDITIONS, Honam Mathematical Journal, 2009, 31, 4, 579
References
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.

2.
H. BLUM AND M. DOBROWOLSKI, On finite element methods for elliptic equations on domains with corners, Computing, 28 (1982), 53-63.

3.
M. BOURLARD, M. DAUGE, M.-S. LUBUMA, AND S. NICAISE, Coefficients 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.

4.
S. C. BRENNER, Multigrid methods for the computation of singular solutions and stress intensity factor I: Corner singularities, Math. Comp., 68 (226), (1999), 559-583.

5.
S. C. BRENNER AND L.-Y. SUNG, Multigrid methods for the computation of singular solutions and stress intensity factors III: Interface singularities, Comput. Methods Appl. Mech. Engrg. 192(2003), 4687-4702.

6.
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.

7.
Z. CAI, S.C. KIM, S.D. KIM AND S. KONG, A finite element method using singular functions for the Poisson equation: Mixed boundary condition, Computer Methods in Applied Mechanics and Engineering, 195:(2006), 2635-2648.

8.
Z. CAI, S. KIM, AND B.-C. SHIN, Solution methods for the Poisson equation: Corner singularities, SIAM J. Sci. Comput., SIAM J. SCL COMPUT., 23:(2001), 672-682.

9.
M. DJAOUA, Equations Integrales pour un Probleme Singulier dans le Plan, These de Troisieme Cycle, Universite Pierre et Marie Curie, Paris, 1977.

10.
M. DOBROWOLSKI, Numerical Approximation of Elliptic Interface and Corner Problems, Habilitationsschrift, Bonn, 1981.

11.
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.

12.
V. GIRAULT AND P. A. RAVIART, Finite element methods for Navier-Stokes equations : theory and algorithms, Springer-Verlag, Berlin, 1986.

13.
P. GRISVARD, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, MA, 1985.

14.
R.B. KELLOGG, Singularities in interface problems, in: B. Hubbard(ED.), Numerical Solution of Partial Differential Equations II, Academic Press, New York, (1971) 351-400.

15.
R.B. KELLOGG, On the Poisson equation with intersecting interfaces, Appl. Anal. 4(1975) 101-129.

16.
S.C. KIM, Z. CAI, J.H. PYO AND S. KONG, A finite element method using singular functions: interface problems, Hokkaido Mathematical Jorunal, To appear.

17.
D. MERCIER, Minimal regularity of the solutions of solutions of some transmission problems, Technical Report 01.7, Universite de Valenciennes et du Hainaut-Cambresis, 2001.

18.
S. NICAISE, Polygonal Interface Problems, Peter Lang, Frankfurt am Main, 1993.

19.
A. SCHATZ AND L. WAHLBIN, Maximum norm estimates in the finite element method on plane polygonal domains, Part 1, Math. Comp., 32 (141), (1978) 73-109

20.
A. SCHATZ AND L. WAHLBIN, Maximum norm estimates in the finite element method on plane polygonal domains, Part 2 (refinements), Math. Comp., 33 (146), (1979) 465-492.

21.
CH. SCHWAB, p- and hp-Finite Element Methods, Oxford University Press, Oxford, 1998.

22.
B. A. SZABO AND I. BABUSKA, Finite Element Analysis, John Wiley & Sons, New York, 1991.