# EFFICIENT PARAMETERS OF DECOUPLED DUAL SINGULAR FUNCTION METHOD

• Kim, Seok-Chan (DEPARTMENT OF APPLIED MATHEMATICS, CHANGWON NATIONAL UNIVERSITY) ;
• Pyo, Jae-Hong (DEPARTMENT OF MATHEMATICS, KANGWON NATIONAL UNIVERSITY)
• Accepted : 2009.11.18
• Published : 2009.12.25

#### Abstract

The solution of the interface problem or Poisson problem with concave corner has singular perturbation at the interface corners or singular corners. The decoupled dual singular function method (DDSFM) which exploits the singular representations of the solutions was suggested in [3, 9] and estimated optimal accuracy in [10]. The convergence rates consist with theoretical results even for the problems with very strong singularity, with the efficiency depending on parameters used in the methods. Furthermore the errors in $L^2$ and $L^\infty$-spaces display some oscillation, in the cases with meshsize not small enough. In this paper, we present an answer to remove the oscillation via numerical experiments. We observe the effects of parameters in DDSFM, and show the consisting efficiency of the method over the strong singularity.

#### Acknowledgement

Supported by : National Research Foundation, Korea Research Foundation

#### References

1. 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. https://doi.org/10.1016/S0045-7825(03)00455-9
2. Z. CHEN AND S. DAI, On the efficiency of adaptive finite element methods for elliptic problems with discontinuous coefficients, SIAM J. Sci. Comput., 24:(2002), 443-462. https://doi.org/10.1137/S1064827501383713
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 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. https://doi.org/10.1016/j.cma.2005.06.004
5. Z. CAI, S. 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
6. P. GRISVARD, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, MA, 1985.
7. 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.
8. R.B. KELLOGG, On the Poisson equation with intersecting interfaces, Appl. Anal. 4(1975) 101-129.
9. S.C. KIM, Z. CAI, J.H. PYO AND S. KONG, A finite element method using singular functions: interface problems, Hokkaido Mathematical Jorunal, 36(2007) 815-836. https://doi.org/10.14492/hokmj/1272848035
10. S.C. KIM AND J.H. PYO, The Optimal Error Estimate of the Decoupled Dual Singular Function Method, submitted.
11. 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.
12. P. MORIN, R.H. NOCHETTO, AND K.G. SIEBERT Data oscillation and convergence of adaptive FEM, SIAM J. Numer. Anal., 38 (2000), pp. 466.488. https://doi.org/10.1137/S0036142999360044
13. S. NICAISE, Polygonal Interface Problems, Peter Lang, Frankfurt am Main, 1993.
14. S. NICAISE, ANNA-MARGARETE SANDIG, General Interface Problems-II, Math. Methods. Appl. Aci. 17(1994) 431-450. https://doi.org/10.1002/mma.1670170603
15. M. PETZOLDT, A posteriori error estimators for elliptic equations with discontinuous coefficients, Adv. Comput. Math. 16 (2002), no. 1, 47–75. https://doi.org/10.1023/A:1014221125034
16. J. XU Counterexamples concerning a weighted $L^{2}$ projection, Math. Comput., 57(1991), 563-568,