PSEUDO-SPECTRAL LEAST-SQUARES METHOD FOR ELLIPTIC INTERFACE PROBLEMS

- Journal title : Journal of the Korean Mathematical Society
- Volume 50, Issue 6, 2013, pp.1291-1310
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/JKMS.2013.50.6.1291

Title & Authors

PSEUDO-SPECTRAL LEAST-SQUARES METHOD FOR ELLIPTIC INTERFACE PROBLEMS

Shin, Byeong-Chun;

Shin, Byeong-Chun;

Abstract

This paper develops least-squares pseudo-spectral collocation methods for elliptic boundary value problems having interface conditions given by discontinuous coefficients and singular source term. From the discontinuities of coefficients and singular source term, we derive the interface conditions and then we impose such interface conditions to solution spaces. We define two types of discrete least-squares functionals summing discontinuous spectral norms of the residual equations over two sub-domains. In this paper, we show that the homogeneous least-squares functionals are equivalent to appropriate product norms and the proposed methods have the spectral convergence. Finally, we present some numerical results to provide evidences for analysis and spectral convergence of the proposed methods.

Keywords

Pseudo-spectral method;least-squares method;interface problem;

Language

English

References

1.

A. K. Aziz, R. B. Kellogg, and A. B. Stephens, Least squares methods for elliptic systems, Math. Comp. 44 (1985), no. 169, 53-70.

2.

C. Bernardi and Y. Maday, Approximations spectrales de problemes aux limites ellip-tiques, vol. 10 of Mathematiques & Applications (Berlin) [Mathematics & Applications], Springer-Verlag, Paris, 1992.

3.

M. Berndt, T. A. Manteuffel, and S. F. McCormick, Analysis of first-order system least squares (FOSLS) for elliptic problems with discontinuous coefficients. II, SIAM J. Numer. Anal. 43 (2005), no. 1, 409-436 (electronic).

4.

M. Berndt, T. A. Manteuffel, S. F. McCormick, and G. Starke, Analysis of first-order system least squares (FOSLS) for elliptic problems with discontinuous coefficients. I, SIAM J. Numer. Anal. 43 (2005), no. 1, 386-408.

5.

P. B. Bochev and M. D. Gunzburger, Analysis of least squares finite element methods for the Stokes equations, Math. Comp. 63 (1994), no. 208, 479-506.

6.

P. B. Bochev and M. D. Gunzburger, Finite element methods of least-squares type, SIAM Rev. 40 (1998), no. 4, 789-837.

7.

P. Boomkamp, B. Boersma, R. Miesen, and G. Beijnon, A chebyshev collocation method for solving two-phase flow stability problems, J. Comput. Phys. 132 (1997), 191-200.

8.

J. H. Bramble, R. D. Lazarov, and J. E. Pasciak, A least-squares approach based on a discrete minus one inner product for first order systems, Math. Comp. 66 (1997), no. 219, 935-955.

9.

Z. Cai, R. Lazarov, T. A. Manteuffel, and S. F. McCormick, First-order system least squares for second-order partial differential equations. I, SIAM J. Numer. Anal. 31 (1994), no. 6, 1785-1799.

10.

Z. Cai, T. A. Manteuffel, and S. F. McCormick, First-order system least squares for second-order partial differential equations. II, SIAM J. Numer. Anal. 34 (1997), no. 2, 425-454.

11.

Z. Cai and B. C. Shin, The discrete first-order system least squares: the second-order elliptic boundary value problem, SIAM J. Numer. Anal. 40 (2002), no. 1, 307-318 (electronic).

12.

C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer Series in Computational Physics, Springer-Verlag, New York, 1988.

13.

Y. Cao and M. D. Gunzburger, Least-squares finite element approximations to solutions of interface problems, SIAM J. Numer. Anal. 35 (1998), no. 1, 393-405 (electronic).

14.

G. J. Fix, M. D. Gunzburger, and R. A. Nicolaides, On finite element methods of the least squares type, Comput. Math. Appl. 5 (1979), no. 2, 87-98.

15.

G. J. Fix and E. Stephan, On the finite element-least squares approximation to higher order elliptic systems, Arch. Rational Mech. Anal. 91 (1985), no. 2, 137-151.

16.

D. Funaro, A variational formulation for the Chebyshev pseudospectral approximation of Neumann problems, SIAM J. Numer. Anal. 27 (1990), no. 3, 695-703.

17.

D. Jesperson, A least squares decomposition method for solving elliptic equations, Math. Comp. 31 (1977), no. 140, 873-880.

18.

B.-N. Jiang, The Least-Squares Finite Element Method, Springer-Verlag, Berlin, 1998.

19.

J.-H. Jung, A note on the spectral collocation approximation of some differential equa- tions with singular source terms, J. Sci. Comput. 39 (2009), no. 1, 49-66.

20.

S. D. Kim, H.-C. Lee, and B. C. Shin, Pseudospectral least-squares method for the second-order elliptic boundary value problem, SIAM J. Numer. Anal. 41 (2003), no. 4, 1370-1387 (electronic).

21.

S. D. Kim, H.-C. Lee, and B. C. Shin, Least-squares spectral collocation method for the Stokes equations, Numer. Methods Partial Differential Equations 20 (2004), no. 1, 128-139.

22.

S. D. Kim and B. C. Shin, Chebyshev weighted norm least-squares spectral methods for the elliptic problem, J. Comput. Math. 24 (2006), no. 4, 451-462.

23.

A. Loubenets, T. Ali, and M. Hanke, Highly accurate finite element method for one- dimensional elliptic interface problems, Appl. Numer. Math. 59 (2009), no. 1, 119-134.

24.

A. I. Pehlivanov, G. F. Carey, and R. D. Lazarov, Least-squares mixed finite elements for second-order elliptic problems, SIAM J. Numer. Anal. 31 (1994), no. 5, 1368-1377.

25.

M. M. J. Proot and M. I. Gerritsma, A least-squares spectral element formulation for the Stokes problem, J. Sci. Comput. 17 (2002), no. 1-4, 285-296.

26.

A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, vol. 23 of Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 1994.

27.

Z. G. Seftel, A general theory of boundary value problems for elliptic systems with discontinuous coefficients, Ukrain. Mat. Z. 18 (1966), no. 3, 132-136.