Browse > Article
http://dx.doi.org/10.4134/JKMS.2013.50.6.1291

PSEUDO-SPECTRAL LEAST-SQUARES METHOD FOR ELLIPTIC INTERFACE PROBLEMS  

Shin, Byeong-Chun (Department of Mathematics Chonnam National University)
Publication Information
Journal of the Korean Mathematical Society / v.50, no.6, 2013 , pp. 1291-1310 More about this Journal
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;
Citations & Related Records
연도 인용수 순위
  • Reference
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.   DOI   ScienceOn
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).   DOI   ScienceOn
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.   DOI   ScienceOn
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.   DOI   ScienceOn
6 P. B. Bochev and M. D. Gunzburger, Finite element methods of least-squares type, SIAM Rev. 40 (1998), no. 4, 789-837.   DOI   ScienceOn
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.   DOI   ScienceOn
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.   DOI   ScienceOn
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.   DOI   ScienceOn
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.   DOI   ScienceOn
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).   DOI   ScienceOn
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).   DOI   ScienceOn
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.   DOI   ScienceOn
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.   DOI   ScienceOn
17 D. Jesperson, A least squares decomposition method for solving elliptic equations, Math. Comp. 31 (1977), no. 140, 873-880.   DOI
18 B.-N. Jiang, The Least-Squares Finite Element Method, Springer-Verlag, Berlin, 1998.
19 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).   DOI   ScienceOn
20 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.   DOI
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.   DOI   ScienceOn
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.   DOI   ScienceOn
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.   DOI   ScienceOn
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.   DOI
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.   DOI
28 B.-C. Shin and J.-H. Jung, Spectral collocation and radial basis function methods for one-dimensional interface problems, Appl. Numer. Math. 61 (2011), no. 8, 911-928.   DOI   ScienceOn
29 A.-K. Tornberg and B. Engquist, Numerical approximations of singular source terms in differential equations, J. Comput. Phys. 200 (2004), no. 2, 462-488.   DOI   ScienceOn