LEAST-SQUARES SPECTRAL COLLOCATION PARALLEL METHODS FOR PARABOLIC PROBLEMS

• Journal title : Honam Mathematical Journal
• Volume 37, Issue 3,  2015, pp.299-315
• Publisher : The Honam Mathematical Society
• DOI : 10.5831/HMJ.2015.37.3.299
Title & Authors
LEAST-SQUARES SPECTRAL COLLOCATION PARALLEL METHODS FOR PARABOLIC PROBLEMS
SEO, JEONG-KWEON; SHIN, BYEONG-CHUN;

Abstract
In this paper, we study the first-order system least-squares (FOSLS) spectral method for parabolic partial differential equations. There were lots of least-squares approaches to solve elliptic partial differential equations using finite element approximation. Also, some approaches using spectral methods have been studied in recent. In order to solve the parabolic partial differential equations in parallel, we consider a parallel numerical method based on a hybrid method of the frequency-domain method and first-order system least-squares method. First, we transform the parabolic problem in the space-time domain to the elliptic problems in the space-frequency domain. Second, we solve each elliptic problem in parallel for some frequencies using the first-order system least-squares method. And then we take the discrete inverse Fourier transforms in order to obtain the approximate solution in the space-time domain. We will introduce such a hybrid method and then present a numerical experiment.
Keywords
first-order least-squares method;parabolic equation;Fourier transform;
Language
English
Cited by
References
1.
P. B. Bochev and M. D. Gunzburger, Finite element methods of least-squares type, SIAM Review, 40 (1998) 789-837.

2.
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. 66-219 (1997) 935-955.

3.
C. Bernardi and Y. Maday, Approximation Spectrales de Problemes aux Limites Elliptiques, Springer-Verlag, Paris (1992).

4.
J. H. Bramble and T. Sun, A negative-norm least squares method for Reissner-Mindlin plates, Math. Comp., 67 (1998) 901-916.

5.
Z. Cai, R. D. Lazarov, T. Manteuffel, and S. McCormick, First-order system least squares for second-order partial differential equations: Part I, SIAM J. Numer. Anal., 31 (1994) 1785-1799.

6.
Z. Cai, T. Manteuffel, and S. McCormick, First-order system least squares for second-order partial differential equations: Part II, SIAM J. Numer. Anal., 34 (1997) 425-454.

7.
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) 307-318.

8.
C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer-Verlag, New York (1988).

9.
C. L. Chang, Finite element approximation for grad-div type systems in the plane, SIAM J. Numer. Anal., 29 (1992) 452-461.

10.
P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, New York, 1978.

11.
J. Douglas, Jr., J.E. Santos and D. Sheen, Approximation of scalar waves in the space-frequency domain, Math. Model Mehtods Appl. Sci., 4 (1994) 509-531.

12.
J. Douglas, Jr., J.E. Santos, D. Sheen and L.S. Bennethum, Frequency domain treatment of one-dimensional scalar waves, Math. Model Mehtods Appl. Sci., 3 (1993) 171-194.

13.
G.J. Fix, M.D. Gunzburger and R.A. Nicolaides, On finite element methods of the least squares type, Comput. Math. Appl. 5-2 (1979) 87-98.

14.
G.J. Fix and E. Stephan, On the finite element-least squares approximation to higher order elliptic systems, Arch. Rational Mech. Anal., 91-2 (1985) 137-151.

15.
X. Feng and D. Sheen, An elliptic regularity estimate for a problem arising from the frequency domain treatment of waves, Trans. Am. Math. Soc., 346 (1994) 475-487.

16.
P. Hessari and B.-C. Shin, The least-squares pseudo-spectral method for Navier-Stokes equations, Comput. Math. Appl., 66 (2013) 318-329.

17.
S.D. Kim, H.-C. Lee and B.-C. Shin, Pseudo-spectral least-squares method for the second-order elliptic boundary value problem, SIAM J. Numer. Anal., 41-4 (2003) 1370-1387.

18.
S.D. Kim, H.-C. Lee and B.-C. Shin, Least-squares spectral collocation method for the Stokes equations, Numer. Meth. PDE., 20 (2004) 128-139.

19.
S.D. Kim and B.-C. Shin, \$H^{-1}\$ least-squares method for the velocity-pressure-stress formulation of Stokes equations, Appl. Numer. Math., 40 (2002) 451-465.

20.
C.-O. Lee, J. Lee, D. Sheen and Y. Yeom, A frequency-domain parallel method for the numerical approximation of parabolic problems, Comput. Meth. Appl. Mech. Engrg., 169 (1999) 19-29.

21.
C.-O. Lee, J. Lee and D. Sheen, Frequency domain formulation of linearized Navier-Stokes equations, Comput. Meth. Appl. Mech. Engrg., 187 (2000) 351-362.

22.
J. Lee and D. Sheen, An accurate numerical inversion of Laplace transforms based on the location of their poles, Comput. Math. Appl., 48 (2004) 1415-1423.

23.
J. Lee and D. Sheen, A parallel method for backward parabolic problems based on the Laplace transformation, SIAM J. Numer. Anal., 44-4 (2006) 1466-1486.

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) 1368-1377.

25.
M. M. J. Proot and M. I. Gerritsma A least-squares spectral element formulation for the Stokes problem, J. of Sci. Comput., 17 (2002) 285-296.

26.
A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Springer-Verlag, Berlin Heidelberg (1994).

27.
D. Sheen, I.H. Sloan and V. Thomee, A parallel method for time-discretization of parabolic problems based on contour integral representation and quadrature, Math. Comp., 69 (2000) 177-195.

28.
D. Sheen, I.H. Sloan and V. Thomee, A parallel method for time-discretization of parabolic equations based on Laplace transformation and quadrature, IMA J. Numer. Anal., 23-2 (2003) 269-299.