Publisher : Department of Mathematics, Kyungpook National University
DOI : 10.5666/KMJ.2010.50.3.419
Title & Authors
Preconditioning Cubic Spline Collocation Methods for a Coupled Elliptic Equation Shin, Byeong-Chun; Kim, Sang-Dong;
A low-order finite element preconditioner is analyzed for a cubic spline collocation method which is used for discretizations of coupled elliptic problems derived from an optimal control problrm subject to an elliptic equation. Some numerical evidences are also provided.
Coupled elliptic equations;Cubic spline collocation methods;Finite element preconditioner;
P. Bochev and M. Gunzburger, Least-squares finite element methods for optimality systems arising in optimization and control problems, SIAM J. Numer. Anal., 43(2006), 2517-2543.
Y. Chen, N. Yi and W. Liu, A Legendre-Galerkin spectral method for optimal control problems governed by elliptic equations, SIAM J. Numer. Anal., 46(2008), 2254-2275.
J. Douglas and T. Dupont, Collocation methods for parabolic equations in a single space variable, Lecture Notee in Mathematics 385, Springer-Verlag Press, Cambridge, UK, 2002.
S. C. Eisenstat, H. C. Elman, and M. H. Schultz, Variational iterative methods for nonsymmetric systems of linear equations, SIAM J. Numer. Anal., 20(1983), 345-357.
A. Greenbaum, Iterative methods for solving linear systems, SIAM, Philadelphia, 1997.
M. Gunzburger, Perspectives in Flow Control and Optimization, Adv. Des. Control 5, SIAM, Philadelphia, 2002.
R. A. Horn and C. R. Johnson, Topics in Matrix Analysis. Cambridge University Press, Cambridge, 1994.
S. D. Kim, Piecewise bilinear preconditioning of high-order finite element methods, ETNA, 26(2007), 228-242.
S. Kim and S. D. Kim, Preconditioning on high-order element methods using Chebyshev-Gauss-Lobatto notes, Applied Numerical Mathematics, 59(2009) 316-333.
S. D. Kim and S. Kim, Exponential decay of $C^1$-cubic splines vanishiing at two symmetric points in each knot interval, Numer. Math., 76(1997), 470-488.
S. D. Kim and S. V. Parter, Preconditioning Chebyshev spectral collocation method for elliptic partial differential equations, SIAM J. Numer. Anal., 33(1996), 2375-2400.
S. D. Kim and S. V. Parter, Preconditioning cubic spline collocation discretizations of elliptic equations, Numer. Math., 72(1995), 39-72.
S. D. Kim and B. C. Shin, On the exponential decay of $C^1$ cubic Lagrange splines on non-uniform meshes and for non-uniform data points, Houston J. of Math., 24(1998), 173-183.
Y. Saad and M. H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric lienar systems, SIAM J. Sci. Comput., 7(1986), 856-869.
L. N. Trefethen and D. B. Bau, Numerical linear algegra, SIAM, Philadelphia, 1997.