JOURNAL BROWSE
Search
Advanced SearchSearch Tips
FINITE ELEMENT APPROXIMATION OF THE DISCRETE FIRST-ORDER SYSTEM LEAST SQUARES FOR ELLIPTIC PROBLEMS
facebook(new window)  Pirnt(new window) E-mail(new window) Excel Download
 Title & Authors
FINITE ELEMENT APPROXIMATION OF THE DISCRETE FIRST-ORDER SYSTEM LEAST SQUARES FOR ELLIPTIC PROBLEMS
SHIN, Byeong-Chun;
  PDF(new window)
 Abstract
In [Z. Cai and B. C. Shin, SIAM J. Numer. Anal. 40 (2002), 307-318], we developed the discrete first-order system least squares method for the second-order elliptic boundary value problem by directly approximating space based on the Helmholtz decomposition. Under general assumptions, error estimates were established in the norms for the vector and scalar variables, respectively. Such error estimates are optimal with respect to the required regularity of the solution. In this paper, we study solution methods for solving the system of linear equations arising from the discretization of variational formulation which possesses discrete biharmonic term and focus on numerical results including the performances of multigrid preconditioners and the finite element accuracy.
 Keywords
least-squares method;multigrid;preconditioner;
 Language
English
 Cited by
 References
1.
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 system, Math. Comp. 66 (1997), 935-955 crossref(new window)

2.
J. H. Bramble and T. Sun, A negative-norm least squares method for Reissner-Mindlin plates, Math. Comp. 67 (1998), 901-916 crossref(new window)

3.
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 crossref(new window)

4.
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 crossref(new window)

5.
Z. Cai and B. C. Shin, The discrete first-order system least-squares: the secondorder elliptic boundary value problem, SIAM J. Numer. Anal. 40 (2002), 307-318 crossref(new window)

6.
C. L. Chang, Finite element approximation for gmd-div type systems in the plane, SIAM J. Numer. Anal. 29 (1992), 452-461 crossref(new window)

7.
S. D. Kim and B. C. Shin, H-1 least-squares methods for the velocity-pressurestress formulation of Stokes equations, Appl. Numer. Math. 40 (2002), no. 4, 451-465 crossref(new window)

8.
A. 1. 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 crossref(new window)

9.
A. Quarteroni and A. Valli, Numerical approximation of partial differential equations, Springer-Verlag, Berlin Heidelberg, 1994