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A least-squares method for second order noncoercive elliptic partial differential equations

Author: JaEun Ku
Journal: Math. Comp. 76 (2007), 97-114
MSC (2000): Primary 65N30; Secondary 65N15
Published electronically: September 28, 2006
MathSciNet review: 2261013
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Abstract: In this paper, we consider a least-squares method proposed by Bramble, Lazarov and Pasciak (1998) which can be thought of as a stabilized Galerkin method for noncoercive problems with unique solutions. We modify their method by weakening the strength of the stabilization terms and present various new error estimates. The modified method has all the desirable properties of the original method; indeed, we shall show some theoretical properties that are not known for the original method. At the same time, our numerical experiments show an improvement of the method due to the modification.

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  • 1. A. K. Aziz, R. B. Kellogg and A. B. Stephens, Least-squares methods for elliptic systems, Math. Comp., 44(1985), pp. 53-70.
  • 2. P. B. Bochev and M. D. Gunzburger, Analysis of least-squares finite element methods for the Stokes equations, Math. Comp., 63(1994), pp. 479-506.
  • 3. 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), pp. 935-955.
  • 4. J. H. Bramble, R. D. Lazarov and J. E. Pasciak, Least squares for 2nd order elliptic problems, Comput. Methods Appl. Mech. Engrg, 152(1998), no. 1-2, pp. 195-210.
  • 5. J. H. Bramble and A. H. Schatz, Least squares for $ 2m$th order elliptic boundary-value problems, Math. Comp., 25(1971), pp. 1-32.
  • 6. S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Springer, 1994.
  • 7. Z. Cai, R. Lazarov, T. A. Manteuffel, and S. F. McCormick, First-order system least squares for second-order partial differential equations: part I, SIAM J. Numer. Anal., 31(1994), pp. 1785-1799.
  • 8. Z. Cai, T. A. Manteuffel, and S. F. McCormick, First-order system least squares for second-order partial differential equations: part II, SIAM J. Numer. Anal., 34(1997), pp. 1727-1741.
  • 9. P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, 1985.
  • 10. J. A. Nitsche and A. H. Schatz, Interior estimates for Ritz-Galerkin methods, Math. Comp., 28(1974), pp. 937-958.
  • 11. A. I. Pehlivanov, G. F. Carey, and P. S. Vassilievski, Least-squares mixed finite element methods for non-selfadjoint elliptic problems: I. Error extimates, Numer. Math. 72(1996), pp. 501-522.
  • 12. A. H. Schatz, An observation concerning Ritz-Galerkin methods with indefinite bilinear forms, Math. Comp., 28(1974), pp. 159-164.
  • 13. A. H. Schatz, Pointwise error estimates for the finite element method and asymptotic error expansion inequalities for the finite element method on irregular grids: Part I. Golobal Estimates, Math. Comp., 67(1998), pp. 877-899.
  • 14. A. H. Schatz and L. B. Wahlbin, Interior maximum norm estimates for finite element methods, Math. Comp., 64(1995), pp. 907-928.
  • 15. L. R. Scott and S, Zhang, Finite element interpolation of non-smooth functions satisfying boundary conditions, Math. Comp., 54(1990), pp. 483-493.
  • 16. G. Strang and G. J. Fox, An Analysis of the Finite Element Method, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1973.
  • 17. L. B. Wahlbin, Superconvergence in Galerkin Finite Element Methods, Lecture Notes in Mathematics 1605, 1995.

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Additional Information

JaEun Ku
Affiliation: Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, Indiana 47907-2067

Keywords: Least-squares, stabilized Galerkin method, error estimates
Received by editor(s): November 2, 2004
Received by editor(s) in revised form: July 7, 2005
Published electronically: September 28, 2006
Additional Notes: Research supported in part by NSF grant DMS-0071412.
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.