Least-squares for second-order elliptic problems

Computer Methods in Applied Mechanics and Engineering

Volumn 152, Issue 1-2, 1998, Pages 195-210

  Bramble, James H ^a   Lazarov, Raytcho D ^a   Pasciak, Joseph E ^a  

^a TEXAS A AND M UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

BOUNDARY CONDITIONS; BOUNDARY VALUE PROBLEMS; CONVERGENCE OF NUMERICAL METHODS; DIFFERENTIAL EQUATIONS; FINITE ELEMENT METHOD;

SECOND ORDER ELLIPTIC PROBLEMS;

LEAST SQUARES APPROXIMATIONS;

EID: 0031695756     PISSN: 00457825     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0045-7825(97)00189-8     Document Type: Article

References (25)

1. R. Adams, Sobolev Spaces (Academic Press, Inc., New York, 1975).
2. A.K. Aziz, R.B. Kellogg and A.B. Stephens, Least-squares methods for elliptic systems. Math. Comput. 44 (1985) 53-70.
3. P.B. Bochev and M.D. Gunzburger, Accuracy of least-squares methods for the Navier-Stokes equations, Comput. Fluids 22 (1993) 549-563.
4. P.B. Bochev and M.D. Gunzburger, Analysis of least-squares finite element methods for the Stokes equations, Math. Comput. 63 (1994) 479-506.
5. J. Bramble and J. Pasciak, Least-squares methods for Stokes equations based on a discrete minus one inner product, J. Comput. Appl. Math. Comput. (1997) to appear.
6. 2 projections, Math. Comput. 56 (1991) 463-476.
7. 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. Comput. 66 (1977) to appear.
8. Z. Cai, R. Lazarov, T. Manteuffel and S. McCormick, First-order system least-squares for second-order differential equations: Part I, SIAM J. Numer. Anal. 31 (1994) 1785-1802.
9. 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) to appear.
10. T.F. Chen and G.J. Fix, Least-squares finite element simulation of transonic flows, Appl. Numer. Math. 2 (1986) 399-408.
11. P. Ciarlet, Basic error estimates for elliptic problems, in: P. Ciarlet and J. Lions, eds., Finite Element Methods: Handbook of Numerical Analysis, Vol. II (North-Holland, New York, 1991) 18-352.
12. R. Falk, A finite element method for the stationary Stokes equations using trial functions which do not satisfy div v = 0, Math. Comput. 30 (1976) 698-702.
13. P. Grisvard, Elliptic Problems in Nonsmooth Domains (Pitman, Boston, 1985).
14. T.J.R. Hughes and L.P. Franca, A new finite element formulation for computational fluid dynamics. VII. The Stokes problems with various well-posed boundary conditions: symmetric formulation that converges for all velocity pressure spaces, Comput. Methods Appl. Mech. Engrg. 65 (1987) 85-96.
15. T.J.R. Hughes, L.P. Franca and M. Bulestra, A new finite element formulation for computational fluid dynamics. V. Circumventing the Babuška-Brezzi condition: A stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations, Comput. Methods Appl. Mech. Engrg. 59 (1986) 85-99.
16. D.C. Jespersen, A least-square decomposition method for solving elliptic systems, Math. Comput. 31 (1977) 873-880.
17. B.N. Jiang and C. Chang, Least-squares finite elements for the Stokes problem, Comput. Methods Appl. Mech. Engrg. 81 (1990) 13-37.
18. P. Neittaanmäki and J. Saranen, On finite element approximation of the gradient for the solution to Poisson equation, Numer. Math. 37 (1981) 131-148.
19. A.I. Pehlivanov, G.F. Carey and P.S. Vassilevski, least-squares mixed finite element methods for non-selfadjoint elliptic problems: I. Error estimates, Numer. Math. 72 (1966) 502-522.
20. 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.
21. A. Schatz, An observation concerning Ritz-Galerkin methods with indefinite bilinear forms, Math. Comput. 28 (1974) 959-962.
22. A. Schatz and L. Wahlbin, Maximum norm estimates in the finite element method on plane polygonal domains. Part 2, Refinements, Math. Comput. 33 (1979) 465-492.
23. A. Schatz and J. Wang, Some new error estimates for Ritz-Galerkin methods with minimal regularity assumptions, Math. Comput. 65 (1996) 19-27.
24. W.L. Wendland, Elliptic Systems in the Plane (Pitman, London, 1979).
25. X. Zhang, Studies in domain decomposition: multi-level methods and the biharmonic Dirichlet problem, Ph.D. Thesis, NYU Courant Inst. Preprint # TR 1991-583.