Local error estimation and adaptive remeshing scheme for least-squares mixed finite elements

Computer Methods in Applied Mechanics and Engineering

Volumn 150, Issue 1-4, 1997, Pages 125-131

  Carey, G F ^a   Pehlivanov, A I ^a  

^a UNIVERSITY OF TEXAS AT AUSTIN   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

BOUNDARY VALUE PROBLEMS; ERROR ANALYSIS; FINITE ELEMENT METHOD; LEAST SQUARES APPROXIMATIONS; MATHEMATICAL MODELS;

ADAPTIVE REMESHING SCHEME; LEAST SQUARES MIXED FINITE ELEMENTS;

COMPUTATIONAL GEOMETRY;

EID: 0031377806     PISSN: 00457825     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0045-7825(97)00098-4     Document Type: Article

References (34)

1. M. Ainsworth and J.T. Oden, A unified approach to a posteriori error estimation using element residual methods, Numer. Math. 65 (1993) 23-50.
2. I. Babuška, The finite element method with Lagrange multipliers, Numer. Math. 20 (1973) 179-392.
3. I. Babuška, I., T. Strouboulis, C.S. Upadhyay and S.K. Gangaraj, A model study of element residual estimators for linear elliptic problems: the equality of the estimators in the interior of meshes of triangles and quadrilaterals, Comput. Struct. 57 (1995) 1009-1028.
4. 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, (1994) to appear.
5. F. Brezzi, On the existence, uniqueness and approximation of saddle point problems arising from Lagrange multipliers, RAIRO Sér. Anal. Numér. 8(R-2) (1974) 129-151.
6. F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods (Springer-Verlag, Berlin, 1991).
7. Z. Cai, R. Lazarov, T.A. Manteuffel and S.F. McCormick, First-order system least squares for second-order pratial differential equations. I, SIAM J. Numer. Anal. 31 (1994) 1785-1799.
8. G.F. Carey, Computational Grids: Generation, Refinement and Solution Strategies (Taylor and Francis, 1997).
9. G.F. Carey, A.I. Pehlivanov and Y. Shen, Least-squares mixed finite elements, In: M. Křížek et al., eds., Finite Element Methods - Proc. Conf. Fifty Years of the Courant Element, Jyväskylä, Finland, 1993 (Marcel Dekker, New York, 1994) 105-117.
10. G.F. Carey, A.I. Pehlivanov and P.S. Vassilevski, Least-squares mixed finite element methods for non-selfadjoint elliptic problems: II. Performance of block-ILU factorization methods, SIAM J. Sci. Comput. 16 (1995) 1126-1136.
11. G.F. Carey and Y. Shen, Convergence studies of least-squares finite elements for first order systems, Comm. Appl. Numer. Methods 5 (1989) 427-434.
12. C.L. Cox and G.J. Fix, On the accuracy of least squares methods in the presence of corner singularities, Comput. Math. Appl. 10 (1984) 463-475.
13. G.J. Fix, M.D. Gunzburger and R.A. Nicolaides, On mixed finite element methods for first order elliptic systems, Number. Math. 37 (1981) 29-48.
14. R. Glowinski and M.F. Wheeler, Domain decomposition and mixed finite element methods for elliptic problems, in: First Int. Symp. on Domain Decomposition Methods for Partial Differential Equations, Paris, 1987 (SIAM, Philadelphia, PA, 1988).
15. J. Haslinger and P. Neittaanmäki, On different finite element methods for approximating the gradient of the solution to the Helmholtz equation, Comput. Methods Appl. Mech. Engrg. 42 (1984) 131-148.
16. D.C. Jespersen, A least squares decomposition method for solving elliptic equations, Math. Comput. 31 (1977) 873-880.
17. B. Jiang and G.F. Carey, Adaptive refinement for least-squares finite elements with element-by-element conjugate gradient solution, Int. J. Numer. Methods Engrg. 24 (1987) 569-580.
18. O.A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flows (Gordon and Breach, London, 1969).
19. J.M. Maubach, Local bisection refinement for n-simplicial grids generated by reflection, SIAM J. Sci. Comput. 16 (1995) 210-227.
20. P. Neittaanmäki and M. Křížek, Conforming FE-method for obtaining the gradient of a solution to the Poisson equation, in: W. Hackbush, ed., Efficient Solution of Elliptic Systems, (Vieweg, Wiesbaden, 1984) 74-86.
21. P. Neitaanmäki and J. Saranen, On finite element approximation of the gradient for the solution of Poisson equation, Numer. Math. 37 (1981) 333-337.
22. J.A. Nitsche and A.H. Schatz, Interior estimates for Ritz-Galerkin methods, Math. Comput. 28 (1974) 937-958.
23. A.I. Pehlivanov and G.F. Carey, Error estimates for least-squares mixed finite elements, RAIRO Math. Model. Numer. Anal. 28 (1994) 499-516.
24. A.I. Pehlivanov and G.F. Carey, Least-squares mixed methods for steady diffusion problem, in: G.F. Carey, ed., Finite Element Modeling of Environmental Problems (John Wiley & Sons, Chichester, 1955) 287-302.
25. A.J. Pehlivanov, and G.F. Carey, Interior error estimates for least-squares fixed finite element methods (1997) submitted.
26. 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.
27. A.I. Pehlivanov, G.F. Carey, R.D. Lazarov and Y. Shen, Convergence analysis of least-squares mixed finite elements, Computing 51 (1993) 111-123.
28. 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 (1996) 501-522.
29. A. Plaza, R. Montenegro and L. Ferragut, An improved derefinement algorithm of nested meshes in: M. Papadrakakis, ed., Advances in Post and Preprocessing for Finite Element Technology (Civil-Comp Ltd., 1994) 175-180.
30. R.A. Raviart and J.M. Thomas, A mixed finite element method for 2nd order elliptic problems, Lect. Notes in Math., Vol. 606 (Springer-Verlag, Berlin, 1977) 292-315.
31. M.C. Rivara, Mesh refinement based on generalized bisection of simplices, SIAM J. Numer. Anal. 2 (1984) 604-613.
32. M.C. Rivara, A grid generator based on 4-triangles conforming mesh refinement algorithms. Int. J. Numer. Methods Engrg. 24 (1987) 1343-1354.
33. S.W. Sloan, Fast algorithm for generating constrained Delaunay triangulations, Comput. Struct. 47 (1993) 441-450.
34. L. Wahlbin, Local Behaviour of Finite Element Methods, in: PG. Ciarlet and J.L. Lions, eds., Handbook of Numerical Analysis, Vol. 2 (Part 1) (North-Holland, Amsterdam, 1991) 353-522.