A mixed-FEM formulation for nonlinear incompressible elasticity in the plane

Numerical Methods for Partial Differential Equations

Volumn 18, Issue 1, 2002, Pages 105-128

  Gatica, Gabriel N ^a   Stephan, Ernst P ^b  

^a UNIVERSITY OF CONCEPCIÓN   (Chile)

^b LEIBNIZ UNIVERSITY HANNOVER   (Germany)

Author keywords

A posteriori error estimate; Hyperelasticity; Mixed finite elements; PEERS

Indexed keywords

EID: 0036142666     PISSN: 0749159X     EISSN: None     Source Type: Journal    
DOI: 10.1002/num.1046     Document Type: Article

References (28)

1. K. J. Bathe, Finite element procedures, Prentice Hall, Englewood Cliffs, NJ, 1996.
2. F. Brezzi and M. Fortin, Mixed and hybrid finite element methods. Springer-Verlag, New York, 1991.
3. D. N. Arnold, F. Brezzi, and J. Douglas, Jr., PEERS: A new mixed finite element method for plane elasticity, Jpn J Appl Math, 1 (1984), 347-367.
4. D. N. Arnold, J. Douglas, Jr., and C. P. Gupta, A family of higher order mixed finite element methods for plane elasticity, Numer Mathematik, 45 (1984), 1-22.
5. U. Brink and E. Stein, A posteriori error estimation in large-strain elasticity using equilibrated local Neumann problems, Comp Methods Appl Mech Eng, 161 (1998), 77-101.
6. U. Brink and E. P. Stephan, Implicit residual error estimators for the coupling of finite elements and boundary elements, Math Methods Appl Sci, 22 (1999), 923-936.
7. U. Brink and E. P. Stephan, Adaptive coupling of boundary elements and mixed finite elements for incompressible elasticity, Numer Methods Part Diff Eq, 17 (2001), 79-92.
8. C. Carstensen and S. A. Funken, A-posteriori error control in low-order finite element discretisations of incompressible stationary flow problems, Math Comput, 70 (2001), 1353-1381.
9. G. N. Gatica and N. Heuer, A dual-dual formulation for the coupling of mixed-FEM and BEM in hyperelasticity, SIAM J Numer Anal, 38 (2000), 380-400.
10. M. A. Barrientos, G. N. Gatica, and N. Heuer, An a-posteriori error estimate for a linear-nonlinear transmission problem in plane elastostatics, Technical Report 00-11, Departamento de Ingeniería Matemática, Universidad de Concepción, (2000), submitted.
11. M. A. Barrientos, G. N. Gatica, and E. P. Stephan, A mixed finite element method for nonlinear elasticity: two-fold saddle point approach and a-posteriori error estimate, Numerische Mathematik, http://dx.dvi.org/10.1007/9009110100337.
12. G. N. Gatica and N. Heuer, Minimum residual iteration for a dual-dual mixed formulation of exterior transmission problems, Numer Linear Algebra Appl, 8 (2001), 147-164.
13. G. N. Gatica and N. Heuer, An expanded mixed finite element method via a dual-dual formulation and the minimum residual method, J Comput Appl Math, 132 (2001), 371-385.
14. G. N. Gatica and S. Meddahi, A dual-dual mixed formulation for nonlinear exterior transmission problems. Math Comput, 70 (2001), 1461-1480.
15. R. E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations, Math Comput, 44 (1985), pp. 283-301.
16. G. N. Gatica, N. Heuer, and E. P. Stephan, An implicit-explicit residual error estimator for the coupling of dual-mixed finite elements and boundary elements in elastostatics, Math Methods Appl Sci, 24 (2001), 179-191.
17. M. Costabel and E. P. Stephan, Coupling of finite and boundary element methods for an elastoplastic interface problem, SIAM J Numer Anal, 27 (1990), 1212-1226.
18. D. Braess, Finite elements. Theory, fast solvers, and applications in solid mechanics, Cambridge University Press, 1997.
19. Z. Chen, Expanded mixed finite element methods for linear second-order elliptic problems, Math Model Numer Anal, 32 (1998), 479-499.
20. Z. Chen, Expanded mixed finite element methods for quasilinear second-order elliptic problems, Math Model Numer Anal, 32 (1998), 501-520.
21. G. N. Gatica, N. Heuer, and S. Meddahi, Solvability and fully discrete Galerkin schemes of nonlinear two-fold saddle point problems, 2001. Submitted. Technical Report 00-03, Departamento de Ingeniería Matemática, Universidad de Concepción, Chile, (2000). (http://www.ing-mat.udec.cl/inf-loc-dim.html)
22. G. N. Gatica, Solvability and Galerkin approximations of a class of nonlinear operator equations, Technical Report 99-03, Departamento de Ingeniería Matemática, Universidad de Concepción, 1999, submitted for publication. (http://www.ing-mat.udec.cl/inf-loc-dim.html)
23. G. N. Gatica, An application of Babuŝka-Brezzi's theory to a class of variational problems, Appl Anal, 75 (2000), 297-303.
24. J. E. Roberts and J.-M. Thomas, Mixed and hybrid methods, P. G. Ciarlet and J. L. Lions, editors, Handbook of numerical analysis, Vol. II, Finite element methods (Part 1), North-Holland, Amsterdam, 1991.
25. E. P. Stephan, Coupling of finite elements and boundary elements for some nonlinear interface problems, Comput Methods Appl Mech Eng, 101 (1992), 61-72.
26. G. N. Gatica and W. L. Wendland, Coupling of mixed finite elements and boundary elements for a hyperelastic interface problem, SIAM J Numer Anal, 34 (1997), 2335-2356.
27. M. Ainsworth and J. T. Oden, A unified approach to a posteriori error estimation using element residual methods, Numer Mathematik, 65 (1993), 23-50.
28. R. A. Araya, G. N. Gatica, and H. Mennickent, A boundary-field equation method for a nonlinear exterior elasticity problem in the plane, J Elasticity, 43 (1996), 45-68.