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Abstract
In this paper, we introduce a new method to analyze the convergence of the standard finite element method for elliptic variational inequalities with noncoercive operators (VI). The method consists of combining the so-called Bensoussan-Lions algorithm with the characterization of the solution, in both the continuous and discrete contexts, as fixed point of contraction. Optimal error estimates are then derived, first between the continuous algorithm and its finite element counterpart, and then between the true solution and the approximate solution.
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References
- Signorini, A. On some static issues of continuous systems. Annals of the Scuola Normale Superiore di Pisa, 1933, 2, 231-251.
- Fichera, G. Elastostatic problems with unilateral constraints; the Signorini problem with ambiguous contingency conditions. National Academy of the Lincei, 1964,7, 91-140
- Stampacchia, G. Bilinear coercive forms on convex sets. Comptes Rendus de l' academie des Sciences, Paris, 1964, 258, 4413-4416 .
- Lions, J.L. and Stampacchia, G. Variational inequalities. Communication in Pure and Applied Mathematics., 1967, 20, 493-519.
- Brezis, H. Equations and nonlinear in dual vector spaces. Annals of Fourier Institute, Grenoble, 1968, 18, 115-175.
- Mosco, U. Implicit variational problems and quasi-variational inequalities. Lecture notes in Mathematics, Springer, Berlin 1975, 543, 83-156.
- Bensoussan, A. and Lions, J.L., Application of variational inequalities to stochastic control problems. Gauthiers-Villars, 1982, Paris.
- Rodrigues, J.F. Obstacle problems in mathematical physics. 1987, North-Holland, New York.
- Cortey-Dumont, P. On variational inequalities with non coercive operators, RAIRO, 1985, 2, 195-212.
- Boulbrachene, M. On The finite element approximation of variational inequalities with noncoercive operators. Numerical Functional Analysis and Optimization, 2015, 36, 1107-1121.
- Cortey Dumont, P. On the finite element approximation in the norm of variational inequalities with nonlinear operators. Numerische Mathematik, 1985,47, 45-57.
- Atkinson, W. and Han, K. Theoretical Numerical Analysis, 3rd Edition, 2010, Springer.
- Lu, C., Huang. and Qiu, J. Maximum principle in linear finite element approximations of anisotropic diffusion--convection--reaction problems. Numerische Mathematik, 2014, 3, 515-537.
- Vejchodsky, T. The discrete maximum principle for Galerkin solutions of elliptic problems. European Journal of Mathematics, 2012, 10, 5-43.
References
Signorini, A. On some static issues of continuous systems. Annals of the Scuola Normale Superiore di Pisa, 1933, 2, 231-251.
Fichera, G. Elastostatic problems with unilateral constraints; the Signorini problem with ambiguous contingency conditions. National Academy of the Lincei, 1964,7, 91-140
Stampacchia, G. Bilinear coercive forms on convex sets. Comptes Rendus de l' academie des Sciences, Paris, 1964, 258, 4413-4416 .
Lions, J.L. and Stampacchia, G. Variational inequalities. Communication in Pure and Applied Mathematics., 1967, 20, 493-519.
Brezis, H. Equations and nonlinear in dual vector spaces. Annals of Fourier Institute, Grenoble, 1968, 18, 115-175.
Mosco, U. Implicit variational problems and quasi-variational inequalities. Lecture notes in Mathematics, Springer, Berlin 1975, 543, 83-156.
Bensoussan, A. and Lions, J.L., Application of variational inequalities to stochastic control problems. Gauthiers-Villars, 1982, Paris.
Rodrigues, J.F. Obstacle problems in mathematical physics. 1987, North-Holland, New York.
Cortey-Dumont, P. On variational inequalities with non coercive operators, RAIRO, 1985, 2, 195-212.
Boulbrachene, M. On The finite element approximation of variational inequalities with noncoercive operators. Numerical Functional Analysis and Optimization, 2015, 36, 1107-1121.
Cortey Dumont, P. On the finite element approximation in the norm of variational inequalities with nonlinear operators. Numerische Mathematik, 1985,47, 45-57.
Atkinson, W. and Han, K. Theoretical Numerical Analysis, 3rd Edition, 2010, Springer.
Lu, C., Huang. and Qiu, J. Maximum principle in linear finite element approximations of anisotropic diffusion--convection--reaction problems. Numerische Mathematik, 2014, 3, 515-537.
Vejchodsky, T. The discrete maximum principle for Galerkin solutions of elliptic problems. European Journal of Mathematics, 2012, 10, 5-43.