A Reduced Basis Method for Parametrized Variational Inequalities
Résumé
Reduced basis methods are an efficient tool for significantly reducing the computational complexity of solving parametrized partial differential equations. Originally introduced for elliptic equations, they have been generalized during the last decade to various types of elliptic, parabolic and hyperbolic systems. In this article, we extend the reduction technique to parametrized variational inequalities. Firstly, we propose a reduced basis variational inequality scheme in a saddle-point form and prove existence and uniqueness of the solution. We state some elementary analytical properties of the scheme such as reproduction of solutions, a-priori stability with respect to the data and Lipschitz- continuity with respect to the parameters. Secondly, we provide rigorous a-posteriori error bounds. An offline/online decomposition guarantees an efficient assembling of the reduced scheme, which can be solved by constrained quadratic programming. The reduction scheme is applied to a one-dimensional obstacle problem with a two-dimensional parameter space. The numerical results confirm the theoret- ical ones and demonstrate the efficiency of the reduction technique.