Variational and quasivariational inequalities with application to a seepage problem through a meshed dam with finite elements

Abstract

The behavior of the pressure of a fluid that filters through the wall of a dike, which is supposed to be made of a porous material, leads to the approach of a boundary problem that involves equations in partial derivatives under Dirichlet and Newman type conditions. ; at some borders while at another it is unknown, that is, it leads to a free border problem. For the case that the Irasversal section of the rectangular wall is considered, the problem is associated with a variational inequality and for non-rectangular flat sections, there is a free form problem associated with a quasivariational inequality. The study of the first case is resolved in reference [4]. In the present work we have considered the numerical study of the second case. To solve this problem, a mathematical treatment is made using the theory of duality. later; by calculating fixed-point iterations, such that in each iteration there is a variational inequality of the second kind, where the bilinear form is not symmetric, but by means of a proposed scheme of minimizing the number of iterations, an equivalent problem of optimization, which allows demonstrating the existence and uniqueness of the solution, under the condition of the existence of the saddle points. Then, to find the solution, we use the finite element method, which finally achieves the numerical simulation of the non-rectangular planar-based Digues seepage problem.