Mathematical simulation of stationary filtration problem with multivalued law in multilayer beds

We consider a stationary filtration problem for incompressible fluid following multivalued filtration law in multilayer beds. Generalized statement of this problem is formulated in the form of mixed variational inequality with monotone operator and separable generally nondifferentiable functional in Hilbert space. We establish the properties of operator (inverse strong monotonicity, coerciveness) and functional (Lipschitz continuity, convexity) contained in this variational inequality. This makes it possible to apply the known results in the theory of monotone operators to prove the existence theorem. To solve the variational inequality, we suggest iterative method that does not require the inversion of the original operator. Each step of the iterative process can essentially be reduced to the solution of the boundary-value problem for the Laplace operator. A convergence of iterative consequence is investigated. This method was realized numerically. The numerical experiments made for the model problems confirmed the efficiency of the iterative method.