Iterative methods for solving seepage problems in multilayer beds in the presence of a point source

This work is devoted to solving steady-state seepage problems of an incompressible fluid that obeys a nonlinear multivalued filtration law with limit gradient in a multilayer bed in the presence of a point source. The seepage problem is formulated as a mixed variational inequality with an inversely strongly monotone operator in a Hilbert space. An iterative splitting method is proposed to solve the variational inequality. Unlike the earliermethods, the method proposed allows one to find not only approximate values of the fluid pressure, but also the filtration rates, in particular, on the sets corresponding to multivalued points in the filtration law. The convergence of the method is analyzed. © 2012 Pleiades Publishing, Ltd

Iterative methods for solving seepage problems in multilayer beds in the presence of a point source

This work is devoted to solving steady-state seepage problems of an incompressible fluid that obeys a nonlinear multivalued filtration law with limit gradient in a multilayer bed in the presence of a point source. The seepage problem is formulated as a mixed variational inequality with an inversely strongly monotone operator in a Hilbert space. An iterative splitting method is proposed to solve the variational inequality. Unlike the earliermethods, the method proposed allows one to find not only approximate values of the fluid pressure, but also the filtration rates, in particular, on the sets corresponding to multivalued points in the filtration law. The convergence of the method is analyzed. © 2012 Pleiades Publishing, Ltd

Iterative methods for solving seepage problems in multilayer beds in the presence of a point source

This work is devoted to solving steady-state seepage problems of an incompressible fluid that obeys a nonlinear multivalued filtration law with limit gradient in a multilayer bed in the presence of a point source. The seepage problem is formulated as a mixed variational inequality with an inversely strongly monotone operator in a Hilbert space. An iterative splitting method is proposed to solve the variational inequality. Unlike the earliermethods, the method proposed allows one to find not only approximate values of the fluid pressure, but also the filtration rates, in particular, on the sets corresponding to multivalued points in the filtration law. The convergence of the method is analyzed. © 2012 Pleiades Publishing, Ltd

On the explicit scheme with variable time steps for solving the parabolic optimal control problem

The paper deals with the optimal control problem, including the linear parabolic equation as a state problem. Pointwise constraints are imposed on the control function. The objective functional involves the observation function in the entire space-time domain. The optimal control problem is approximated by a finite dimensional problem with mesh approximation of the state equation by the explicit (forward Euler) mesh scheme with variable time steps. The existence of unique solutions for the continuous and mesh optimal control problems is proved. The Uzawa-type iterative method is used for solving the finite dimensional optimal control problem. The results of numerical experiments are presented