Approximations of parabolic variational inequalities

On Tracking of Solutions of Parabolic Variational Inequalities

The problem of constructing a feedback control algorithm for a parabolic variational inequality is considered. This algorithm should provide tracking a prescribed trajectory by a solution of the given inequality. Two solving algorithms, which are stable with respect to informational noises, are designed. The algorithms are based on the method of extremal shift, which is known in the theory of g...

Controllability for Variational Inequalities of Parabolic Type with Nonlinear Perturbation

Let H and V be two complex Hilbert spaces. Assume that V is a dense subspace in H and the injection of V into H is continuous. If H is identified with its dual space, we may write V ⊂ H ⊂ V ∗ densely and the corresponding injections are continuous. The norm on V , H, and V ∗ will be denoted by ‖ · ‖, | · |, and ‖ · ‖∗, respectively. The duality pairing between the element v1 of V ∗ and the elem...

Solution of Degenerate Parabolic Variational Inequalities with Convection

Abstract. Degenerate parabolic variational inequalities with convection are solved by means of a combined relaxation method and method of characteristics. The mathematical problem is motivated by Richard’s equation, modelling the unsaturated – saturated flow in porous media. By means of the relaxation method we control the degeneracy. The dominance of the convection is controlled by the method ...

Stable Approximations of a Minimal Surface Problem with Variational Inequalities

In this paper we develop a new approach for the stable approximation of a minimal surface problem associated with a relaxed Dirichlet problem in the space BV(Ω) of functions of bounded variation. The problem can be reformulated as an unconstrained minimization problem of a functional J on BV(Ω) defined by J (u) = A(u) + ∫ ∂Ω |Tu − φ|, where A(u) is the “area integral” of u with respect to Ω, T ...

Internal and subspace correction approximations of implicit variational inequalities