On the uniqueness and numerical approximation of solutions to certain parabolic quasi-variational inequalities

Abstract

A class of abstract nonlinear evolution quasi-variational inequality (QVI) problems in function space is considered. The framework developed includes constraint sets of obstacle and gradient type. The paper addresses the existence, uniqueness and approximation of solutions when the constraint set mapping is of a special form. Uniqueness is addressed through contractive behavior of a nonlinear mapping whose fixed points are solutions to the QVI. An axiomatic semi-discrete approximation scheme is developed, which is proven to be convergent and is numerically implemented. The paper ends by a report on numerical tests for several nonlinear constraints of gradient-type.