Numerical Analysis of Selected Variational and Quasi-variational Inequalities

University Of Tennessee Knoxville, Knoxville TN

Investigators

Abstract

Unilateral and constrained phenomena are ubiquitous in science and engineering. The quantities that govern a physical process are often subject to constraints: Be it of mechanical, physical, thermodynamical or practical nature. Examples of these can be impenetrability conditions (two bodies cannot be at the same place at the same time), the fact that mass cannot be negative and that entropy cannot decrease or simply the fact that we are not able to produce more than a fixed amount of forcing. In addition, many of these constraints might depend on the quantities of interest themselves (e.g. friction). One final example is mechanical contact which is, in fact, the only mechanism through which we can exert any mechanical action on another body. These examples show that the derivation of realistic models for the description of these phenomena is of fundamental importance in applications. However these models will be, as a rule, nonlinear. The development and analysis of numerical techniques for approximating the solution of these problems is of practical relevance. The models that govern constrained phenomena carry the name of variational and quasi-variational inequalities. This proposal aims at the study of numerical techniques for a collection of variational and quasi-variational inequalities which have not been considered before and to which the classical ideas and techniques do not apply. They include: variational inequalities for nonlocal operators, evolution problems for degenerate parabolic equations and the general study of time discretization techniques for quasi-variational inequalities. The numerical methods that will result from this project will be of interest to a wide range of practitioners, since the proposed problems have a wide range of applications. For instance, obstacle problems with fractional diffusion appear in control theory, fluid mechanics and even finance; the degenerate parabolic equations that will be considered find applications in imaging and materials science; the prototypical example of a quasi-variational inequality is friction, but they also arise in game theory - when trying to find Nash points - and in control theory when dealing with impulse controls.