Regularization and Approximation of a Class of Evolution Problems in AppliedMathematics

摘要

The major effort of this project has been the development of the foundations forregularization techniques related to equations and some possibly far-reaching contributions to this area. The approach that has been taken is a departure from the usual artificial viscosity type of strategies which are produced on a somewhat adhoc basis. The basic strategy is to regularize locally by a micro-structured parabolic system. A mathematical analysis of the regularized equations has been developed to support our approach. Supporting approximate analysis and numerical experiments have been made. The development and the mathematical foundations of these microstructure models have been primary achievements of the project . The relevant nonlinear systems of partial differential equations have also shown to provide good models of diffusion or convection of fluid or gas through a heterogeneous porous medium. Examples include flow in fissured media, problems with adsorption, heat diffusion with freezing-melting, and models for semiconductors. We have established that the problems are well posed and developed the theory of the regularity and dependence of the solutions on data. Such information will aid approximation theory and the design of algorithms to numerically simulate solutions to these types of problems.