Parabolic Partial Integro-Differential Equations: Superconvergence Estimates and Applications

Abstract

Partial integro-differential equations of parabolic type arise naturally in the modeling of many phenomena in various fields of physics, engineering, and economics. The main aim of this thesis is to study finite element methods with numerical quadrature for this class of equations. Both one- and two-dimensional problems are considered. We investigate the stability and convergence properties of the schemes and obtain superconvergence error estimates. It is important to note that these superconvergence results hold also for the equivalent finite difference methods and, in this context, they stand without restrictions on the spatial mesh. In the derivation of these results, we introduce an approach to error analysis that deviates from the traditional one. The significant advantage of this modified strategy is that less regularity for the solution of the continuous problem is needed. The discretization in time using an implicit-explicit method is also addressed, and stability and convergence estimates are derived. The mathematical modeling and numerical simulation of non-reactive solute transport in porous media is also in the scope of this thesis. Among many other applications, this fluid dynamic problem plays a major role in hydrology, medical science, and the petroleum industry. Fick’s law is the underlying principle for obtaining the traditional partial differential equation that describes the solute concentration profile; however, several deviations from this law have been reported. With foundations in the non-Fickian dispersion theory, an integro-differential model is proposed in this thesis. The accuracy of the model is tested in one dimension, and the results indicate that the model is much improved over the conventional one. In fact, even in laboratory-scale homogeneous porous media, these transport processes may exhibit anomalous non-Fickian behavior that only the alternative model correctly reproduces. A robust numerical discretization is also presented and some numerical experiments are conducted. These experiments illustrate the applicability and computational feasibility of the proposed model to simulate two-dimensional problems. A natural extension of our model would allow concentration-dependent viscosity. In this thesis, we also study finite element methods with numerical quadrature for coupled problems that include a simplified version of such a model as a particular case. Again, our numerical method allows the derivation of superconvergence approximations for the variables involved.