FINITE ELEMENT ANALYSIS AND COMPUTATIONAL MODELING OF SOME NON-LINEAR PARABOLIC PDEs

Abstract

This work deals with analysis and approximations of some non-linear parabolic partial differential equations (PDEs) using nite element method. Such di erential equations arise frequently in science and engineering. For instance, heat conduction, weather prediction, option pricing, gas dynamics in an exhaust pipe, waves in deep water, some chemical reactions like BZ reaction, iodine clock reaction etc. give rise to non-linear parabolic PDEs. The non-linearity in these equations poses a di cult task in analysing and approximating the solutions to such di erential equations. With the advent of high speed computers, however, engineers and mathematicians are devising techniques which enable us to approximate solutions to such di erential equations to a su cient degree of accuracy. In this work, we consider some problems of non-linear nature and try to establish their existence, uniqueness and also approximate their solutions using Galerkin nite element method. A priori error estimates are also derived for such approximations. In Chapter 1, some basic concepts regarding this work are introduced and a brief literature survey is presented. In Chapter 2, we consider Burgers0-Fisher equation. The existence and uniqueness of the solution is proved using Faedo-Galerkin approximations. Further, some a priori error estimates are given for semi-discrete and fully-discrete solutions. Also, since we often model a physical situation by this di erential equation, it is, therefore, desirable to require a positive solution in such cases. Hence, we present a positivity analysis of the solution and give a bound on time step to ensure the solution remains positive if started with a positive initial solution. The time discretization of the system is done using Euler backward scheme which is unconditionally stable. The non-linearity in the system is resolved by lagging it to the previous known level. Some numerical examples are also considered and the results are compared with the results from literature. i In Chapter 3, a coupled version of the non-linear parabolic PDEs is considered. Using Banach xed point theorem, the existence and uniqueness of the solution is established. We also prove an a priori error estimate for the approximation. Neumann type boundary conditions are taken in this chapter. The time discretization of the system is done using Crank-Nicolson scheme (C-N scheme) and the non-linearity is resolved by the predictorcorrector scheme (P-C scheme). Since the C-N scheme and the P-C scheme are second order convergent, we get an overall second order convergence which is demonstrated in numerical examples. In Chapter 4, we consider the Brusselator model where the cross-di usion is allowed. The presence of cross-di usion a ects the stability of equilibrium. As we know, the di usion in reaction di usion equations may destabilize the equilibrium, which is called `Turing instability'. Similarly, we investigate the e ect of cross-di usion on the stability. We nd that the cross-di usion increases the wave number associated to the solution. Some Turing patterns of the model are also plotted in the chapter. In Chapter 5, we consider the Schr odinger equation. Some new soliton-type solutions are given for the equation. Further, since these soliton-type solutions peter out for large spatial values, we may truncate the in nite domain to some nite sub-domain. Therefore, the truncation analysis is performed for the soliton solutions so that we may truncate the domain without loosing much information about the solution. Some examples are also considered in this chapter where we see the interaction of solitons. The C-N scheme together with P-C scheme is used for this purpose. In Chapter 6, a general reaction di usion advection equation is considered and is analyzed for the existence of solution. Further, a priori error estimates are discussed for approximation error and second order convergence is found. Some 1D and 2D examples are considered in this chapter and their computational aspects are discussed.