A Semilinear Parabolic Equation With CNG Format

The parabolic equation has a profound physical background,which comes from many physical and engineering problems.There are common processes of heat con-duction between materials,and heat exchange of heat mating in nuclear reactors.But these problems usually can't get exact solution.At present,it is an effective method to approximate the original problem by numerical method and obtain the approximate solution of the original problem.The finite element method is a com-mon numerical method.This paper deals with a reaction diffusion problem.It is a mathematical model based on the heat exchange within a nuclear reactor.The model describes how in an insulated containment vessel of a nuclear reactor,and a process in which neutrons react with other substances to produce large amounts of heat,which is produced by the diffusion of heat within a container.In this thesis,we mainly give a uncoordinated format of heat exchanging in the nuclear reactor.we give the error estimations of the full-discrete formate,and discuss the stability of the finite element soluntion in detail.In the second chapter of this paper,An uncoordinated finite element space V~his structured,and we estab-lish the full-discrete finite element approximation of semilinear parabolic equations with Crank-N icolson-Galerkin format.At present,the numerical process of discrete equations in this format is given in detail.We study the error estimations for the uncoordinated element methods.A projection operator was introducted.Meanwhile,the error?U~n-u(t_n)?is divided into two parts.?U~n-P u(t_n)?and?P u(t_n)-u(t_n)?.The former term is divided into linear term and nonlinear term through simplification and combination,and all of parts are estimated successively,while the latter term is directly estimated by interpolation approximation theorem.In Chapter 3,we study stability of the system of equations.By subtracting the original problem and its initial value perturbation,The recurrence relation is derived from young inequality and triangle inequality.Finally the stability of the finite element solution is proved.