Application Of The Finite Element Method On A Kind Of Nonlinear Parabolic Equation

Nonlinear parabolic equations is a kind of very important partial differential equation, and it plays a important role in the physical,mechanical and other natural science research. But solution of such equations can not express in analysis-theformula form, in the 20 century, finite element method became an important tool to get the approximate solution. In this paper, we mainly apply the B spline finite element and least-square B spline finite element to nonlinear parabolic equations on the basis of finite element method.In chapter 1, we introduces the research background of nonlinear parabolic equations,the finite element method and spline function, then introduces the development of B-spline finite element method and the least-square B-spline finite element method, and finally we illustrate the innovation points and research conclusion of this paper.In chapter 2, we gives B-spline finite element, the least-square B-spline theory of finite element method, and some lemma that the chapter used.In chapter 3, we applied the B-spline finite element to the Fishers, and according to the equations, we give the proof of convergence, finally the numerical experiments are consistent with theoretical analysis.In chapter 4, we applied the the least -squares B-spline finite element method to the Fishers equation, and gives the stability analysis, Finally the numerical experiment shows the effectiveness of the proposed method.In chapter 5, in order to illustrate the least-square B-spline finite element method that can be applied to the similar nonlinear parabolic equations, we put it into Burgers-Fisher equation, and also we give stability analysis as the the chapter 4 proved, finally the numerical experiments verify the assumptions.