Numerical Methods For Solving A Class Of Fourth Order Nonlinear Parabolic Equations

High order nonlinear differential equations are important mathematical models,which can describe phenomena in many fields.They have attracted much attention for their practicability.In this paper,numerical solutions for a class of fourth order nonlinear parabolic equations with different practical backgrounds are studied.We use the B-spline finite element method to solve two fourth order nonlinear equations for the case of variable coefficient and the finite volume element method to solve a fourth order nonlinear equation.For the former,the model with variable coefficient is more widely used,however the theoretical analysis is more difficult.With new methods and skills,we deal with the variable coefficient to solve the difficulties.For the latter one,since it is difficult,a suitable finite volume element scheme is constructed.Firstly,we study the cubic B-spline finite element method for the fourth order non-linear parabolic equationSome researchers have analyzed the Hermite cubic finite element method for the above equation with constant coefficient.We extend the model to the case of variable coeffi-cient,which enlarges the range of application of equation.Furthermore,the bandwidth of stiffness matrix of cubic B-spline finite element scheme is seven,and the order of stiffness matrix is only half of that of Hermite cubic finite element method.It is difficult to derive the boundedness of the semi-discrete solution,we deal with the fourth order main term by firstly integrating and then scaling,then the difficulty is solved.Based on the bound-ness,we prove that the convergence accuracy of L~2norm error is of third order and the approximate solution in H~2semi-norm reaches the second order convergence.Regarding to the discretization of time variable,the linearized backward Euler scheme is chosen,which can reduce the difficulty in dealing with the nonlinear term and improve the speed of numerical calculation.Using the boundedness and Sobolev space embedding theorem,we obtain that the convergence order of L~2norm error is O(?t+h~3)(?t is the time step and h is the space step).The numerical experiment is also presented to demonstrate the theoretical results.Secondly,we discuss the finite element method based on cubic B-splines for anoth-er high order nonlinear differential equation describing thin-film epitaxial growth with variable coefficientSome researchers have analyzed the backward Euler scheme based on the Hermite cubic finite element method,but the coefficient of fourth order term is a constant.We not only broaden the application of the model,but also get a higher convergence accuracy.In order to prove the boundedness of the solution of the semi-discrete problem,we introduce the energy function.The energy function is different from the case of constant coefficient,so it needs new skills to deal with difficulties.For example,we analyze the relationship between the energy function and the B-spline finite element scheme,calculate the derivative of E_h(t)and then estimate it by using the properties of the variable coefficient.Further,we prove that the solution of the semi-discrete scheme achieves the fourth order convergent in L~2norm and the second order convergent in H~2semi-norm,respectively.The variable coefficient makes the fully schemes more diverse.We choose the Crank-Nicolson scheme,and the nonlinear term is replaced by(?).Variable coefficient increases the difficulty to proof the H~2semi-norm boundedness.By the boundedness,the error estimates of the solution in L~2norm and H~2semi-norm are deduced.Notice that the error estimate in H~2semi-norm is a difficult problem in the theoretical analysis of higher order nonlinear parabolic equation.The numerical results show that the scheme is effective.Comparing with the backward Euler scheme,the solution of the Crank-Nicolson scheme can obtain a higher time convergence rate O((?t)~2)in L~2norm,and the stiffness matrix is a smaller sparse matrix.Finally,we consider the Hermite cubic finite volume element method for the nonlinear parabolic equation describing crystal surface growthThe numerical methods for this equation involve the finite difference method and the finite element method.Because of the particularity of the finite volume element method,the test functions are the piecewise linear functions,and the generalized functions are needed.The integration by parts is not used for the nonlinear term as the traditional Ritz-Galerkin method,but carries out the inner product operation directly.Then a linearized backward Euler fully discrete scheme is established.The numerical experiment indicates that the convergence order of the numerical solution in H~2semi-norm is O(?t+h~2).