Finite Element Analysis For A Kind Of Fourth-order Nonlinear Partial Differential Equations With Variable Coefficients

Partial differential equations have a wide range of applications in interpreting and predicting natural phenomena, especially the high-order parabolic equation. It re-ceived widespread attention because its profound physical background. The numerical methods for solving partial differential equations has evolved with the advancement of computer technology.This paper discusses a partial differential equation which describes the growth of epitaxial films with initial boundary value in the one-dimensional space. This equation is a fourth-order parabolic equation with nonlinear terms, and what is to be emphasized is that the coefficient of fourth-order terms changes when time and spatial variables change.Equations are as follows:This paper briefly introduced the background and theoretical knowledge, and then semi-discrete format of (1) was analyzed. First, the semi-discrete scheme was derived based on the equivalent variational equations and got the H-2norm stability of the semi-discrete solutions on the initial value. Subsequently, introduced the elliptic operators, decompose the error and process the nonlinear term. Eventually we got the error estimate of semi-discrete solution.Then take advantage of Euler differential equations, we established the half-hidden-half-explicit format of full discretization. Similarly with the analysis of semi-discrete scheme, the error of the solutions reached O(Δt+h3).Finally,using MATLAB we did the numerical experiments of the full discretiza-tion format.The experimental results is identical with the theoretical analysis which verified the correctness of the theoretical analysis.