Energy-stable Finite Element Methods For Two Classes Of Nonlinear Fourth-order Partial Differential Equations

The finite element method is a numerical method for solving partial differential equa-tions based on the classical variational method,using piecewise interpolation polynomials combined with the development and popularization of computer.Parabolic equation and hyperbolic equation are important parts of partial differential equation.In this paper,the energy-stable finite element methods for two kinds of nonlinear fourth-order partial differential equations are studied.An energy-conserving finite ele-ment method is developed and intensively analyzed for a class of nonlinear fourth-order wave equations,where the Crank-Nicolson type of temporal discretization scheme is de-signed to cooperate with the finite element approximation in space in order to achieve the conservation of discrete energy at each time step.And,the optimal spatial conver-gence properties in both L~2-and H~1-norm and the second-order temporal approxima-tion rate are obtained in both semi-and fully discrete scheme for finite element solu-tions of u(deflection),u(bending moment)and/or u_t(deflection speed)at the same time.In addition,an energy-stable finite element method with the Crank-Nicolson type of temporal discretization scheme is developed and analyzed for a class of nonlinear fourth-order parabolic equations,including the Swift-Hohenberg(SH)equation and the extended Fisher-Kolmogorov(EFK)equation.Except the energy stability(energy dissi-pation)properties,the optimal spatial convergence properties in both L~2-and H~1-norm and the second-order temporal approximation rate are also obtained in both semi-and fully discrete scheme.Finally,numerical experiments are carried out to validate all at-tained theoretical results.