An energy-conserving finite element method for nonlinear fourth-order wave equations

Abstract

In this paper, an energy-conserving finite element method is developed and intensively analyzed for a class of nonlinear fourth-order wave equations in a general sense for the first time, where the two-level, Crank-Nicolson type of temporal discretization scheme is designed to cooperate with the Lagrange finite element approximation in space in order to achieve the conservation of discrete energy at each time step. The energy conservation is crucial in engineering field to preserve the total energy as constant for dynamic vibration problems of beams and thin plates that can be modeled by the presented wave equations. In addition, the optimal spatial convergence properties in both L2- and H1-norm and the second-order temporal approximation rate are obtained for finite element solutions of u (deflection), Δu (bending moment) and/or ut (deflection speed) at the same time. Numerical experiments are carried out to validate all attained theoretical results.