A new energy dissipation-preserving Crank-Nicolson type nonconforming FEM for damped wave equation with cubic nonlinearity

Abstract

In this article, an energy dissipative Crank-Nicolson (C-N) type fully discrete nonconforming finite element method (FEM) is developed for the damped wave equation with cubic nonlinearity, and its unconditional superconvergence behavior is rigorously analyzed for the nonconforming $E{Q}_1^{rot}$ element. By introducing an auxiliary variable $p=u_t$, the problem is converted to a proper parabolic system, a new C-N type fully discrete scheme is presented, which preserves the energy dissipation property so as to ensure the boundedness of the numerical solutions in the broken $H^1$-norm. Then, the existence and uniqueness of the numerical solutions are proved strictly. Also, thanks to the dissipation property mentioned above, along with specific characteristics of $E{Q}_1^{rot}$ element and interpolation post-processing technique, the unconditional superclose and superconvergence estimates of order $O(h^2+{\tau }^2)$ for original variable u and auxiliary variable p in the broken $H^1$-norm are derived, respectively, without any restriction between the mesh size h and the time step τ that is usually required in the previous literature. Finally, the theoretical findings and good performance of the proposed method are confirmed by some numerical results.