C0 interior penalty methods for a dynamic nonlinear beam model

In this work, we aim to develop efficient numerical schemes for a nonlinear fourth-order partial differential equation arising from the so-called dynamic Gao beam model. We use C0 interior penalty finite element methods over the spatial domain to set up the semi-discrete formulations. Convergence results for the semi-discrete case are shown, based on a truncated variational formulation and its equivalent abstract formulations. We combine time discretizations to derive fully discrete numerical formulations. Newton's method is applied to compute one time step numerical solutions of a nonlinear system. Two numerical examples are provided: one supports our theoretical results and the other presents a buckling state of the Gao beams.