Dissipative Numerical Method for a Flexible Euler-Bernoulli Beam with a Force Control in Rotation and Velocity Rotation

In this paper, we study a flexible Euler-Bernoulli beam clamped at one end and subjected to a force control in rotation and velocity rotation. We develop a finite element method, stable and convergent which preserves the property of time decay of energy in the continuous case. We prove firstly the existence and uniqueness of the weak solution. Then, we discretize the system in two steps: in the first step, a semi-discrete scheme is obtained for discretization in space and, in the second step, a fully-discrete scheme is obtained for discretization in time by the Crank-Nicolson scheme. At each step of the discretization, the a-priori error estimates are obtained.