\( H^1\)-Galerkin mixed finite element method for the vibration equation of beam with structural damping

The vibration equation of the damped beam can describe the deformation and bending of the beam with damping means such as dampers, which has important engineering practical value and scientific significance for the design of the numerical solution of the vibration equation of the damped beam. In this paper, we develop the \(H^1\) $H^1$ -Galerkin mixed finite element numerical approximation scheme for the vibrational equation of structured damped beams by introducing intermediate variables and transforming the original fourth order problem into a system of four lower order partial differential equations. The existence, uniqueness and stability of the solution are analyzed and the optimal order error estimate are analyzed. Finally, numerical examples are presented to verify the validity of the theoretical results and to test the effect of the variation of the damping coefficient on the vibration of the beam.