A Mixed Finite Element Method for Vibration Equations of Structurally Damped Beam and Plate

ABSTRACT This paper employs a mixed finite element method to study the initial‐boundary value problems of structurally damped beam‐plate vibration. By introducing variables, we transform the fourth‐order spatial and second‐order temporal differential equations into second‐order spatial and first‐order temporal coupled equations. We use the elliptic projection operator to establish the mixed finite element semidiscrete and fully discrete formats, prove the existence and uniqueness of solutions, and complete optimal error estimations in the ‐norm and ‐norm. When the mixed finite element space is a polynomial of degree , the convergence order of the numerical solution in the ‐norm is , and the convergence order in the ‐norm is . The effectiveness of the mixed finite element format is verified through numerical examples, and the impact of the damping coefficient on beam and plate vibration is investigated. Simultaneously, the occurrence of strong damping effects is observed.