A Priori Error Estimates for Biharmonic Eigenvalue Problems with Simply Supported Boundary Conditions

The biharmonic eigenvalue problem is a classical fourth-order partial differential equation and a subject of significant research interest, particularly in applied fields such as elasticity, fluid mechanics, and quantum mechanics. Specifically, the biharmonic eigenvalue problem under simply supported boundary conditions finds wide applications in thin plate vibration modeling. To accurately solve such problems, numerical methods play a crucial role. Among them, the discontinuous Galerkin finite element method offers high mesh flexibility and adaptability, enabling arbitrary high-order accuracy and flexible handling of complex boundary conditions. As a result, DG methods have become an essential numerical tool for solving various partial differential equations and practical problems. In the context of simply supported boundary conditions, a priori error estimates for the biharmonic eigenvalue problem serve as a vital tool for assessing the discrepancy between numerical and exact solutions. These estimates provide deeper insights into the convergence and stability of numerical methods. This paper investigates a priori error estimates for the biharmonic eigenvalue problem with simply supported boundary conditions. By employing the discontinuous Galerkin method, we analyze its error behavior in solving the biharmonic eigenvalue problem and derive corresponding error estimates. These estimates not only help evaluate the accuracy of numerical schemes but also provide theoretical support for practical applications.