Fundamental Theorem for Explicit Eigenvalue Bounds

This chapter details a Galerkin projection-based method for obtaining explicit eigenvalue bounds in the form: \(\lambda _{k} \leq \lambda _{k,h}/(1+C_{h}^{2}\lambda _{k,h})\) . It thoroughly analyzes the eigenvalue problem formalized as \(a(u,v)=\lambda b(u,v)\) , covering cases where either \(a(\cdot ,\cdot )\) or \(b(\cdot ,\cdot )\) is positive semi-definite. The general settings of function spaces in eigenvalue approximation allow for the use of both conforming and non-conforming FEMs to solve various problems, such as the biharmonic eigenvalue problem, the Steklov eigenvalue problem, and the Maxwell eigenvalue problem, with diverse function settings.