Accurate Solution to Elliptic Eigenvalue Problems using Finite Elements

where Ω is an open connected subset of R with a sufficiently smooth boundary, ρ and θ are sufficiently smooth scalar functions on Ω bounded below by a positive number, and u is a function on Ω that does not vanish uniformly. This eigenvalue PDE is the standard idealization of finding the vibrational modes of a thin plate clamped at the edges, in which the stiffness of the plate is given by θ, the density is ρ, and the displacement associated with the vibration is u. It is valid for small displacements only. This eigenvalue problem is derived by considering solutions to the hyperbolic partial differential equation for the unforced, undamped motion of the plate ρÜ +∇ · (θ∇U) = 0 that have the form of a standing vibration: U(x, t) = sin(λt)u(x). Here are some reasonable smoothness assumptions to make: ∂Ω is composed of a finite number of C arcs that are joined at a finite number of vertices by angles strictly between 0 and 2π. Functions θ and ρ are piecewise continuous, meaning that there is a partition of Ω into a finite number of pieces Ω1, . . . ,Ωk, such that each Ωi satisfies the same smoothness assumption as Ω and such that θ and ρ are both continuous on each Ωi. Following the approach in [1], we use the isoparametric finite element method to reduce (1) to the matrix eigenvalue problem Kx = λMx, (2)