On the Accurate Finite Element Solution of a Class of Fourth Order Eigenvalue Problems

TL;DR

The paper addresses accurate numerical approximation of biharmonic eigenproblems in two dimensions, focusing on corner-caused oscillations and parity behavior under domain deformation. It employs a conforming $C^0$ mixed finite element method with unstructured meshes to solve both the clamped plate problem $\Delta^2 u = \lambda u$ and the buckling problem $\Delta^2 u = \lambda\,\Delta u$, using inverse iteration on a Schur-complement system and direct sparse Cholesky solves to achieve high accuracy. Key contributions include numerical verification of the corner asymptotics $p=\alpha+i\beta$ from $p+1+\frac{\sin((p+1)\theta)}{\sin\theta}=0$, identification of a critical angle $\theta_c \approx 0.8128\pi$ beyond which oscillations vanish, and evidence that the parity of the least biharmonic eigenfunction can change with domain geometry (via crossing of $\lambda_{even}(c)$ and $\lambda_{odd}(c)$). The work extends analysis to circular-sector domains and non-convex shapes, providing high-precision data and suggesting directions for rigorous eigenfunction enclosures, higher-dimensional extensions, and broader geometric applications.

Abstract

This paper is concerned with the accurate numerical approximation of the spectral properties of the biharmonic operator on various domains in two dimensions. A number of analytic results concerning the eigenfunctions of this operator are summarized and their implications for numerical approximation are discussed. In particular, the asymptotic behaviour of the first eigenfunction is studied since it is known that this has an unbounded number of oscillations when approaching certain types of corner on domain boundaries. Recent computational results of Bjørstad and Tjøstheim, using a highly accurate spectral Legendre-Galerkin method, have demonstrated that a number of these sign changes may be accurately computed on a square domain provided sufficient care is taken with the numerical method. We demonstrate that similar accuracy is also achieved using an unstructured finite element solver which may be applied to problems on domains with arbitrary geometries. A number of results obtained from this mixed finite element approach are then presented for a variety of domains. These include a family of circular sector regions, for which the oscillatory behaviour is studied as a function of the internal angle, and another family of (symmetric and non-convex) domains, for which the parity of the least eigenfunction is investigated. The paper not only verifies existing asymptotic theory, but also allows us to make a new conjecture concerning the eigenfunctions of the biharmonic operator.

Figures (1)

• Figure 1: The dumb-bell shape domain for $c=1,\; c=0.5,\; c=0.1$