Speaker: Zhimin Zhang, Wayne State University

Title: How well can we hear the shape of a drum by computer algorithms?

Abstract: Can we determine the shape of a domain by its Laplacian eigenvalues? The question puzzled us for many years until 1992, when three mathematicians surprised everyone by a counterexample. However, this is not the end of the story to applied mathematicians, since in most cases we are unable to obtain exact eigenvalues and the numerical approximation by computer algorithms is necessary. Naturally, another question arises: How many numerical eigenvalues can we trust? When approximating PDE eigenvalue problems by numerical methods such as finite difference and finite element, it is common knowledge that only a small portion of numerical eigenvalues are reliable. However, this knowledge is only qualitative rather than quantitative in the literature. In this talk, I will first survey some theoretical results from pure mathematics regarding eigenvalue problems. Then I will investigate the number of “trusted” eigenvalues by the finite element (and the related finite difference method results obtained from mass lumping) approximation of 2mth order elliptic PDE eigenvalue problems. Our two model problems are the Laplace and bi-harmonic operators, for which a solid knowledge regarding magnitudes of eigenvalues are available in the literature. Combining this knowledge witha priorierror estimates of the finite element method, we are able to figure out roughly how many “reliable” eigenvalues can be obtained from numerical approximation under a pre-determined convergence rate.