The Laplacian eigenvalues of a polygon

“The difficulties are almost always at the boundary.” That statement applies to the solution of partial differential equations (with a given boundary) and also to shape optimization (with an unknown boundary). These problems require two decisions, closely related but not identical: 1. How to discretize the boundary conditions. 2. How to discretize the boundary itself. That second problem is the one we discuss here. The region Ω is frequently replaced by a polygon or polyhedron. The approximate boundary ΩN may be only a linear interpolation of the true boundary Ω.Q. A perturbation theory that applies to smooth changes of domain is often less successful for a polygon. This paper concentrates on a model problem-the simplest we could find—and we look at eigenvalues of the Laplacian. The boundary Ω will be the unit circle. The approximate boundary ΩN is the regular inscribed polygon with N equal sides. It seems impossible that the eigenvalues of regular polygons have not been intensively studied., but we have not yet located an authoritative reference. The problem will be approached numerically from three directions, without attempting a general theory. Those directions are: 1. Finite-element discretizations of the polygons ΩN. 2. A Taylor series based on piecewise smooth perturbations of the circle. 3. A series expansion of the eigenvalues in powers of 1/N. The second author particularly wishes that we could have consulted George Fix about this problem.His Harvard thesis demonstrated the tremendous improvement that “singular elements” can bring to the finite-element method (particularly when Ω has a reentrant corner, or even a crack). His numerical experiments in [1] came at the beginning of a long and successful career in applied mathematics. We only wish it had been longer.