The Calculus of Moving Surfaces and Laplace Eigenvalues on an Ellipse with Low Eccentricity

Our goal is to demonstrate the utility of the calculus of moving surfaces (CMS) in boundary variation problems. We discuss the relative advantages of the CMS compared to the alternative approach of interior variations. We illustrate the technique by calculating the two leading terms of a power series for the Laplace eigenvalues on an ellipse with semi-axes 1+a and 1+b, where a and b are small. We compare the CMS estimates with those obtained by the conventional finite element method with Richardson extrapolation. The comparison confirms the cubic rate of convergence for the CMS estimates.