Visualization of the eigenvalue problems of the Laplacian for embedded surfaces and its applications (Q2869276)

This article is based on a conference presentation, and there is significant overlap between it and the authors' paper [Interdiscip. Inf. Sci. 14, No. 2, 191--223 (2008; Zbl 1167.65065)].NEWLINENEWLINEFor a Riemannian manifold \((M, g)\) without boundary, a well-studied partial differential equation is the eigenvalue problem \(\Delta \varphi = \lambda \varphi\) on \(M\), where \(\Delta\) is the Laplacian of \((M,g)\) acting on \(C^\infty(M)\). When \(M\) has boundary, there are various boundary conditions that one may consider; the article under review considers both Dirichlet and Neumann boundary conditions. The focus is on numerical solutions to these problems for compact surfaces \(M\) embedded in \(\mathbb{R}^3\). Note that the only such \(M\) for which explicit eigenvalues and eigenfunctions are known is the sphere. The authors employ the finite element method: using a triangulation of \(M\), computation of the eigenvalues and eigenfunctions is reduced to the computation of two matrices. Similar techniques allow them to treat the Dirichlet or Neumann problem of the Poisson equation for a given bounded plane domain, as well as the Cauchy problem of the heat and wave equations. The bulk of the paper is dedicated to concrete applications of their techniques to certain surfaces and domains. They give tables of computed eigenvalues and pictures of corresponding eigenfunctions, graphs showing the behavior of the eigenvalues under deformations, pictures of solutions of the Poisson equation, and pictures of heat conduction and wave propagation. Finally, numerical counterexamples to the hot spots conjecture are included.NEWLINENEWLINEFor the entire collection see [Zbl 1257.53002].