Courant's Nodal Line Theorem and Its Discrete Counterparts

Abstract

Summary Courant's nodal line theorem (CNLT) states that if the Dirichlet eigenvalues of the Helmholtz equationu + λρ u = 0 for D ∈ R m are ordered increasingly, the nodal set of the nth eigenfunction un, which consists of hypersurfaces of dimension m − 1, divides D into no more than n sign domains in which un has one sign. We formulate and prove a discrete CNLT for a piecewise linear finite element discretization on a triangular/tetrahedral mesh.