Courant's Nodal Line Theorem and Its Discrete Counterparts
G. M. L. Gladwell · The Quarterly Journal of Mechanics and Applied Mathematics · 2002
Courant's nodal line theorem (CNLT) states that if the Dirichlet eigenvalues of the Helmholtz equation Δu + λρu = 0 for D ∈ Rm 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.