Courant’s nodal line theorem and its discrete counterparts

Abstract

A study of Courant's nodal line theorem (CNLT) and its discrete counterparts was performed. A discrete CNLT for a piecewise linear finite element discretization on a triangular/tetrahedral mesh was formulated and proved. The triangular finite element discretization of Helmholtz equation exhibited properties which were analogues to those of continuous equation, provided all triangles were acute-angled. The analogues were found to be straightforward for eigenmodes corresponding to simple eigenvalues.