Courant's Nodal Line Theorem and its discrete counterparts

Abstract

Courant's Nodal Line Theorem (CNLT) relates to the Dirichlet/Neumann eigenfunctions u(x) of elliptic equations, the simplest and most important of which is the Helmholtz equation Au+ >.pu = 0 for DC Rm. CNLT states that if the eigenvalues are ordered increasingly, the nodal set of the nth eigenfunction Un divide D into no more than n nodal domains in which Un has a fixed sign. We investigate whether the numerical solutions approximated by finite clement method (FEM) retain this sign characteristic stated in CNLT. We derive various properties of the FEM solutions. then formulate and prove discrete analogues of CNLT for piecewise linear FEM solutions on a triangular/ tetrahedral mesh.

For linear combinations of eigenfunctions, CNLT is replaced by Courant Herrmann conjecture (CHC). CHC states that the nodal set of a combination v = L~i c;ui also divides D into at most n nodal domains. We exhibit numerical counterexamples. We find that even linear combinations of the first two eigenfunctions can have three, four or more nodal domains. Also. we show that the discrete version of CHC is false in general. A restricted theorem is proved, which holds for both continuous and discrete cases. Although CHC is false in general, We conjecture that CHC is true for some convex domains, particularly for rectangles.