  This paper proposes a method for rigorously analyzing the sign-change
structure of solutions of elliptic partial differential equations subject to
one of the three types of homogeneous boundary conditions: Dirichlet, Neumann,
and mixed. Given explicitly estimated error bounds between an exact solution $
u $ and a numerically computed approximate solution $ \hat{u} $, we evaluate
the number of sign-changes of $ u $ (the number of nodal domains) and determine
the location of zero level-sets of $ u $ (the location of the nodal line). We
apply this method to the Dirichlet problem of the Allen-Cahn equation. The
nodal line of solutions of this equation represents the interface between two
coexisting phases.
