A posteriori verification for the sign-change structure of solutions of elliptic partial differential equations
Kazuaki Tanaka · Japan Journal of Industrial and Applied Mathematics · 2021
Abstract 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 solutionuand a numerically computed approximate solution $${\hat{u}}$$ u^ , we evaluate the number of sign-changes ofu(the number of nodal domains) and determine the location of zero level-sets ofu(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.