A posteriori verification for the sign-change structure of solutions of elliptic partial differential equations

TL;DR

This paper introduces a rigorous method to analyze the sign-change structure of elliptic PDE solutions, providing bounds on nodal domains and zero level-sets, with applications to phase interface problems like the Allen-Cahn equation.

Contribution

The paper presents a novel a posteriori verification technique for determining the sign-change structure of elliptic PDE solutions based on explicit error bounds.

Findings

Successfully applied to Allen-Cahn equation

Accurately identifies nodal lines and domains

Provides rigorous bounds on solution sign changes

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 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.