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.

本文提出了一种严格分析椭圆偏微分方程解的符号变化结构的方法，该方程受三种齐次边界条件之一（狄利克雷特，诺伊曼和混合）的约束。给定精确解u和数值计算的近似解\（{\ hat {u}} \）之间的显式估计误差范围，我们评估u的正负号变化数（节点域的数量）并确定其位置u的零级集（节点线的位置）。我们将此方法应用于Allen-Cahn方程的Dirichlet问题。该方程解的节点线表示两个共存相之间的界面。