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.
翻译:本文建议了一种方法,用于严格分析按三种相同边界条件之一:Dirichlet、Neumann和混合条件之一的椭圆部分差分方程解决方案的信号交换结构。根据明确估计的精确解决方案 $ $ 和 $ $ $ 和 $ $ $ 数字计算近似解决方案之间的误差界限,我们评估了 $ u (节点域数) 的信号交换数量,并确定了 $ u (节点线位置) 的零级方位位置。我们将这种方法应用于 Allen-Cahn 方程的 Dirichlet 问题。 这个方程的节点方程代表了两个共存阶段之间的界面 。