论文标题
椭圆形偏微分方程解的标志变换结构的后验验证
A posteriori verification for the sign-change structure of solutions of elliptic partial differential equations
论文作者
论文摘要
本文提出了一种严格分析椭圆形偏微分方程解的标志变换结构，遵循三种类型的均匀边界条件之一：dirichlet，neumann和混合。给定精确解决方案$ u $和数值计算的近似解决方案$ \ hat {u} $之间明确估计的误差界，我们评估了$ u $的签名变换数（节点域的数量）（nodal域的数量），并确定$ u $ $ u $的零级别的位置（Nodal Line的位置）。我们将此方法应用于Allen-Cahn方程的Dirichlet问题。该方程式的溶液的节点线表示两个共存阶段之间的接口。
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.
