发表机构
School of Mathematics, Sichuan University; School of Mathematics, Southwest Jiaotong University（四川大学数学学院; 西南交通大学数学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一个抽象传递定理，从连续Carleman不等式推导全离散后向欧拉格式的Carleman估计，适用于热方程离散化，并得到松弛可观测性和近似零控制结果。
AI 中文摘要
直接处理离散抛物型Carleman估计的方法通常依赖于针对离散算子定制的加权恒等式。我们证明了一个抽象的传递定理，该定理从连续Carleman不等式以及适当的加权逼近和相容性性质出发，为具有相容空间重构的全离散后向欧拉格式推导出此类估计。对于一类与时间无关的正自伴算子，关键构造是为每个离散状态分配一个平移的连续辅助问题，其离散解恰好是指定状态。加权时空误差估计在整个离散载荷空间上一致成立，从而在保留原始格式的残差项和观测项的同时传递不等式。这将特定于PDE的Carleman分析与每种离散化所需的数值验证分离开来。我们验证了在二维和三维局部分级网格上使用一致质量$P_1$有限元离散的热方程，以及在任意固定维度下使用标准笛卡尔有限差分离散的热方程的假设。对于热方程，传递的估计在Carleman参数高达$\min\{h^{-4/5},(\delta t)^{-2/5}\}$的尺度内有效。对于有界实值时空势，所得估计在独立的空间和时间细化下产生松弛可观测性和近似零控制，具有一致有界的成本和指数小的终端误差；重构控制的弱聚点是连续零控制。
英文摘要
Direct approaches to discrete parabolic Carleman estimates often rely on weighted identities tailored to the discrete operator. We prove an abstract transfer theorem that derives such estimates for fully discrete backward Euler schemes with conforming spatial reconstructions from a continuous Carleman inequality together with suitable weighted approximation and compatibility properties. For a class of time-independent positive self-adjoint operators, the key construction assigns to each discrete state a shifted continuous auxiliary problem whose discrete solution is exactly the prescribed state. Weighted space-time error estimates, uniform over the full discrete load space, then transfer the inequality while preserving the residual and observation terms of the original scheme. This separates the PDE-specific Carleman analysis from the numerical verification required for each discretization. We verify the hypotheses for the heat equation discretized by consistent-mass $P_1$ finite elements on locally graded meshes in two and three dimensions and by standard Cartesian finite differences in any fixed dimension. For the heat equation, the transferred estimate is valid for Carleman parameters up to the scale $\min\{h^{-4/5},(δt)^{-2/5}\}$. For bounded real-valued space-time potentials, the resulting estimates yield relaxed observability and approximate null controls with uniformly bounded cost and exponentially small terminal errors under independent spatial and temporal refinement; weak accumulation points of the reconstructed controls are continuous null controls.
