From Continuous to Fully Discrete Carleman Estimates: A Transfer Principle for Parabolic Schemes

Abstract

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.