An Explicit A Posteriori Error Estimate-Based Iterative Method for Parabolic Cauchy Problems

Abstract

In this paper, we introduce an innovative iterative approach to solve the Cauchy problem for the heat equation, utilizing a prediction-correction strategy founded on an explicit a posteriori error estimation. This estimation technique stabilizes and refines the approximate solution while enhancing the convergence speed. We prove convergence theorems and outline resolution algorithms. Numerical simulations are presented to validate the efficacy of our methods. To address the temporal dynamics, we implement a time discretization scheme that converts the parabolic challenge into a succession of quasi-elliptic problems at discrete time intervals. The novelty of our contribution consists in adapting the data completion framework reliant on explicit error estimates from elliptic to parabolic contexts, verifying that the stabilization and acceleration features endure under time discretization.