Error Analysis of Fully Discrete Data Assimilation Algorithms for Reaction-Diffusion Equation (Advances in Applied Mathematics and Mechanics, 2025)
In this paper we propose a continuous downscaling data assimilation algorithm for solving reaction-diffusion equations with a critical parameter. For the spatialdiscretization we consider the finite element methods. Two backward differentiationformulae (BDF), a backward Euler method and a two-step backward differentiationformula, are employed for the time discretization. Employing the dissipativity property of the underlying reaction-diffusion equation, under suitable conditions on therelaxation (nudging) parameter and the critical parameter, we obtain uniform-in-timeerror estimates for all the methods for the error between the fully discrete approximation and the reference solution corresponding to the measurements given on a coarsemesh by an interpolation operator. Numerical experiments verify and complementour theoretical results.