Error Analysis of Fully Discrete Data Assimilation Algorithms for Reaction-Diffusion Equation

In this paper we propose a continuous downscaling data assimilation algorithm for solving reaction-diffusion equations with a critical parameter. For the spatial discretization we consider the finite element methods. Two backward differentiation formulae (BDF), a backward Euler method and a two-step backward differentiation formula, are employed for the time discretization. Employing the dissipativity property of the underlying reaction-diffusion equation, under suitable conditions on the relaxation (nudging) parameter and the critical parameter, we obtain uniform-in-time error estimates for all the methods for the error between the fully discrete approximation and the reference solution corresponding to the measurements given on a coarse mesh by an interpolation operator. Numerical experiments verify and complement our theoretical results.