TY - JOUR
T1 - Error Analysis of Fully Discrete Data Assimilation Algorithms for Reaction-Diffusion Equation
AU - Wang , Wansheng
AU - Jin , Chengyu
AU - Huang , Yi
JO - Advances in Applied Mathematics and Mechanics
VL - 3
SP - 840
EP - 866
PY - 2025
DA - 2025/03
SN - 17
DO - http://doi.org/10.4208/aamm.OA-2023-0150
UR - https://global-sci.org/intro/article_detail/aamm/23900.html
KW - Data assimilation, reaction-diffusion equation, finite element method, BDF methods, fully discrete, uniform-in-time error estimates, Allen-Cahn equation.
AB - <p style="text-align: justify;">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.</p>