In this paper we develop and analyze a variational data assimilation method with efficient decoupled iterative numerical algorithms for the Stokes--Darcy equations with the Beavers-Joseph interface condition. By using Tikhonov regularization and formulating the variational data assimilation into an optimization problem, we establish the existence, uniqueness, and stability of the optimal solution. Based on the weak formulation of the Stokes--Darcy equations, the Lagrange multiplier rule is utilized to derive the first order optimality system for both the continuous and discrete variational data assimilation problems, where the discrete data assimilation is based on a finite element discretization in space and the backward Euler scheme in time. By rescaling the optimality system and then analyzing its corresponding bilinear forms, we prove the optimal finite element convergence rate with special attention paid to recovering uncertainties missed in the optimality system. To solve the discrete optimality system efficiently, three decoupled iterative algorithms are proposed to address the computational cost for both well-conditioned and ill-conditioned variational data assimilation problems, respectively. Finally, numerical results are provided to validate the proposed methods.

Publication:

SIAM Journal on Scientific Computing Volume 46, Issue 2, July 18, 2023

https://doi.org/10.1137/22M1492994

Author:

XUEJIAN LI

Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409 USA

Email: xlcdt@mst.edu

WEI GONG

LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Haidian, Beijing 100190,People's Republic of China

Email: wgong@lsec.cc.ac.cn

XIAOMING HE

Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409 USA

Email: hex@mst.edu

AND TAO LIN

Department of Mathematics, Virginia Tech, Blacksburg, VA 24061 USA

Email: tlin@vt.edu