Variational data assimilation and its decoupled iterative numerical algorithms for Stokes-Darcy model. (English) Zbl 1540.35422
Summary: 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.
MSC:
| 35Q93 | PDEs in connection with control and optimization |
| 35Q35 | PDEs in connection with fluid mechanics |
| 76S05 | Flows in porous media; filtration; seepage |
| 76D07 | Stokes and related (Oseen, etc.) flows |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65K10 | Numerical optimization and variational techniques |
| 65J20 | Numerical solutions of ill-posed problems in abstract spaces; regularization |
| 65F22 | Ill-posedness and regularization problems in numerical linear algebra |
| 65F10 | Iterative numerical methods for linear systems |
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 76M20 | Finite difference methods applied to problems in fluid mechanics |
| 49M41 | PDE constrained optimization (numerical aspects) |
| 93C20 | Control/observation systems governed by partial differential equations |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
Keywords:
data assimilation; Stokes-Darcy equations; optimization; gradient-based iterative method; finite element methodSoftware:
EnKFReferences:
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