"Variational Data Assimilation with Finite-element Discretization for S" by Xuejian Li, Xiaoming He et al.
 

Variational Data Assimilation with Finite-element Discretization for Second-order Parabolic Interface Equation

Abstract

In this paper, we propose and analyze a finite-element method of variational data assimilation for a second-order parabolic interface equation on a two-dimensional bounded domain. The Tikhonov regularization plays a key role in translating the data assimilation problem into an optimization problem. Then the existence, uniqueness and stability are analyzed for the solution of the optimization problem. We utilize the finite-element method for spatial discretization and backward Euler method for the temporal discretization. Then based on the Lagrange multiplier idea, we derive the optimality systems for both the continuous and the discrete data assimilation problems for the second-order parabolic interface equation. The convergence and the optimal error estimate are proved with the recovery of Galerkin orthogonality. Moreover, three iterative methods, which decouple the optimality system and significantly save computational cost, are developed to solve the discrete time evolution optimality system. Finally, numerical results are provided to validate the proposed method.

Department(s)

Mathematics and Statistics

Comments

National Natural Science Foundation of China, Grant 12071468

Keywords and Phrases

data assimilation; finite-element method optimization; gradient-based iterative method; second-order parabolic interface equation

International Standard Serial Number (ISSN)

1464-3642; 0272-4979

Document Type

Article - Journal

Document Version

Citation

File Type

text

Language(s)

English

Rights

© 2025 Oxford University Press; Institute of Mathematics and its Applications, All rights reserved.

Publication Date

01 Jan 2025

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