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.

中文翻译：

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二阶抛物型界面方程的有限元离散变分数据同化

在本文中，我们提出并分析了二维有界域上二阶抛物线界面方程的变分数据同化的有限元方法。吉洪诺夫正则化在将数据同化问题转化为优化问题方面发挥着关键作用。然后分析优化问题解的存在性、唯一性和稳定性。我们利用有限元方法进行空间离散化，并利用后向欧拉方法进行时间离散化。然后基于拉格朗日乘子思想，推导了二阶抛物型界面方程的连续和离散数据同化问题的最优性系统。通过伽辽金正交性的恢复证明了收敛性和最优误差估计。此外，还开发了三种解耦最优系统并显着节省计算成本的迭代方法来求解离散时间演化最优系统。最后，提供数值结果来验证所提出的方法。