Fully discrete finite element data assimilation method for the heat equation
Fully discrete finite element data assimilation method for the heat equation
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DOI：
10.1051/m2an/2018030
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发表时间：
2017-07
期刊：
影响因子：
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通讯作者：
E. Burman;J. Ish-Horowicz;L. Oksanen
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文献类型：
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作者：
E. Burman;J. Ish-Horowicz;L. Oksanen
我们考虑有限元离散重建的最终状态的热方程，当初始数据是未知的，但额外的数据是在一个子域中的空间时间。对于空间离散化，我们考虑标准连续仿射有限元逼近，时间导数使用向后微分离散化。我们正则化的离散系统，通过增加一个惩罚的H2-半范数的初始数据，缩放的网格参数。该方法的分析使用了E. Burman和L. Oksanen [Numer. 139（2018）505-528]，将数值方法的离散稳定性与物理问题的尖锐Carleman估计相结合，以导出近似解的最佳误差估计。对于自然时空能量范数，远离t = 0，收敛性与已知初始数据的经典问题相同，但与经典情况相反，我们在最终时刻没有获得L2范数的更快收敛。
We consider a finite element discretization for the reconstruction of the final state of the heat equation, when the initial data is unknown, but additional data is given in a sub domain in the space time. For the discretization in space we consider standard continuous affine finite element approximation, and the time derivative is discretized using a backward differentiation. We regularize the discrete system by adding a penalty on the H2-semi-norm of the initial data, scaled with the mesh-parameter. The analysis of the method uses techniques developed in E. Burman and L. Oksanen [Numer. Math. 139 (2018) 505–528], combining discrete stability of the numerical method with sharp Carleman estimates for the physical problem, to derive optimal error estimates for the approximate solution. For the natural space time energy norm, away from t = 0, the convergence is the same as for the classical problem with known initial data, but contrary to the classical case, we do not obtain faster convergence for the L2-norm at the final time.