Primal-Dual Mixed Finite Element Methods for the Elliptic Cauchy Problem
Primal-Dual Mixed Finite Element Methods for the Elliptic Cauchy Problem
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DOI：
10.1137/17m1163335
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发表时间：
2017-12
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影响因子：
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通讯作者：
E. Burman;M. Larson;L. Oksanen
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文献类型：
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作者：
E. Burman;M. Larson;L. Oksanen
我们考虑用原始-对偶混合有限元方法来求解椭圆型柯西问题或其他相关的数据同化问题。该方法具有局部守恒性。我们利用已知的条件稳定性估计得到了先验误差估计，并确定了产生光滑精确解最优收敛的弱一致镇定和Tikhonov正则化的最小数量。数据中的扰动的影响也被考虑在内。通过选择对偶变量的特殊稳定化，该方法的一个简化版本可以看作是Dard e，Hannukainen和Hyv Onen在SIAM J.Numer提出的基于混合拟可逆法求解椭圆{C}Auchy问题的最小二乘混合有限元方法的变体。分析，51(4)2013。主要区别在于，我们选择的正则化不依赖于辅助参数，网格大小是唯一的渐近参数。最后，我们证明了简化方法可以用于缺陷校正迭代来确定完整方法的解。通过一些数值算例说明了理论的正确性。
We consider primal-dual mixed finite element methods for the solution of the elliptic Cauchy problem, or other related data assimilation problems. The method has a local conservation property. We derive a priori error estimates using known conditional stability estimates and determine the minimal amount of weakly consistent stabilization and Tikhonov regularization that yields optimal convergence for smooth exact solutions. The effect of perturbations in data is also accounted for. A reduced version of the method, obtained by choosing a special stabilization of the dual variable, can be viewed as a variant of the least squares mixed finite element method introduced by Dard\'e, Hannukainen and Hyv\"onen in \emph{An {$H\sb {\sf{div}}$}-based mixed quasi-reversibility method for solving elliptic {C}auchy problems}, SIAM J. Numer. Anal., 51(4) 2013. The main difference is that our choice of regularization does not depend on auxiliary parameters, the mesh size being the only asymptotic parameter. Finally, we show that the reduced method can be used for defect correction iteration to determine the solution of the full method. The theory is illustrated by some numerical examples.