Functional A Posteriori Error Equalities for Conforming Mixed Approximations of Elliptic Problems
Functional A Posteriori Error Equalities for Conforming Mixed Approximations of Elliptic Problems
复制标题
椭圆问题混合近似的泛函后验误差方程
DOI：
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
D. Pauly
中科院分区：
文献类型：
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作者：
I. Anjam;D. Pauly
在本文中，我们展示了如何使用雷宾精神中的函数型后验误差估计来找到符合混合近似的精确误差（而不仅仅是误差的估计）。误差是在混合范数中测量的，它同时考虑了原始变量和对偶变量。对于一类椭圆型偏微分方程，我们得到了这个结果。我们首先通过使用简化的反应扩散问题推导出我们主要结果的一个特殊版本，以证明与经典的泛函后验误差估计的强联系。在此之后，我们在抽象设置中推导出主要结果。我们的主要结果表明，为了获得一个符合混合近似的精确全局误差值，只需要问题数据和精确解的混合近似。不需要计算任何辅助数据。精确误差的计算包括简单地计算两个（通常是积分）量，其中所有的量都是已知的，通过任何符合的方法得到近似解。最后给出了数值计算结果。
In this paper we show how to find the exact error (not just an estimate of the error) of a conforming mixed approximation by using the functional type a posteriori error estimates in the spirit of Repin. The error is measured in a mixed norm which takes into account both the primal and dual variables. We derive this result for elliptic partial differential equations of a certain class. We first derive a special version of our main result by using a simplified reaction-diffusion problem to demonstrate the strong connection to the classical functional a posteriori error estimates of Repin. After this we derive the main result in an abstract setting. Our main result states that in order to obtain the exact global error value of a conforming mixed approximation one only needs the problem data and the mixed approximation of the exact solution. There is no need for calculating any auxiliary data. The calculation of the exact error consists of simply calculating two (usually integral) quantities where all the quantities are known after the approximate solution has been obtained by any conforming method. We also show some numerical computations to confirm the results.