后验误差估计在有限元法中用于介电常数的重建
A posteriori error estimation in a finite element method for reconstruction of dielectric permittivity
AI总结：
本文基于最小化方法研究了系数反问题的后验误差估计，通过有限元方法重建介质的介电常数，并给出了计算系数与真实解之间的误差估计。
摘要：
我们为有限元近似在最小化方法的系数反问题中提供了后验误差估计。该问题是从电场的边界测量中重建介电常数ε=ε(x)，x∈Ω⊂R³。电场通过麦克斯韦方程与介电常数相关联。重建过程基于最小化一个Tikhonov泛函，其中介电常数、电场和拉格朗日乘子函数被近似为分段多项式。我们的主要结果是将计算系数ε_h与真实极小解ε之间的差异用计算函数表示的估计。
英文摘要：
We present a posteriori error estimates for finite element approximations in a minimization approach to a coefficient inverse problem. The problem is that of reconstructing the dielectric permittivity $\varepsilon = \varepsilon(\mathbf{x})$, $\mathbf{x}\inΩ\subset\mathbb{R}^3$, from boundary measurements of the electric field. The electric field is related to the permittivity via Maxwell's equations. The reconstruction procedure is based on minimization of a Tikhonov functional where the permittivity, the electric field and a Lagrangian multiplier function are approximated by peicewise polynomials. Our main result is an estimate for the difference between the computed coefficient $\varepsilon_h$ and the true minimizer $\varepsilon$, in terms of the computed functions.
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