SIAM Journal on Numerical Analysis, Volume 59, Issue 1, Page 119-142, January 2021.
In this work, we present a novel error analysis for recovering a spatially dependent diffusion coefficient in an elliptic or parabolic problem. It is based on the standard regularized output least-squares formulation with an $H^1(\Omega)$ seminorm penalty and then discretized using the Galerkin finite element method with conforming piecewise linear finite elements for both state and coefficient and backward Euler in time in the parabolic case. We derive a priori weighted $L^2(\Omega)$ estimates where the constants depend only on the given problem data for both elliptic and parabolic cases. Further, these estimates also allow deriving standard $L^2(\Omega)$ error estimates under a positivity condition that can be verified for certain problem data. Numerical experiments are provided to complement the error analysis.

中文翻译：

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椭圆和抛物线问题扩散系数识别的有限元逼近误差分析

SIAM数值分析学报，第59卷，第1期，第119-142页，2021年1月。
在这项工作中，我们提出了一种新颖的误差分析，用于恢复椭圆或抛物线问题中空间相关的扩散系数。它基于具有$ H ^ 1（\ Omega）$半范数罚分的标准正则输出最小二乘公式，然后使用Galerkin有限元方法离散化，其中状态和系数均符合分段线性有限元，并且在时间上向后欧拉在抛物线的情况下。我们得出先验加权$ L ^ 2（\ Omega）$估计，其中对于椭圆和抛物线情况，常数仅取决于给定的问题数据。此外，这些估计还允许在可针对某些问题数据进行验证的正性条件下得出标准的$ L ^ 2（\ Omega）$误差估计。提供了数值实验以补充误差分析。