论文标题
椭圆和抛物线问题的扩散系数鉴定的有限元近似值的错误分析
Error Analysis of Finite Element Approximations of Diffusion Coefficient Identification for Elliptic and Parabolic Problems
论文作者
论文摘要
在这项工作中，我们提出了一个新的错误分析，用于恢复椭圆形或抛物线问题中的空间依赖扩散系数。它基于标准的正则最小二乘公式公式，具有$ H^1（ω）$ seminorm惩罚，然后使用Galerkin有限元法离散，并在寄生虫中使用符合状态和系数的分段线性有限元，并在时间上向后退Euler。我们得出\ textit {先验}加权$ l^2（ω）$估计，而常数仅依赖椭圆形和抛物线案例的给定问题数据。此外，这些估计还允许在阳性条件下得出标准的$ l^2（ω）$错误估计值，该条件可以验证某些问题数据。提供数值实验以补充误差分析。
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 $H^1(Ω)$ 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 \textit{a priori} weighted $L^2(Ω)$ estimates where the constants depend only on the given problem data for both elliptic and parabolic cases. Further, these estimates also allow deriving standard $L^2(Ω)$ error estimates, under a positivity condition that can be verified for certain problem data. Numerical experiments are provided to complement the error analysis.
