Stable reconstruction of discontinuous solutions to the Cauchy problem in steady-state anisotropic heat conduction with non-smooth coefficients

IF 1.9 3区 数学 Q2 Mathematics
M. Bucataru, Iulian Cîmpean, L. Marin
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引用次数: 0

Abstract

We study the recovery of the missing discontinuous/non-smooth thermal boundary conditions on an inaccessible portion of the boundary of the domain occupied by a solid from Cauchy data prescribed on the remaining boundary assumed to be accessible, in case of stationary anisotropic heat conduction with non-smooth/discontinuous conductivity coefficients. This inverse boundary value problem is ill-posed and, therefore, should be regularized. Consequently, a stabilising method is developed based on a priori  knowledge on the solution to this inverse problem and the smoothing feature of the direct problems involved. The original problem is transformed into a control one which reduces to solving an appropriate minimisation problem in a suitable function space. The latter problem is tackled by employing an appropriate variational method which yields a gradient-type iterative algorithm that consists of two direct problems and their corresponding adjoint ones. This approach yields an algorithm designed to approximate specifically merely L 2 -boundary data in the context of a non-smooth/discontinuous anisotropic conductivity tensor, hence both the notion of solution to the direct problems involved and the convergence analysis of the approximate solutions generated by the algorithm proposed require special attention. The numerical implementation is realised for two-dimensional homogeneous anisotropic solids using the finite element method, whilst regularization is achieved by terminating the iteration according to two stopping criteria.
非光滑系数稳态各向异性热传导Cauchy问题不连续解的稳定重构
我们研究了在非光滑/不连续导热系数的稳态各向异性热传导情况下,从假定可访问的剩余边界上规定的Cauchy数据中恢复被固体占据的区域边界不可访问部分缺失的不连续/非光滑热边界条件。这个反边值问题是不适定的,因此应该正则化。因此,基于该反问题解的先验知识和所涉及的直接问题的平滑特征,开发了一种稳定方法。将原问题转化为控制问题,简化为在合适的函数空间中求解合适的最小化问题。后一个问题采用适当的变分方法来解决,该方法产生一个梯度型迭代算法,该算法由两个直接问题及其相应的伴随问题组成。这种方法产生了一种算法,专门用于在非光滑/不连续各向异性电导率张量的背景下近似l2边界数据,因此,所涉及的直接问题的解的概念和所提出的算法产生的近似解的收敛性分析都需要特别注意。采用有限元法实现了二维均匀各向异性固体的数值实现,并根据两个停止准则终止迭代,实现了正则化。
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来源期刊
CiteScore
2.70
自引率
5.30%
发文量
27
审稿时长
6-12 weeks
期刊介绍: M2AN publishes original research papers of high scientific quality in two areas: Mathematical Modelling, and Numerical Analysis. Mathematical Modelling comprises the development and study of a mathematical formulation of a problem. Numerical Analysis comprises the formulation and study of a numerical approximation or solution approach to a mathematically formulated problem. Papers should be of interest to researchers and practitioners that value both rigorous theoretical analysis and solid evidence of computational relevance.
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