Numerical multiscale methods to determine the coefficient in diffusion problems

Articles in Press

Document Type : Research Paper

Authors

1 Department of Applied Mathematics, Amirkabir University of Technology, No. 350, Hafez Ave, Valiasr Square, Tehran, Iran 1591634311.

2 Department of Mathematical Sciences, Sharif University of Technology, Tehran 11365-9415, Iran.

3 School of Science, Engineering & Environment, University of Salford, Salford, M5 4WT, UK.

Abstract

Here we study the inverse problem of determining the highly oscillatory coefficient $a^\varepsilon$ in some PDEs of the form $ u^\varepsilon_t - \nabla .(a^\varepsilon(x) \nabla u^\varepsilon)=0$, in a bounded domain $\Omega \subset\mathbb{R}^d $; $\varepsilon$ indicates the smallest characteristic wavelength in the problem ($0 < \varepsilon \ll 1$).
Assume that $g(t, x)$ is given input data for $(t, x) \in (0,T) \times\partial \Omega$ and the associated output is the thermal flux $a^\varepsilon(x)\nabla u(T_0,x)\cdot n(x)$ measured on the boundary at a given time $T_0$. Because of ill-posedness of the inverse problem, we reduce the dimension by seeking effective parameters. For forward solver, we apply either analytic homogenization or some numerical multiscale methods such as FE-HMM and LOD method.

Keywords

Main Subjects

Computational Methods for Differential Equations

Articles in Press, Accepted Manuscript
Available Online from 03 January 2025

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History

  • Receive Date: 24 December 2023
  • Revise Date: 14 November 2024
  • Accept Date: 23 December 2024

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