Numerical multiscale methods to determine the coefficient in diffusion problems

Tavakolian, Marzieh; Hatam, Ali; Fotouhi, Morteza; Chadwick, Edmund

doi:10.22034/cmde.2024.59745.2547

Numerical multiscale methods to determine the coefficient in diffusion problems

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.

10.22034/cmde.2024.59745.2547

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 $T_0$. Due to the ill-posedness of the inverse problem, we reduce the dimension by seeking effective parameters. For the forward solver, we apply either analytic homogenization or some numerical multiscale methods such as the FE-HMM and LOD method.

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