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

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	Numerical multiscale methods to determine the coefficient in diffusion problems
Computational Methods for Differential Equations
مقاله 7، دوره 13، شماره 4، دی 2025، صفحه 1162-1176 اصل مقاله (2.47 M)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22034/cmde.2024.59745.2547
نویسندگان
Marzieh Tavakolian¹؛ Ali Hatam^* ¹؛ Morteza Fotouhi²؛ Edmund Chadwick³
¹Department of Applied Mathematics, Amirkabir University of Technology, No. 350, Hafez Ave, Valiasr Square, Tehran, Iran 1591634311.
²Department of Mathematical Sciences, Sharif University of Technology, Tehran 11365-9415, Iran.
³School of Science, Engineering & Environment, University of Salford, Salford, M5 4WT, UK.
چکیده
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.
کلیدواژه‌ها
Heterogeneous multiscale method؛ Homogenization؛ Inverse problem؛ Localized orthogonal decomposition method؛ Parabolic partial differential equations

مراجع
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Numerical multiscale methods to determine the coefficient in diffusion problems