Numerical multiscale methods to determine the coefficient in diffusion problems
نویسندگان
1 Department of Mathematical Sciences, Sharif University of Technology, Tehran 11365-9415, Iran.
2 School of Science, Engineering & Environment, University of Salford, Salford, M5 4WT, UK.
3 Department of Applied Mathematics, Amirkabir University of Technology, No. 350, Hafez Ave, Valiasr Square, Tehran, Iran 1591634311.
4 Department of Applied Mathematics, Amirkabir University of Technology, No. 350, Hafez Ave, Valiasr Square, Tehran, Iran 1591634311.
doi
10.22034/cmde.2024.59745.2547چکیده
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.
