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Zbl 1149.65083
Sidi Ammi, Moulay Rchid; Torres, Delfim F.M.
Numerical analysis of a nonlocal parabolic problem resulting from thermistor problem.
(English)
[J] Math. Comput. Simul. 77, No. 2-3, 291-300 (2008). ISSN 0378-4754

The authors study the spatially semidiscrete piecewise linear finite element method for the nonlinear parabolic problem $$\partial u/\partial t - \nabla \cdot (k(u) \nabla u) = \lambda f(u) \bigg/ \biggl(\int_\Omega f(u)\,dx\biggr)^2$$ in $\Omega \times (0,\bar{t})$, $u = 0$ on $\partial \Omega \times (0,\bar{t})$, $u(0) = u_0$ in $\Omega$, where $\Omega$ is a bounded domain in $\mathbb{R}^2$, $\bar{t}$ a positive number, $f$ and $k$ given functions from $\mathbb{R}$ to $\mathbb{R}$, $\lambda >0$. This parabolic problem describes the temperature profile of a thermistor device with electrical resistivity $f$. The full discrete backward Euler method and the Crank-Nicolson-Galerkin scheme are also considered. An algorithm for solving the fully discrete problem is proposed but no numerical results are presented.
[Dinh Nho Hao (Hanoi)]
MSC 2000:
*65M60 Finite numerical methods (IVP of PDE)
65M20 Method of lines (IVP of PDE)
35K55 Nonlinear parabolic equations
65M06 Finite difference methods (IVP of PDE)
65M15 Error bounds (IVP of PDE)

Keywords: finite element method; nonlocal parabolic equation; elliptic projection; error estimates; semidiscretization

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