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Numerical analysis of a nonlocal parabolic problem resulting from thermistor problem. (English) Zbl 1149.65083

The authors study the spatially semidiscrete piecewise linear finite element method for the nonlinear parabolic problem

$\partial u/\partial t-\nabla ·\left(k\left(u\right)\nabla u\right)=\lambda f\left(u\right)/{\left({\int }_{{\Omega }}f\left(u\right)\phantom{\rule{0.166667em}{0ex}}dx\right)}^{2}$

in ${\Omega }×\left(0,\overline{t}\right)$, $u=0$ on $\partial {\Omega }×\left(0,\overline{t}\right)$, $u\left(0\right)={u}_{0}$ in ${\Omega }$, where ${\Omega }$ is a bounded domain in ${ℝ}^{2}$, $\overline{t}$ a positive number, $f$ and $k$ given functions from $ℝ$ to $ℝ$, $\lambda >0$. 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.

MSC:
 65M60 Finite elements, Rayleigh-Ritz and Galerkin methods, finite 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)
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