Numerical solutions of the thermistor problem by spline finite elements (Q1763296)

This paper deals with the numerical resolution of the following boundary value problem: \[ \begin{gathered} {\partial U\over\partial t}= {\partial^2U\over\partial t^2}+ \alpha\cdot\sigma\Biggl({\partial\phi\over\partial x}\Biggr)^2,\quad 0< x< 1,\quad t> 0,\\ {\partial\over\partial x} \Biggl(\sigma{\partial\phi\over\partial x}\Biggr)= 0,\quad 0< x< 1,\quad t> 0,\\ {\partial U\over\partial x}= 0,\quad x= 0,\quad t> 0,\\ {\partial U\over\partial x}+ \beta U= 0,\quad x= 1,\quad t> 0,\\ \phi(0, t)= 0,\quad t> 0,\\ \phi(1,t)= 1,\quad t> 0,\\ U(x,0)= 0,\quad 0\leq x\leq 1,\\ \phi(x,0)= x,\quad 0\leq x\leq 1,\end{gathered} \] where \[ \Phi(U)= \begin{cases} 1 & 0\leq u\leq 1\\ \delta & U> 1\end{cases} \] \[ \delta= 10^{-5},\quad \beta> 0, \] using a subdomain collocation and a Petrov-Galerkin method based on spline-finite elements. The resulting system of ordinary differential equations is solved by the usual Crank-Nicolson finite difference method using a variant of Thomas algorithm.