Numerical solutions of the thermistor problem by spline finite elements

This paper presents approximate steady-state solutions of a one-dimensional positive temperature coefficient (PTC) thermistor problem, having a step function electrical conductivity that is a highly non-linear function of the temperature, using subdomain collocation and Petrov-Galerkin methods 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. It is shown that the numerical solutions obtained by the present methods exhibit the correct physical characteristics of the problem and, they are in very good agreement with the exact solution. A Fourier stability analysis of each method is also investigated.