Numerical solution of a non-local elliptic problem modeling a thermistor with a finite element and a finite volume method

− $w''(x) = \lambda (f(w(x)))/((\eq^1_(-1) f(w(z)) dz)^2) \all x \in$ (−1, 1),
$w'(1) + \alpha w(1) = 0$, $w'$(−1) − $\alpha w$(−1)$ = $0,

where $\alpha$ and $\lambda$ are positive constants and $f$ is a function satisfying $f(s)$ > 0, $f'(s) < 0, f''(s) > 0$ for $s > 0, \eq^\infty_0 f(s)ds < \infty.$ The solution of the equation represents the steady state of a thermistor device. The problem has a unique solution for a critical value $\lambda$* of the parameter $\lambda$, at least two solutions for $\lambda < \lambda$* and has no solution for $\lambda > \lambda$*. We apply a finite element and a finite volume method in order to find a numerical approximation of the solution of the problem from the space of continuous piecewise quadratic functions, for the case that $\lambda < \lambda$* and for the stable branch of the bifurcation diagram. A comparison of these two methods is made regarding their order of convergence for $f(s) = e^( - s)$ and $f(s) = (1 + s)^( - 2)$. Also, for the same equation but with Dirichlet boundary conditions, a situation where the solution is unique for $\lambda < \lambda$*, a similar comparison of the finite element and the finite volume method is presented.