On the box method for a non-local parabolic variational inequality (Q5942860)

The authors study a box scheme or finite volume element method for the following non-local nonlinear parabolic variational inequality arising in the study of thermistor problems: Find \(u\in K=\{v\in H^1_0(\Omega):v\geq 0\}\), such that \(\forall v\in K\) NEWLINE\[NEWLINE(u_t,v-u)+(k(u)\nabla u,\nabla(v-u)+\eta\left(\int_\Omega G(x,y)u(y) dy,v-u\right)+\alpha(u^4,v-u)\geq (\sigma(u)|\nabla\phi|^2,v-u),NEWLINE\]NEWLINE NEWLINE\[NEWLINE(\sigma(u)\nabla\phi,\nabla v)=0,\quad \forall v\in H^1_0(\Omega).NEWLINE\]NEWLINE Under some assumptions on the data and regularity of the solution, optimal error estimates in the \(H^1\)-norm are given.