We consider a curve which propagates in the normal direction with velocity proportional to its curvature plus a forcing term. This geometric problem is first approximated by a singularly perturbed parabolic double obstacle problem with small parameter epsilon>0 . Conforming piecewise linear finite elements over a quasi-uniform and strongly acute mesh of size h are further used for space discretization, and combined with backward differences for time discretization with uniform time-step tau . We show convergence and linear error estimates for the zero level set of the fully discrete solution to the true interface, even past singularities.