  In this paper we consider finite element approaches to computing the mean
curvature vector and normal at the vertices of piecewise linear triangulated
surfaces. In particular, we adopt a stabilization technique which allows for
first order $L^2$-convergence of the mean curvature vector and apply this
stabilization technique also to the computation of continuous, recovered,
normals using $L^2$-projections of the piecewise constant face normals.
Finally, we use our projected normals to define an adaptive mesh refinement
approach to geometry resolution where we also employ spline techniques to
reconstruct the surface before refinement. We compare or results to previously
proposed approaches.
