On Wk, p-manifolds and Wk, p-surfaces for analysis of the convergence of mesh-based approximation

To categorize the convergence properties of mesh-based approximations to manifolds and surfaces, this paper defines these approximations as “Wk,p-manifolds” and “Wk,psurfaces.” In particular, this paper examines the importance of these classifications in the convergence in L1-norm of interpolants, built on the approximate manifold or surface, of functions defined on the approximated manifold or surface. To provide context, the applicability of an interpolation framework established by Nédélec involving the convergence of metric determinants is examined. An extension of Nédélec’s framework to Wk,p-surfaces is presented.