Diffusion on a Boundaryless Manifold and Tesselation

Question

Suppose we are dealing with diffusion on a boundaryless manifold $M$. When people use the finite-element method to find an approximate solution, I always see them use \begin{align} \int_{M^*} W \Delta U \mathrm{d} x & = - \int_{M^*} \nabla W \cdot \nabla U \mathrm{d} x, \end{align} in the weak form. Here $\Delta$ is the Beltrami-operator and $\nabla$ is the surface gradient, $M^*$ is usually a triangularly tesselated approximation of $M$, meaning locally planar. I was wondering if the usage of above formula (which is essentially product rule + divergence theorem) is actually valid in this case since $M^*$ is not a smooth object, meaning that it contains `kinks' in the transition form one triangle to the other. Or should the replacement $M \rightarrow M^*$ be regarded as part of the discretization ?