# Weak form of the Laplace-Beltrami operator on closed manifolds

Suppose we are dealing with diffusion over a boundaryless manifold $M$ (for simplicity let's say it's a surface). In that case, we have \begin{align} \int_M W \Delta U \mathrm{d} x & = -\int_M \nabla W \cdot \nabla U \mathrm{d} x, \end{align} where $\Delta$ is the Laplace-Beltrami operator and $\nabla$ the surface gradient. What I have seen plenty of times in papers is that people simply take this identity and use it over a non-smooth surface $M^*$ that approximates $M$, for example via tessellation to derive a system of equations for a finite-element approximation. The problem is that on non-smooth surfaces the above identity is not applicable. Here is what I believe happens: I believe that you should carry out the integral over all triangles $T_i$ that make up $\mathcal{M}^*$ \begin{align} \int_{M^*} W \Delta U \mathrm{d} x & = \sum_{i = 1}^n \int_{T_i} W \Delta U \mathrm{d} x \nonumber \\ & = -\int_{M^*} \nabla W \cdot \nabla U \mathrm{d} x + \sum_{i = 1}^n \int_{\partial T_i} W \nabla U \cdot \mathbf{N}_{\partial T_i} \mathrm{d} \Gamma. \end{align} So essentially they have weakly imposed that \begin{align} \sum_{i = 1}^n \int_{\partial T_i} W \nabla U \cdot \mathbf{N}_{\partial T_i} \mathrm{d} \Gamma = 0. \end{align} So I believe that the solutions of the FEM-approach weakly satisfy \begin{align} \nabla U \cdot \mathbf{N}_{\partial T_i^j} = - \nabla U \cdot \mathbf{N}_{\partial T_k^l}, \end{align} whenever $\partial T_i^j$ and $\partial T_k^l$ overlap on $M^*$ (which any strong solution automatically satisfies). Is this correct ? And if so, can someone give a rigorous proof or at least a reference to a source where this is explained ? Thanks in advance.