Consider the problem
\begin{align} \Delta u &= f \;\text{ in } \Omega \subset \mathbb{R}^3 \\ u &= g \;\,\text{on} \; \,\partial \Omega \end{align}
Suppose I have a tetrahedral mesh of $\Omega$ and a triangle mesh of $\partial \Omega$. If the surface mesh and the tetrahedral mesh aren't related (e.g. they were generated separately, one is more refined etc.) are there any methods to enforce essential BCs?
Natural BCs seem straightforward since an integral over the surface isn't related to the volume mesh. I've heard that it's possible to impose essential boundary conditions in the weak formulation, would this be a way forward?
OR
Is the only possible way to extract the surface nodes of the tetrahedral mesh and then apply BCs to the stiffness matrix?
Thanks.
Edit: I'm also curious about the case when your surface is represented implicitly/parametrically. Are there any algorithms besides generating a surface mesh/conforming volume mesh?