In FEM we use the continuum model of the system and mesh it to solve for the solution but what is the lower limit for the mesh so that this continuum approach still holds and at what point we should apply Molecular Dynamics (MD) approach?
For example in the simulation of a microfluidic-channel having largest dimension of 10 micro-meters, the mesh size can go to as small as nano-meter, will it be reasonable to mesh this system with such small elements? OR the continuum approach no more holds and one will get a wrong solution.
For the continuum approximation to be valid, the finite element mesh size must be much greater than the mean free path of the fluid particles. The Knudsen number can provide a measure of this validity as well as providing corrections to the viscosity model where the regions of validity overlap.
http://en.wikipedia.org/wiki/Knudsen_number
Note that you should differentiate between the "modelling" and "solving" phases. That is while continuum mechanics approximations may not be strictly valid in the nano realm (and even totally wrong in the pico region where atomic and molecular "particle" features dominate), it is for sure that using smaller meshes in solving the continuum equation (i.e. via FEM), gives more accurate answer than using a coarser mesh grid. That said, if one could theoretically solve a continuum mechanics problem, i.e. Navier-Stokes equations, using sub-nano sized elements by a stable method, the solutions are guaranteed to have less distance to the original N-S PDE's solutions with respect to coarser meshes (for example see the Consistency section in http://en.wikipedia.org/wiki/Numerical_methods_for_ordinary_differential_equations). However, one should not be deluded to reveal sub-nano physics from these meshes. That is he/she should only consider integrated quantities obtained from the solution on "large enough" integration domains.
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