Role of mixed function spaces in coupled differential equations

Question

What is the importance of mixed function spaces?

Could I solve a set of coupled partial differential equations without having to make a mixed function space?

Computationally what does mixed function space do?

Answer 1

Conceptually, mixed function spaces account for the structure of, say, a PDE and a constraint (which might be a second PDE) that are coupled together through a Lagrangian Multiplier.

One can also solve both equations separately without mixing the FE spaces and realize the coupling in a different way, maybe by adding some algebraic constraints (the interface conditions) to the system.