Why Continuous Finite Elements are Good

At the highest level, polynomials used to approximate solutions to partial differential equations come in two flavors: continuous with respect to the mesh, and discontinuous with respect to the mesh. One of the key advantages of the continuous ones is that they produce conforming approximations of Sobolev Spaces, and so one can enjoy density properties and whatnot. This is discussed in more detail, with specific toy examples, here: https://joshuasiktarcomputationalarchive.weebly.com/uploads/2/8/6/2/28629309/why_continuous_finite_elements_are_good.pdf.