The first part of the paper is an overview of the theory of approximation of wave equations by Galerkin methods. It treats convergence theory for linear second order evolution equations and includes studies of consistency and eigenvalue approximation.
We emphasize differential operators, such as the curl, which have large kernels and use L2 stable interpolators preserving them.
The second part is devoted to a framework for the construction of finite element spaces of differential forms on cellular complexes.
Material on homological and tensor algebra as well as differential and discrete geometry is included. Whitney forms, their duals, their high order versions, their tensor products and their hp-versions all fit.