Three ways to represent the unknown
The three methods differ by the location of the degrees of freedom and by the continuity imposed between neighbouring cells. The rest of the pipeline (assembling the system, solving it, refining the mesh) is shared between the three methods.
FEM: shared nodes
Each degree of freedom is located at a node and is shared by all the cells that contain this node. The solution is continuous by construction. The matrix is sparse and, for a diffusion problem, symmetric.
DG: one block per cell
Each cell has its own degrees of freedom, which are not shared with other cells. Continuity is enforced weakly by a numerical flux at every face, as in finite volume schemes.
FV: one value per cell
This is the simplest of the three methods, with a single value per cell (the cell average) and no local polynomial. As for DG, cells are coupled by a flux through each face, and FV can be seen as the order 0 member of the DG family.
Where to go next
FEM solver
The base method, illustrated on a 2D Laplacian problem, and its links with multigrid and basis functions.
DG solver
The same problem, with one block per cell and a numerical flux between cells.
Finite volumes
Cell averages and a two-point flux. This is the cheapest discretization in the library.
Multigrid
A preconditioner that keeps the number of Krylov iterations constant as the mesh is refined, available for FEM and DG.
Basis functions
Lagrange, Taylor and B-spline: the three local polynomial families used by FEM and DG.
Nonlinear solvers
Fixed point, Newton, Newton preconditioned by multigrid and FAS, for nonlinear PDEs.
Inverse problems
Recovering an unknown coefficient from data, by differentiating through the forward solve.