I was looking into the Book of Riviere " Discontinuous Galerkin Methods for solving Elliptic and Parabolic Equations". In the comparaison of section 2.12 (copied below), the example of rectangular mesh indicated that the DG is more economic (has les DOFs) then the CG when using a certain space of elements. Can any one explain in more details please ? What is the difference between the two spaces Qk and Pk ?
Thank you for your help.
" Size of problem:For DG, the total number of degrees of freedom is proportional to the number of elements in the mesh. The constant of proportionality is a function of the polynomial degree. For CG, the degrees of freedom depend on the number of vertices and possibly the number of vertices and elements in the mesh. For instance, consider a structured mesh of 5 × 5 rectangular elements. The degrees of freedom for a DG approximation of degree 1, 2, 3, 4 are 75, 150, 250, 375, respectively, whereas the degrees of freedom for a CG approximation of degree 1, 2, 3, 4 are 36, 121, 256, 441, respectively. Thus, on such small mesh, if k ≥ 3, the CG method is more costly than DG. The reason is that we have to use the space Qk on rectangular elements for CG, but we can still use the space Pk on rectangular elements for DG."