# Intergrid transfer operator for Discontinuous Galerkin method (multigrid)

I am studying DG and I've seen that it's possible to define a multigrid method. Let's consider two nested grids $$\Omega_{l-1}$$ and $$\Omega_{l}$$ and $$V_{l-1}$$ and $$V_l$$ classical DG spaces, both of order $$p$$, with basis $$\left\{\varphi_1^l, \dots, \varphi_{n_{l}}^l\right\}$$ and $$\left\{\varphi_1^{l-1}, \dots, \varphi_{n_{l-1}}^{l-1}\right\}$$, respectively. For a multigrid method, we need to define transfer operators between the two spaces. I've read in some papers that one could use the canonical injection $$\mathcal{I}: V_{l-1} \rightarrow V_l$$

I guess this amounts to write a function $$v=\sum_{i=1}^{n_l-1}c_i\varphi_i^{l-1} \in V_{l-1}$$ in terms of basis function of $$V_l$$. Is this correct? If so, how can we determine an expression for such a transfer matrix? Intuitively, I'd say that the generic entry should be:

$$\varphi_i^{n_{l-1}}(p_j)$$

where $$p_j$$ is a support point for the finer finite element space $$V_l$$

• And for what concerns the restriction operator, am I allowed to take it just as the transpose (adjoint) of the prolongation? I mean, can I simply take $R=P^T$, where $P$ is the injection matrix? Or do I have to modify such a $R$? (Thanks for your answer). @WolfgangBangerth Commented Mar 30 at 15:08
• Yes, you will want to take the transpose. This guarantees that $RAP$ remains positive definite (or has all of the other properties you need for multigrid to succeed, if $A$ is not positive definite). Commented Apr 1 at 16:26