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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 $V_l$

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There is really nothing different between continuous and discontinuous finite element spaces: The injection mapping you state is exact in the sense that you can represent a coarse function on the fine grid exactly in both cases. As a consequence, you do the same thing: You need to evaluate the coarse shape functions at the fine support points just as you state, again in both cases, to define the injection matrix.

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  • $\begingroup$ 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 $\endgroup$
    – FEGirl
    Commented Mar 30 at 15:08
  • $\begingroup$ 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). $\endgroup$ Commented Apr 1 at 16:26

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