Homogeneous multigrid for embedded discontinuous Galerkin methods

Abstract

We formulate a multigrid method for an embedded discontinuous Galerkin (EDG) discretization scheme for Poisson’s equation. This multigrid method is homogeneous in the sense that it uses the same discretization scheme on all levels. In particular, we use the injection operator developed in Lu et al. (IMA J Numer Anal, 2021) for HDG and show optimal convergence rates under the assumption of elliptic regularity. Our analytical findings are underlined by numerical experiments.

Appendix A. Used results

Here, we summarize the results from other sources that we used in the proofs of our propositions.

Lemma A.1

Let $\mu$ be any function in ${{\tilde{M}}}_\ell$. The following statement holds:

$$\begin{aligned} \vert \! \vert \! \vert \sqrt{\tau _\ell } ({\mathcal {U}}_\ell \mu - \mu ) \vert \! \vert \! \vert _\ell \lesssim \sqrt{ h_\ell \tau _\ell } \Vert \varvec{{\mathcal {Q}}}_\ell \mu \Vert _0. \end{aligned}$$

(A.1)

Thus, if $\tau _\ell h_\ell \lesssim 1$,

$$\begin{aligned} \Vert {\mathcal {U}}_\ell \mu - \mu \Vert _\ell \lesssim h_\ell \Vert \varvec{{\mathcal {Q}}}_\ell \mu \Vert _0. \end{aligned}$$

(A.2)

Proof

The first inequality is [5, Lemma 3.4 (iv)] whose right hand side is estimated using [5, Lemma 3.4 (v)]. The second inequality follows after multiplication with $h_\ell$ and exploiting the definitions of $\vert \! \vert \! \vert \cdot \vert \! \vert \! \vert _\ell$ and $\Vert \cdot \Vert _\ell$. $\square$

Lemma A.2

If $\tau _\ell h_\ell \lesssim 1$, the local solution operators obeys

$$\begin{aligned} \Vert {\mathcal {U}}_\ell \mu \Vert _0 \lesssim \Vert \mu \Vert _\ell \end{aligned}$$

(A.3)

Proof

This is Theorem 3.1 in [5], where we use that the constant becomes independent of $h_\ell$ if $\tau _\ell h_\ell \lesssim 1$. $\square$

Lemma A.3

(Lemma 4.3 in [19]) The DG reconstructions of the injection operator admits the estimate

$$\begin{aligned} \Vert {\mathcal {U}}_{\ell -1} \mu - {\mathcal {U}}_\ell I_\ell \mu \Vert _0 \lesssim h_\ell \Vert \mu \Vert _{a_{\ell -1}}, \qquad \forall \mu \in {{\tilde{M}}}_{\ell -1}. \end{aligned}$$

(A.4)