V-cycle Multigrid Algorithms for Discontinuous Galerkin Methods on Non-nested Polytopic Meshes

Abstract

In this paper we analyze the convergence properties of V-cycle multigrid algorithms for the numerical solution of the linear system of equations stemming from discontinuous Galerkin discretization of second-order elliptic partial differential equations on polytopic meshes. Here, the sequence of spaces that stands at the basis of the multigrid scheme is possibly non-nested and is obtained based on employing agglomeration algorithms with possible edge/face coarsening. We prove that the method converges uniformly with respect to the granularity of the grid and the polynomial approximation degree p, provided that the minimum number of smoothing steps, which depends on p, is chosen sufficiently large.

Appendix: Proof of Lemma 7

In order to show Lemma 7, we follow the analysis presented in [43]. We first show two preliminary results making use of the properties of Sect. 3.

Lemma 8

Let Assumptions 1–4 hold and let ${\widetilde{\varPi }}_j$ be the projection operator on $V_j$ as defined in Lemma 4, for $j=J,J-1$. Then

Proof

Using the triangular inequality, Remark 7 and the approximation estimates of Lemma 4 we have:

where in the last inequality we also used hypotheses (3) and (4). $\square$

Lemma 9

Let Assumptions 1–4 hold. Let and denote by $w_j \in V_j$ the solution of $\forall v \in V_j$ with $j=J-1,J$. Then the following inequality holds:

(28)

Proof

Consider the unique solution $w \in V$ of the problem

$$\begin{aligned} \mathcal {A}(w,v) = (g,v)_{L^2(\varOmega )} \quad \forall v \in V. \end{aligned}$$

Using Corollary 1, we have

(29)

Using the triangular inequality and Remark 7 we have:

Using (29), Lemmas 4 and 8, we have

From the elliptic regularity assumption (2) and hypotheses (3) and (4), we can write

(30)

Now, let $z_j \in V_j$ be the solution of:

Using (30) we get the following estimate:

Then, we have:

from which, together with (30), inequality (28) follows. $\square$

Proof (of Lemma 7)

For any $v_J \in V_J$ we consider the following equality:

(31)

Next, consider the solution $z_j$ of the following problems

By using the definition of $P_J^{J-1}$ and Lemma 9, we have:

Using the last inequality together with (31) we get (26). $\square$