[Paper Review] V-cycle multigrid algorithms for discontinuous Galerkin methods on non-nested polytopic meshes

Paola F. Antonietti, Giorgio Pennesi|arXiv (Cornell University)|Sep 26, 2017

Advanced Numerical Methods in Computational Mathematics36 references3 citations

TL;DR

This paper proposes V-cycle multigrid algorithms for discontinuous Galerkin methods on non-nested polytopic meshes, using agglomeration with edge/face coarsening to avoid small faces in coarse levels. It proves uniform convergence with respect to mesh granularity and polynomial degree $p$, provided sufficient smoothing steps are used, overcoming conditioning issues from small faces in standard agglomeration schemes.

ABSTRACT

In this paper we analyse the convergence properties of V-cycle multigrid algorithms for the numerical solution of the linear system of equations arising from discontinuous Galerkin discretization of second-order elliptic partial differential equations on polytopal 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 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 number of smoothing steps, which depends on p, is chosen sufficiently large.

Motivation & Objective

• To address the poor conditioning of linear systems arising from high-order DG discretizations on polytopic meshes with small faces.
• To develop a multigrid solver that avoids inheriting small faces from fine to coarse levels by using non-nested, agglomerated meshes.
• To prove uniform convergence of the V-cycle multigrid method with respect to mesh refinement and polynomial degree $p$.
• To establish convergence under a general non-nested multigrid framework applicable to DG methods.
• To overcome limitations of the BPX framework by relaxing the assumption $\mathcal{A}_j(I_{j-1}^j u, I_{j-1}^j u) \leq \mathcal{A}_{j-1}(u,u)$, which may not hold in DG settings.

Proposed method

• Construct a sequence of non-nested discrete polynomial spaces via agglomeration with possible edge/face coarsening to prevent small faces from propagating to coarse levels.
• Define distinct discrete bilinear forms on each level to accommodate non-nested spaces, ensuring stability and accuracy.
• Employ a V-cycle multigrid algorithm with Richardson smoothing and subspace correction steps, where smoothing steps depend on the polynomial degree $p$.
• Use interpolation operators $I_{j-1}^j$ and restriction operators $P_j^{j-1}$ between non-nested spaces to transfer corrections across levels.
• Apply a non-nested multigrid framework based on the BPX framework, adapted to DG settings by modifying the stability and approximation assumptions.
• Prove convergence using a combination of elliptic regularity, approximation error estimates, and stability bounds involving $h_j^2/p_j^{2-\mu}$ terms.

Experimental results

Research questions

• RQ1 Can V-cycle multigrid methods achieve uniform convergence for high-order DG discretizations on non-nested polytopic meshes?
• RQ2 Does edge/face coarsening in agglomeration prevent the conditioning degradation caused by small faces in coarse-level meshes?
• RQ3 Is the convergence of the multigrid method uniform with respect to both mesh size $h$ and polynomial degree $p$?
• RQ4 Can the standard BPX multigrid convergence theory be adapted to DG methods when the interpolation stability condition $\mathcal{A}_j(I_{j-1}^j u, I_{j-1}^j u) \leq \mathcal{A}_{j-1}(u,u)$ fails?
• RQ5 What is the minimal number of smoothing steps required for uniform convergence, and how does it scale with $p$?

Key findings

• The V-cycle multigrid method converges uniformly with respect to mesh refinement and polynomial degree $p$, provided the number of smoothing steps is sufficiently large and depends on $p$.
• The convergence rate is bounded by $\|w_{J} - I_{J-1}^{J}w_{J-1}\|_{L^2(\Omega)} \lesssim \frac{h_J^2}{p_J^{2-\mu}}\|g\|_{L^2(\Omega)}$, indicating optimal convergence behavior.
• The method avoids the conditioning issues of standard agglomeration by using edge/face coarsening, preventing small faces from appearing in coarse-level meshes.
• The analysis establishes that the non-nested multigrid framework remains robust even when the standard BPX stability assumption fails in DG settings.
• The key estimate $\|z_{J-1} - I_{J-1}^J z_{J-1}\|_{L^2(\Omega)} \lesssim \frac{h_J^2}{p_J^{2-\mu}}\|w_{J-1} - P_J^{J-1}w_J\|_{L^2(\Omega)}$ underpins the convergence proof.
• The method ensures uniform convergence by combining approximation error bounds with stability estimates involving the $L^2$-projection and interpolation operators.