Matrix-free implementation of the non-nested multigrid method

TL;DR

This work develops a parallel, matrix-free non-nested geometric multigrid method for continuous Lagrange finite elements, enabling arbitrarily overlapping and independently partitioned levels within deal.II. It introduces a matrix-free intergrid transfer operator $\mathcal{P}^{(c,f)}$ and uses adaptive coarse-grid solvers, including polynomial multigrid with $p^{(c)}=p^{(f)}-1$ and AMG on the coarsest level. A detailed implementation addresses distributed geometric search, efficient evaluation at transfer points via sum-factorization, and boundary-curvature issues, with extensive 2D/3D numerical experiments (Poisson and linear elasticity) validating robustness and performance. Compared to nested MG and AMG/pMG baselines, the non-nested approach delivers competitive or superior performance at higher polynomial orders, highlighting the potential for flexible hierarchies and future extensions to more aggressive coarsening and enhanced transfer operators.

Abstract

Traditionally, the geometric multigrid method is used with nested levels. However, the construction of a suitable hierarchy for very fine and unstructured grids is, in general, highly non-trivial. In this scenario, the non-nested multigrid method could be exploited in order to handle the burden of hierarchy generation, allowing some flexibility on the choice of the levels. We present a parallel, matrix-free, implementation of the non-nested multigrid method for continuous Lagrange finite elements, where each level may consist of independently partitioned triangulations. Our algorithm has been added to the multigrid framework of the C++ finite-element library deal.II. Several 2D and 3D numerical experiments are presented, ranging from Poisson problems to linear elasticity. We test the robustness and performance of the proposed implementation with different polynomial degrees and geometries.

Figures (21)

• Figure 1: Two overlapping cells coming from consecutive levels. Green dots: DoFs associated to a $\mathcal{Q}^1$ element on the coarser cell $K$. Blue squares and red stars: DoFs associated to a $\mathcal{Q}^1$ element on the finer cell $T$. Red stars correspond to the DoFs $(\bm{p}_2,\bm{p}_4)$ falling inside $K \in \mathcal{T}_l$, while blue squares are the ones that are falling outside. Each $\bm{q}_i$ will evaluate on $(\bm{p}_2,\bm{p}_4)$ only.
• Figure 2: Schematic illustration of $hp$-multigrid scheme for a $\mathcal{Q}^3$ element. Support points corresponding to continuous Lagrangian elements are represented with white dots. Left: Classical nested setting. Right: Non-nested $hp$ variant where the hierarchy of levels is non-matching. Notice how the $\mathtt{CoarseGridSolver}$ is polynomial multigrid (p-MG).
• Figure 3: UML diagram of transfer operators available in the $\mathtt{MGTransferGlobalCoarsening}$ framework in deal.II. The new abstract class delegates the implementation of the intergrid transfers to the derived class $\mathtt{MGTwoLevelTransfer}$ (used in case of nested meshes) or to the new $\mathtt{MGTwoLevelTransferNonNested}$ in case of a non-nested multigrid method. Each two-level transfer object is specific to consecutive levels $l$ and $l+1$.
• Figure 4: Coupling between processes for two overlapped and distributed triangulations (each color represents a different rank). (a) The two partitioned triangulations. The cube is displayed with a wireframe view in order to show the inner ball. (b) Clip view to highlight the geometric intersection of mesh elements belonging to different processes. Notice that grids are discretizing two different geometries only for the purpose of showing the issue. In practice, they will both discretize the same domain $\Omega$.
• Figure 5: (a) Gauss-Lobatto points for a quadrature rule of order $p=4$ on a cell $T \in \mathcal{T}_{l+1}$. (b) Evaluation points seen from the coarser cell $K \in \mathcal{T}_l$ (red stars) do not have a tensor-product structure.

Theorems & Definitions (1)

• Definition 5.1: Vertical communication efficiency for non-matching levels