Monolithic Multigrid Preconditioners for High-Order Discretizations of Stokes Equations

TL;DR

This work develops and evaluates a monolithic phMG framework for high-order Stokes discretizations using Taylor-Hood and Scott-Vogelius elements, combining $p$-coarsening with geometric $h$-multigrid to reduce memory and computation. For Taylor-Hood, phMG delivers substantial speedups over $h$MG at higher discretization orders and on unstructured domains, driven by cheaper coarse-grid relaxations and smaller coarse-grid stencils; for Scott-Vogelius, phMG achieves competitive solve times but incurs higher setup costs due to larger mixed-field patches, with full-block factorization (FBF) variants often outperforming phMG in setup efficiency. Across 2D and 3D problems, the results highlight a trade-off: phMG triangles at high $k$ gain efficiency, while FBF-based strategies provide robust setup performance for SV, especially in 3D. The findings guide solver selection for high-order Stokes simulations and motivate future work on patch-assembly optimization, more gradual $p$-coarsening, and extensions to broader coupled systems.

Abstract

This work introduces and assesses the efficiency of a monolithic $ph$MG multigrid framework designed for high-order discretizations of stationary Stokes systems using Taylor-Hood and Scott-Vogelius elements. The proposed approach integrates coarsening in both approximation order ($p$) and mesh resolution ($h$), to address the computational and memory efficiency challenges that are often encountered in conventional high-order numerical simulations. Our numerical results reveal that $ph$MG offers significant improvements over traditional spatial-coarsening-only multigrid ($h$MG) techniques for problems discretized with Taylor-Hood elements across a variety of problem sizes and discretization orders. In particular, the $ph$MG method exhibits superior performance in reducing setup and solve times, particularly when dealing with higher discretization orders and unstructured problem domains. For Scott-Vogelius discretizations, while monolithic $ph$MG delivers low iteration counts and competitive solve phase timings, it exhibits a discernibly slower setup phase when compared to a multilevel (non-monolithic) full-block-factorization (FBF) preconditioner where $ph$MG is employed only for the velocity unknowns. This is primarily due to the setup costs of the larger mixed-field relaxation patches with monolithic $ph$MG versus the patch setup costs with a single unknown type for FBF.

Figures (10)

• Figure 1: Sequential Mesh Refinement Processes: The initial image (left) illustrates the triangulation of a triangular domain. This is followed by a quadrisection refinement ($\mathcal{R}_u$), resulting in the intermediate mesh (middle). The final image (right) shows the result of a barycentric mesh refinement ($\mathcal{R}_b$), employing the Alfeld split, further refining the intermediate mesh.
• Figure 1: Illustration of the $h$MG cycle, showcasing the rediscretization process on coarser meshes with the same discretization order. The relaxation stages are represented by gray dots at each level of the V-cycle. Black lines signify the grid-transfer points, while the red dot at the base of the V-cycle indicates an exact coarse-grid solve.
• Figure 1: Overview of the high-order preconditioners used for solving \ref{['eq:saddle']}.
• Figure 2: Illustration of $ph$MG V-cycles applied to $\pmb{\mathbb{P}}_{k}/\mathbb{P}_{k-1}$ discretizations, deviating from \ref{['fig:hmg_cycle']} only in the modal restriction and interpolation represented by dashed lines. These cycles are also compatible with Scott-Vogelius discretizations ($\pmb{\mathbb{P}}_{k}/\mathbb{P}_{k-1}^{disc}$) on $\ell=0$. Figure (a) depicts a Direct $ph$MG cycle where the $\pmb{\mathbb{P}}_{k}/\mathbb{P}_{k-1}$ discretization directly transitions to $\pmb{\mathbb{P}}_{2}/\mathbb{P}_{1}$. Figure (b) shows a Gradual $ph$MG cycle, where $\pmb{\mathbb{P}}_{k}/\mathbb{P}_{k-1}$ discretization first steps down to an intermediate $\pmb{\mathbb{P}}_{\hat{k}}/\mathbb{P}_{\hat{k}-1}$ with $k > \hat{k} > 2$, before reducing the approximation order to $\pmb{\mathbb{P}}_{2}/\mathbb{P}_{1}$.
• Figure 2: $p$MG coarsening schedule for velocity field. For $\pmb{ \mathbb{P}}_{k}$ with $k\in[2,5]$, coarsening maps directly to $\pmb{ \mathbb{P}}_{2}$. For $k\in[6,7]$, we first coarsened to $\pmb{ \mathbb{P}}_{4}$ and then to $\pmb{ \mathbb{P}}_{2}$. For $k\in[8,10]$, we initially coarsen to $\pmb{ \mathbb{P}}_{5}$, followed by $\pmb{ \mathbb{P}}_{2}$.

Theorems & Definitions (4)

• Remark 3.1: Comparison of Vertex-star Patch Setup Approaches
• Remark 4.1: Bias in the Convergence and Timings Results
• Remark 4.2: $ph$MG Relaxation at Low Discretization Order
• Remark 4.3: Setup Costs