A non-nested unstructured mesh perspective on highly parallel multilevel smoothed Schwarz preconditioner for linear parametric PDEs

TL;DR

This work tackles the challenge of scalable preconditioning for linear parametric PDEs on complex, non-nested unstructured meshes. It introduces a non-nested multilevel smoothed Schwarz preconditioner built from two core innovations: a geometry-preserving, parallel mesh coarsening strategy and a multiphysics-oriented, parallel non-nested MLS interpolation, both integrated into a V-cycle framework. The approach enables reuse of a single coarse mesh hierarchy across parameter variations and demonstrates substantial improvements in GMRES convergence and parallel efficiency, achieving scalability up to about $10^3$ processors across 2D and 3D problems, including convection–diffusion and Stokes/elastodynamics settings. The results indicate strong potential for large-scale, multiphysics simulations on complex geometries, with future work directed at theoretical analyses of the method.

Abstract

The multilevel Schwarz preconditioner is one of the most popular parallel preconditioners for enhancing convergence and improving parallel efficiency. However, its parallel implementation on arbitrary unstructured triangular/tetrahedral meshes remains challenging. The challenges mainly arise from the inability to ensure that mesh hierarchies are nested, which complicates parallelization efforts. This paper systematically investigates the non-nested unstructured case of parallel multilevel algorithms and develops a highly parallel non-nested multilevel smoothed Schwarz preconditioner. The proposed multilevel preconditioner incorporates two key techniques. The first is a new parallel coarsening algorithm that preserves the geometric features of the computational domain. The second is a corresponding parallel non-nested interpolation method designed for non-nested mesh hierarchies. This new preconditioner is applied to a broad range of linear parametric problems, benefiting from the reusability of the same coarse mesh hierarchy for problems with different parameters. Several numerical experiments validate the outstanding convergence and parallel efficiency of the proposed preconditioner, demonstrating effective scalability up to 1,000 processors.

Figures (16)

• Figure 1: 4 examples of $\mathcal{N}$: (a) Finite volume method for 2D convection-diffusion equation; (b) Finite element method $[\mathcal{P}_2]^3$ for 3D linear elasticity equations; (c) Mixed finite element method $[\mathcal{P}_2]^2$-$\mathcal{P}_1$ for 2D Stokes flow; (d) Mixed finite element method $[\mathcal{P}_2]^3$-$\mathcal{P}_1$ for 3D Stokes flow;
• Figure 2: Schematic description of V-cycle multilevel smoothed Schwarz framework. Here, $\mathcal{S}(B^{-1}_{\mathcal{M}_{i}})$ represents the smoother (both pre-smoothing and post-smoothing) based on RAS preconditioner $B^{-1}_{\mathcal{M}_{i}}$, and $\mathcal{C}(\mathcal{M}_{i})$ represents the error correction, for $i = 0, \dots, L-1$.
• Figure 3: Schematic description of the geometry-preserving mesh coarsening using $np = 2$ processors in one iteration. Yellow dashed lines represent the boundaries between two subdomains. Red and blue parts respectively represent the domains processed by the two processors with CPAFT algorithm. The white parts represent the elements that are not processed in this iteration.
• Figure 4: The 3 iterations of Algorithm \ref{['alg: coarsening']} using $np = 2$ processors. Here, yellow dashed lines represent the boundaries between two subdomains. The algorithm terminates after three iterations because all interior vertices are removed.
• Figure 5: Illustration of the splitting of $\mathcal{M}$. Here, we use Figure \ref{['multiphysics']}(c) as an example. $\mathcal{N} = (3,2,0) = (1, 1, 0) + (1, 1, 0) + (1, 0, 0)$ represents the split into two velocity components and one pressure component. Then, the unknowns on $\mathcal{M}^{1}$, $\mathcal{M}^{2}$, and $\mathcal{M}^{3}$ correspond to the three physical variables are interpolated separately.

Theorems & Definitions (5)

• Remark 1: Main difficulties
• Lemma 1
• proof
• Theorem 1
• proof