Geometric Multigrid solvers for Hybrid High-Order methods on polytopal meshes

1. Problem Statement

The paper addresses the challenge of developing optimal geometric multigrid (GMG) solvers for Hybrid High-Order (HHO) discretizations on arbitrary polytopal meshes in both 2D and 3D.

• Context: HHO methods are non-conforming hybrid methods that use both cell-based and face-based unknowns. They offer high-order accuracy and flexibility on general meshes (including polytopes) and allow for static condensation to eliminate cell unknowns, reducing the global system size to only face unknowns (skeletal unknowns).
• The Challenge: Existing GMG solvers for hybrid methods (like HDG or HHO) typically rely on nested mesh hierarchies with planar faces. When using agglomeration (merging fine elements into coarse polytopal elements) to create multigrid levels, the resulting coarse interfaces are often non-planar (curved or composed of multiple planar segments).
  • Standard approaches fail here because they either require complex geometric rediscretization to force planar faces (limiting applicability to 2D) or suffer from poor convergence because the number of face unknowns does not decrease optimally on coarse levels.
  • There was no known GMG solver capable of handling arbitrary polytopal agglomeration hierarchies with robust convergence in 3D.

2. Methodology

The authors propose a novel GMG framework specifically designed to handle non-planar interfaces arising from agglomeration without requiring geometric rediscretization.

A. Modified Skeleton Spaces

The core innovation is the use of modified skeleton spaces for the face unknowns on coarse levels.

• Instead of using standard discontinuous polynomials on every sub-face of a coarse interface (which leads to redundant degrees of freedom), the authors define the polynomial space on an interface $F$ as:
  $$P_k(F) = P_0(F) + \nabla P_{k+1}(\Omega) \cdot n_F$$
• Key Benefit: This space has a maximum dimension independent of the number of planar faces composing the interface. It naturally accommodates non-planar interfaces while reducing the number of degrees of freedom (DOFs) on coarse levels, ensuring efficient coarsening.

B. Intergrid Transfer Operators (Prolongation)

Two specific prolongation operators are defined to map coarse-level skeletal functions to fine-level functions:

1. $I^U_\ell$: Based on the local inverse static condensation operator ($U_{\ell-1}$) followed by an averaging projection onto fine interfaces.
2. $I^R_\ell$: Based on the local reconstruction operator ($R_{\ell-1}$) followed by the averaging projection.

• These operators reconstruct higher-order polynomials on coarse cells and project them onto the fine interfaces, handling the geometry of non-planar interfaces via weighted $L^2$-orthogonal projections.

C. Smoothers

The authors employ star-patch smoothers (block-Jacobi relaxation) acting on the skeletal unknowns:

• Face-star patch: Aggregates DOFs on a specific interface.
• Vertex-star patch: Aggregates DOFs on all interfaces connected to a vertex (and edge-star in 3D).
• These are implemented as parallel subspace correction methods, solving local problems on patches of elements surrounding topological entities.

D. Convergence Analysis

The paper provides a rigorous convergence analysis based on the framework of space decomposition methods (Xu's theory).

• Assumptions: The authors prove that the proposed prolongation operators satisfy necessary stability and approximation properties (boundedness, consistency error bounds, and regularity assumptions).
• Result: They prove that the resulting V-cycle multigrid method exhibits robust convergence that is independent of:
  • The mesh size ($h$).
  • The number of multigrid levels ($L$).
  • The polynomial order ($k$) (observed numerically, though the theory focuses on $h$ and $L$).

3. Key Contributions

1. First Optimal GMG for Arbitrary Polytopal Agglomeration: This is the first GMG solver for hybrid discretizations (HHO) that handles arbitrary polytopal agglomeration hierarchies in both 2D and 3D without requiring planar coarse faces.
2. Modified Interface Spaces: Introduction of a specific polynomial space for coarse interfaces that reduces DOFs and maintains efficiency during agglomeration, inspired by curved domain HHO methods.
3. Rigorous Analysis: A complete theoretical proof of robust convergence for the proposed geometric multigrid method, extending the analysis to non-nested, non-planar hierarchies.
4. Extensibility: The methodology is shown to be applicable to other hybrid methods like Hybridizable Discontinuous Galerkin (HDG) and Weak Galerkin (WG) due to their equivalence with HHO.

4. Numerical Results

The authors validated the method using the Julia-based Gridap ecosystem on various mesh families:

• Mesh Types:
  • 2D: Cartesian, Rep-tile (L-shaped polygons with non-straight interfaces), and Agglomerated Voronoi meshes.
  • 3D: Cartesian meshes.
• Performance Metrics:
  • Iteration Counts: The number of FGMRES iterations required to reduce the residual by $10^{-8}$ remained constant regardless of the number of levels ($L=2$ to $6$) and the finest mesh size ($n_c$).
  • Robustness: The solver demonstrated $h$-independence and level-independence.
  • Comparison:
    • Vertex-star patches consistently outperformed face-star patches in terms of iteration counts (though at a higher computational cost per iteration).
    • Both prolongation operators ($I^U$ and $I^R$) showed similar performance.
    • The method remained stable for polynomial orders $k=0, 1, 2$.

5. Significance

• Bridging the Gap: This work bridges the gap between the flexibility of polytopal methods (which excel at mesh agglomeration for complex geometries) and the efficiency of geometric multigrid solvers.
• Scalability: By enabling optimal multigrid solvers on agglomerated meshes, this approach makes HHO methods significantly more scalable for large-scale 3D simulations, particularly in applications involving complex geometries where traditional conforming mesh generation is difficult.
• Foundation for Future Work: The framework provides a theoretical and practical foundation for extending multigrid solvers to other hybrid discretizations and complex physical problems (e.g., elasticity, fluid dynamics) on general polytopal meshes.