Cut-Cell Geometric Multigrid Method

• Cut-cell geometric multigrid is a solver for PDEs that embeds the physical domain in a regular grid, enabling efficient treatment of irregular boundary intersections.
• The method organizes hierarchical grids with hp-refinement and adaptive meshing to accurately capture cut-cell geometries and ensure effective operator transfer.
• Robust smoothing techniques such as additive Schwarz and Vanka-type smoothers mitigate near-singular modes, achieving mesh-independent convergence and optimal computational costs.

The cut-cell geometric multigrid method is a class of solvers for partial differential equations (PDEs) designed for unfitted, immersed, or embedded boundary discretizations. Cut-cell methods replace the need for boundary-conforming meshes by embedding the physical domain in a regular computational grid (Cartesian or simplicial) and determining domain geometry via analytic or level-set functions. Cut cells—elements of the computational grid intersected by the boundary—often induce near-singular, ill-conditioned submatrices in the discrete operator. The geometric multigrid framework, when properly tailored to the cut-cell structure, ensures both optimal complexity, mesh-size robustness, and resilience to the pathological conditioning introduced by small or irregular cut cells. This article summarizes the core mathematical principles, hierarchical grid organization, smoothing techniques, restriction/prolongation strategies, scalability, and numerical results for state-of-the-art cut-cell geometric multigrid algorithms, emphasizing approaches for finite cell (FEM), ghost-FEM, discontinuous Galerkin (DG), and high-order finite-volume discretizations.

1. Mathematical Foundation and Cut-Cell Discretization

Cut-cell solvers are formulated on an embedding computational domain Ω_emb ⊃ Ω_phys, with the physical boundary ∂Ω_phys specified by analytic C^1 curves or implicitly via a level-set function Φ(x). The cut-cell mesh results from intersecting the regular grid (structured or AMR) with Ω_phys, classifying cells as internal, external, or cut. Cut elements are characterized by K ∈ Ω_emb with measure(K ∩ Ω_phys) ∈ (0, |K|) (Jomo et al., 2020, Zhu et al., 1 Apr 2025, Qian et al., 2024).

Discrete weak forms leverage various strategies:

• Finite Cell Method (FCM): Embedded domain with penalization weighting α(x)=1 for x∈Ω_phys, α(x)=ε≪1 for x∈Ω_emb∖Ω_phys (enforcing near-zero fictitious region solution), with bases of hierarchical integrated Legendre polynomials (Jomo et al., 2020).
• Ghost Finite Element Method: Variational formulation with Dirichlet/Neumann conditions imposed via Nitsche's method and local penalties λ_K, yielding symmetric positive-definite operators even on cut cells (Dilip et al., 8 May 2025).
• CutFEM and Extended DG: Polynomial spaces on arbitrary cut subcells, discontinuous across interfaces, symmetric interior penalty bilinear forms with explicit stabilization terms proportional to the local interface geometry (Cui et al., 15 Aug 2025, Kummer et al., 2020).

High-order spatial discretization for cut cells is achieved by poised lattice generation (PLG) algorithms, assembling finite-volume/fitted stencils via weighted least squares on minimal polynomial interpolation lattices (see Section 2) (Zhu et al., 1 Apr 2025, Liu et al., 6 Jan 2026, Qian et al., 2024). For embedded boundaries, cut cells with very small measure are merged into neighbors above a prescribed threshold ε to avoid degeneracy (Qian et al., 2024, Zhu et al., 1 Apr 2025, Liu et al., 6 Jan 2026).

2. Multigrid Hierarchy: Grid Organization and Space Construction

Geometric multigrid for cut-cell problems relies on hierarchically nested meshes and associated FE/FV spaces:

• Uniform and multi-level hp-refined grids: Hierarchies consist of successive p-coarsening (polynomial degree reduction) and h-coarsening (overlay mesh or AMR refinement depth). Each level ℓ corresponds to a FE space 𝒱_ℓ, nested via hierarchical Legendre bases (Jomo et al., 2020).
• AMR and octree/quadtree frameworks: Forest-of-trees structures enforce 2:1 balance and permit adaptive refinement, crucial near ∂Ω_phys (Saberi et al., 2024, Saberi et al., 2021).
• Explicit block structure: For FV/PLG discretizations, the operator decomposes into blocks for regular cells (SFV) and boundary/cut cells (PLG), e.g., block matrix

$$\begin{pmatrix} L_{11} & L_{12} \ L_{21} & L_{22} \end{pmatrix}$$

where the interface block L_{22} is treated by direct or sparse LU (Liu et al., 6 Jan 2026, Zhu et al., 1 Apr 2025, Qian et al., 2024).

Meshes at each level are generated by coarsening (merging of fine sibling cells into parent, preserving cut-cell status), with restriction/prolongation operators mapping between nested subspaces (in FCM via trivial selection masks; in FV/FEM as weighted injections and interpolations) (Jomo et al., 2020, Dilip et al., 8 May 2025).

3. Algebraic Transfer Operators: Restriction and Prolongation

Restriction (R) and prolongation (R^T) operators in cut-cell geometric multigrid are governed by the nested nature of discretization spaces:

• For hierarchical bases (Legendre, ghost-FE) with strict nesting, transfer matrices are identity-plus-zero block selections of degrees of freedom (DOFs), realized algebraically as dropping or reincorporating DOFs (Jomo et al., 2020, Dilip et al., 8 May 2025). No explicit storage is required.
• In FV/PLG schemes, restriction is volume-weighted injection for regular cells and zero-fill for boundaries/cut cells; prolongation is constant or piecewise injection from coarse to fine grid (Liu et al., 6 Jan 2026, Qian et al., 2024, Zhu et al., 1 Apr 2025).
• DG/XDG methods use orthonormalization-enhanced prolongation/restriction matrices computed from the basis functions; Galerkin product yields coarse-grid operators (Kummer et al., 2020).
• All transfer operators are constructed to respect the variational structure and penalty weights of the cut region.

4. Smoother Design and Cut-Cell Mode Elimination

The critical challenge of cut-cell multigrid methods is the robust damping of near-singular error modes induced by small or irregular cut elements. Standard pointwise smoothers (Jacobi, Gauss–Seidel) fail to address these modes. Consequently:

• Additive Schwarz (AS) and Vanka-type smoothers: Elementwise or patchwise blocks are extracted, each containing all functions overlapping the element or patch/node. The local block matrix is inverted (or solved via direct LU), and corrections are injected with relaxation parameter ω based on maximum block overlaps (Jomo et al., 2020, Saberi et al., 2024, Saberi et al., 2021). Restricted Additive Vanka (RAV) smoothers exploit exact parallelization, with patches assigned to unique MPI ranks (Saberi et al., 2024).
• Multiplicative vertex-patch smoothers: For CutFEM, vertex-centered patches are solved via direct factorization on each patch; repeated sweeps over cut patches enhance boundary mode damping (Cui et al., 15 Aug 2025).
• Blockwise Jacobi–LU smoothing: In PLG/FV, regular cell blocks are damped by Jacobi, whereas interface/cut-cell blocks are solved exactly by efficiently banded LU (Liu et al., 6 Jan 2026, Qian et al., 2024, Zhu et al., 1 Apr 2025).
• DG p-multigrid block solvers: Schwarz smoothers are combined with local p-level splitting (low/high-order modes) (Kummer et al., 2020).

Smoother selection is pivotal for iteration count independence from mesh-size, cut-cell configuration, polynomial degree, and parallel process count (Jomo et al., 2020, Saberi et al., 2024, Saberi et al., 2021).

5. Coarse-Grid Solves and Optimal Complexity

On the coarsest grid, operators admit direct LU, Cholesky, or sparse factorization:

• FV/PLG interface blocks L_{22} and their nested dissection reordering; sparsity and separator structure yield O(n^{3/2}) LU in 3D, O(h^{-3}) complexity per V-cycle (Qian et al., 2024).
• For high-order FV/DG discretization, coarse-grid solvers preserve the mesh-invariant convergence properties and enable O(N) or O(h^{-2}) arithmetic complexity in practice (Liu et al., 6 Jan 2026, Zhu et al., 1 Apr 2025).
• The hierarchical FCM approach achieves O(N) V-cycle cost per iteration for uniform grids or O(N/p) on hp-adapted zones (Jomo et al., 2020).
• In large-scale flow problems, additive patch smoothing and memory-efficient cache policies preserve scalability up to hundreds of millions DOFs (Saberi et al., 2024).

6. Robustness, Scalability, and Numerical Performance

Cut-cell geometric multigrid methods deliver robust residual reduction and optimal convergence factors (e.g., contraction numbers ρ≃0.1–0.2 per V-cycle), as established in large-scale numerical studies and theoretical analysis:

• Iteration counts: Remain bounded and mesh-independent up to multi-billion DOFs (Jomo et al., 2020, Saberi et al., 2024), typically ≤10–20 iterations to reduce to solver tolerance.
• Accuracy: Fourth-order solution errors in L^p norms are confirmed on irregular domains with arbitrary topology and even C¹ discontinuities in the boundary (Liu et al., 6 Jan 2026, Zhu et al., 1 Apr 2025, Qian et al., 2024). FV/PLG truncation gives O(h³) local errors but achieves O(h⁴) global accuracy.
• Efficiency: Multigrid solve times scale as O(N) or O(h^{-2}), with per-V-cycle cost dominated by interior smoothing and relatively cheap banded LU on interface blocks. Weak and strong scaling studies demonstrated near-ideal parallel efficiency up to 98,304 cores (Jomo et al., 2020, Saberi et al., 2024).
• GPU Implementation: In CutFEM, GPU-optimized kernels deliver ≈1×10^10 DOF/s for fitted domains, and cut-mesh solvers achieve comparable throughput as grid sizes grow (Cui et al., 15 Aug 2025).
• DG/XDG methods: Multigrid struggles with ill-conditioning unless cell agglomeration and residual minimization are incorporated; otherwise, iteration counts and errors become geometry-dependent (Kummer et al., 2020).
• Practical benchmarks: Popcorn (complex 3D interfaces), aluminum rod (3D elasticity), cylinder-in-channel (2D flow), rotated squares, flower domains, and complex "panda" topologies confirm multigrid robustness in analytically challenging settings (Jomo et al., 2020, Zhu et al., 1 Apr 2025, Liu et al., 6 Jan 2026, Qian et al., 2024, Saberi et al., 2024).

7. Algorithmic Summary and Implementation Highlights

The standard algebraic V-cycle for cut-cell geometric multigrid is recursive and encompasses the following steps (Jomo et al., 2020, Liu et al., 6 Jan 2026, Zhu et al., 1 Apr 2025, Saberi et al., 2024):

• For all levels ℓ > 0: apply n_s pre-smoothing steps with the prescribed Schwarz/block smoother, restrict residual via trivial mask or injection operator, recursively call coarse-level V-cycle (possibly direct solve at ℓ=0), prolongate correction (inject or mask), apply n_s post-smoothing steps.
• Smoother implementations are memory-efficient (cache_none, cache_matrix, cache_inverse), highly parallelizable (replication per patch, negligible halo exchange), and can be tuned for target architectures (CPU, GPU) (Saberi et al., 2024, Cui et al., 15 Aug 2025).
• Coarse-grid builds enforce 2:1 refinement balance, merging of extreme cut-cells, and exact representation of merged volumes in operator assembly.
• Interface blocks are ordered and banded for efficient sparse direct solves; restriction/prolongation operations remain O(N) in complexity.

The framework supports extension to arbitrary boundary conditions (Dirichlet, Neumann, Robin), general elliptic operators (variable coefficients, cross-derivatives), and all forms of high-order discretizations (Liu et al., 6 Jan 2026, Zhu et al., 1 Apr 2025, Qian et al., 2024).

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References:

• (Jomo et al., 2020) Hierarchical multigrid approaches for the finite cell method on uniform and multi-level hp-refined grids
• (Cui et al., 15 Aug 2025) A multigrid method for CutFEM and its implementation on GPU
• (Qian et al., 2024) A fourth-order, multigrid cut-cell method for solving Poisson's equation in three-dimensional irregular domains
• (Saberi et al., 2024) A restricted additive smoother for finite cell flow problems
• (Saberi et al., 2021) Adaptive geometric multigrid for the mixed finite cell formulation of Stokes and Navier-Stokes equations
• (Dilip et al., 8 May 2025) Multigrid methods for the ghost finite element approximation of elliptic problems
• (Liu et al., 6 Jan 2026) A Fourth-Order Cut-cell Multigrid Method for Generic Elliptic Equations on Arbitrary Domains
• (Kummer et al., 2020) BoSSS: a package for multigrid extended discontinuous Galerkin methods
• (Zhu et al., 1 Apr 2025) A Fast Fourth-Order Cut Cell Method for Solving Elliptic Equations in Two-Dimensional Irregular Domains