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A Multigrid Method for CutFEM and its Convergence

Michal Wichrowski

arXiv:2609.04067v1math.NA

TL;DR

CutFEM multigrid lacks a convergence theory that directly handles non-inherited level forms and the mismatch between physical and active domains. The paper analyzes geometric multigrid with vertex-patch smoothers using discrete extension stability, proving cut- and mesh-uniform two-level bounds and conditional W-cycle bounds. The theory tracks degree and ghost-penalty effects, while the stronger V-cycle experiments remain outside the proven scope.

  • Problem

    The paper addresses the absence of convergence theory for CutFEM geometric multigrid with vertex-patch smoothing when level forms are non-inherited and physical and active domains differ.

  • Method

    The analysis uses stabilized energy-norm transfer, the discrete extension property, vertex-patch decompositions, and explicit treatment of boundary-strip ghost-penalty coupling.

  • Results

    The two-level contraction is uniform in mesh size and cut geometry, and W-cycle bounds hold under an additional smallness condition on the two-level rate.

  • Takeaways & Limitations

    The theory identifies boundary-strip coupling and ghost-penalty design as central to degree-dependent convergence, while lowering the ghost penalty improves iteration counts in the reported experiments.

  • Takeaways & Limitations

    The numerical experiments use a stronger V-cycle for which no convergence bound is claimed, and the sharp p-dependence of the strip rate remains open.

Abstract

from arXiv · show

We develop a convergence theory for geometric multigrid with vertex-patch smoothers applied to cut finite element discretizations of the Poisson problem. The framework addresses non-inherited level forms and the mismatch between the physical and active domains. Using the discrete extension property, we prove two-level convergence bounds uniform in the mesh size and the cut geometry, and W-cycle bounds under an additional smallness assumption on the two-level rate. The numerical experiments intentionally use the stronger V-cycle, for which no convergence bound is claimed here. The convergence constants degrade with the degree $p$. Lowering the ghost penalty improves iteration counts. An aligned two-cell model exhibits a semidefiniteness threshold of order $p^{-2}$, whereas the visibility scale of a degree-$p$ cut mode decreases exponentially. Experiments at the model threshold reduce the iteration counts, but do not establish an assembled-operator threshold.

1 Introduction

The paper develops geometric multigrid convergence theory for CutFEM Poisson discretizations, addressing non-inherited forms, active-domain mismatch, and boundary-strip coupling. It proves cut- and mesh-uniform two-level convergence, conditional W-cycle convergence, and analyzes degree and ghost-penalty effects.

  • Motivation: CutFEM embeds complex or evolving domains in a fixed Cartesian background mesh, avoiding body-fitted remeshing while requiring stabilization for poorly cut cells.Ghost penalties let poorly cut cells borrow stability from neighboring cells with substantial physical support.
  • Problem and method: The analysis targets geometric multigrid with vertex-patch smoothing for Poisson CutFEM discretizations on nested Cartesian active meshes.The active domain contains the physical domain, and the level forms combine Poisson, Nitsche, and ghost-penalty terms.
  • Problem and method: Non-inherited ghost penalties and boundary-strip coupling prevent standard same-color patches from being exactly orthogonal or simultaneously multiplicative.The coupling is associated with the observed polynomial-degree dependence and enters the convergence analysis.
  • Convergence theory: The two-level contraction is uniform in mesh size and cut geometry, while the W-cycle inherits a bound under the additional smallness condition q ≤ 1/4.The proof uses stable energy-norm injection and a cut-independent vertex-patch splitting based on the discrete extension property.
  • Parallel smoothing: A semi-multiplicative smoother accounts for thin-strip coupling with 2d coloring, while stride-3 coloring restores exact orthogonality on the strip.The numerical experiments use the undamped semi-multiplicative variant, beyond the stated corollary.
  • Degree dependence: The convergence constants degrade with p, and the analysis localizes this dependence to the boundary-strip contraction σℓ(p).Within the derivative-jump family, no weight choice makes the assembled bounds polynomial in p.

2 Problem setting and assumptions

The problem setting uses tensor-product Qp spaces on active Cartesian meshes under geometric and CutFEM stability assumptions. These assumptions provide cut-independent coercivity, boundedness, and discrete extension control while tracking dependence on p and the global ghost weight.

  • Framework: The analysis concerns a vertex-patch smoother for CutFEM and fixes the geometric and stability framework used in its convergence proof.Constants are independent of level, mesh size, and boundary position, but may depend on p, geometry, and the Nitsche parameter.
  • Discrete setting: The meshes are Cartesian, with axis-parallel cells and faces, and the local finite element space is the tensor-product space Qp.Normal derivatives are coordinate derivatives, enabling one-dimensional polynomial inequalities in the normal direction.
  • Geometric assumptions: The boundary is C2 with positive reach and satisfies cut-cell resolution conditions that exclude mesh vertices and control how Γ intersects cells and faces.These assumptions support the local geometric constructions used in the analysis.
  • Stability assumptions: CutFEM stability assumes a sufficiently large cut-independent Nitsche parameter and gives coercivity, boundedness, and a discrete-extension estimate with cut-independent constants.These properties yield equivalence between the stabilized norm and H1-type norms on the active domain.
  • Ghost penalty: The ghost penalty uses standard derivative-jump weights h^(2k−1)/(k!)^2 multiplied by a single global factor γ ∈ (0, 1].The analysis tracks how both p and γ enter the resulting constants.

3 Stability of the intergrid transfer

The intergrid transfer remains stable despite non-inherited level forms and changing ghost-face sets. The transfer itself is p-uniform in mesh-dependent norms, while stable decomposition introduces the dominant tracked degree dependence through polynomial trace and Markov estimates.

  • Prolongation stability: The level forms are non-inherited because ghost penalties use different face sets and weights on each level, but injection remains stable in the stabilized energy norm.The active-domain nesting makes the coarse function well defined on the fine active domain.
  • Ghost-face comparison: Fine ghost faces map to coarse ghost faces through coarse parents, allowing fine ghost contributions to be bounded by coarse-level ghost penalties.The argument relies on nested Cartesian cells and the classification of fine faces relative to coarse faces.
  • Prolongation stability: The prolongated coarse ghost energy is controlled on corresponding coarse faces without trace, Markov, or inverse inequalities.The mesh-norm estimate is independent of polynomial degree; energy-norm dependence enters only through norm equivalence.
  • Stable decomposition: The stable decomposition instead bounds single-level ghost energy by the H1-seminorm, introducing p-dependent trace and iterated Markov factors.Tensor-product structure permits one-dimensional inequalities in the normal coordinate, with finite overlap across ghost-face neighborhoods.
  • Degree dependence: The tracked p-growth is dominated by a superexponential factor from the truncated Markov-based estimate, although the bound is loose because the full-series dominant index lies beyond k = p.The transfer contributes no p-growth in mesh-dependent norms, while the decomposition constant remains the main source of tracked dependence.

4 Vertex-patch smoother and method implementation

The smoother uses admissible vertex patches designed to cover every degree of freedom, with classical and full-residual local spaces differing only in which confined degrees of freedom they update. Numerical comparisons show that enlarging the local spaces reduces iterations, but does not remove their growth with polynomial degree.

  • Two local-space variants: The classical construction imposes homogeneous Dirichlet conditions on the full patch boundary, whereas the full-residual space also updates degrees of freedom whose complete active-cell support is confined to the patch.The full-residual variant avoids imposing an artificial Dirichlet condition on confined active-domain boundary degrees of freedom.
  • Two local-space variants: Both variants use the same stabilized operator, global residual, penalty weights, patch centers, and sweep; only the local patch-space restriction differs.This isolates the effect of the local-space definition in the comparison.
  • Patch construction: Admissible patches include interior and selected exterior vertices so every degree of freedom is updated by at least one local space.Exterior centers are necessary for covering degrees of freedom outside the physical domain but inside the active domain.
  • Numerical comparison: Updating all confined degrees of freedom lowers GMRES iteration counts at every measured degree by a factor of 1.6–1.8 from Q3 onward.The comparison uses L = 5, γ = 10^-1, and the same assembled problem for both variants.
  • Numerical comparison: At γ = 10^-1, iteration counts still grow by an order of magnitude between Q3 and Q6, and neither variant converges within 200 iterations at Q7.Thus, overconstraint explains only a constant factor of the degree dependence, not its growth.

5 Cut-robust two-level vertex-patch Schwarz theory

The analysis establishes cut-uniform two-level convergence for vertex-patch Schwarz methods by connecting the physical-domain energy to the active-domain finite element space through discrete extension estimates. It also identifies explicit degree dependence and limits the stronger cycle claims to W-cycles under an additional rate condition.

  • Working hypotheses: The extension property connects the physical-domain energy to norms on the active domain, enabling cut-independent estimates for the Schwarz analysis.The active mesh covers the enlarged domain Ωℓ, while the energy is defined on Ω and Γ.
  • Working hypotheses: The adjacent-interval extrapolation constant CE(q) is the smallest uniform polynomial-extension factor across a shared face and grows exponentially with q.The lower-bound construction using Chebyshev polynomials shows that this exponential growth is intrinsic to the estimate.
  • Stable decomposition: The stable decomposition has a constant independent of the mesh level and cut configuration, but it may depend on p, the global ghost weight, and the Nitsche parameter.The decomposition combines a coarse space with vertex-patch subspaces and uses exact local and coarse solves.
  • Two-level convergence: The two-level method converges uniformly in the level and interface position for every admitted cut configuration, with exact local and coarse solves.The result applies to additive and multiplicative formulations, including the stated V-cycle ordering bound.
  • Scope and limitations: The proven rate is uniform in mesh size and cut configuration, but its tracked dependence on p is superexponential and dominated by the ghost-energy factor Θ′p.The theory is presented as the first convergence proof for a vertex-patch multigrid smoother for CutFEM with ghost penalties.
  • Scope and limitations: W-cycle convergence requires a smallness assumption on the two-level rate, while the stronger V-cycle used in experiments remains outside the proven convergence theory.The experiments should therefore be interpreted as numerical evidence beyond the theorem’s scope.

6 The p-dependence of the method

The proven two-level rate is cut- and level-uniform but worsens rapidly with polynomial degree, motivating a separate analysis of boundary-strip contraction and ghost-weight choices. Experiments show that extra cut sweeps and reduced ghost weights can improve iteration counts, while the two-cell model exposes incompatible algebraic scales without certifying an assembled-operator threshold.

  • Degree dependence: The two-level estimate is uniform in level and cut configuration, but its proven dependence on p is superexponential.The leading tracked growth comes from the ghost-energy comparison constant rather than the polynomial trace, Markov, or transfer factors.
  • Numerical degree dependence: Q1 and Q2 require at most six GMRES iterations, while from Q3 onward the GMRES count roughly doubles per degree and the stationary V-cycle reaches the cap at Q5.The observed growth appears exponential over p = 3, …, 6, but cannot be separated from the onset of the theoretical bound.
  • Strip contraction: A cut-strip contraction σ(p) < 1 is uniform in level, cut position, and sweep ordering for each fixed degree, but its sharp rate remains open.The bound is derived by combining strip-local Friedrichs control with stable splitting, yet it provides neither a sharp rate nor a polynomial lower bound for 1 − σ(p).
  • Strip sweeps: Repeated cut sweeps reduce the required work, although the inferred effective rates are diagnostic rather than bounds for σℓ(p).At Q6, calibration gives σeff ≈ 0.95 and predicts about 30 sweeps; this agrees at Q4 but underestimates work at Q5–Q7.
  • Ghost-weight variation: Decreasing the global ghost weight substantially reduces iteration counts for p ≥ 3, but the smallest successful weights lie outside the range certified by the theory.GMRES nonsingularity in these runs does not establish coercivity.
  • Two-cell model: The two-cell model has a semidefiniteness threshold of order p^-2, whereas the degree-p mode visibility scale decays exponentially, making the scales incompatible as p grows.Experiments at the model threshold reduce counts, but the model is necessary rather than sufficient and does not certify the assembled operator.

7 Conclusion

The analysis establishes cut- and mesh-uniform two-level convergence and a conditional W-cycle result, while the experiments use the stronger but unanalyzed V-cycle. Degree dependence remains tied to the boundary strip, and the ghost-penalty and two-cell investigations leave important sharpness questions open.

  • Convergence theory: Two-level convergence is uniform in mesh size and interface position, while the W-cycle result requires a smallness assumption on the two-level rate.The framework also handles non-inherited forms through stable prolongation and vertex-patch decompositions.
  • Convergence theory: The experiments intentionally test the stronger V-cycle, for which no convergence bound is claimed.
  • Parallel smoothers: The ghost penalty couples same-color boundary-strip patches, so standard 2d coloring is not exactly multiplicative.A stride-3 strip coloring yields orthogonal color classes, while the proven semi-multiplicative bound uses damping.
  • Degree dependence: The degree dependence is governed by the boundary-strip contraction σℓ(p), and repeated strip sweeps can mitigate deterioration at vanishing relative refinement cost.The diagnostic sweep count underestimates the required number above Q4, and no present derivative-jump profile yields polynomial p-bounds.
  • Open questions: The two-cell analysis identifies incompatible ghost-weight scales but does not determine a threshold for the assembled operator.

A Proof of the strip contraction (Proposition 3)

The strip-sweep proof confines the relevant error to a boundary band, establishes a cut- and level-uniform decomposition into strip patches, and applies multiplicative Schwarz estimates to obtain a uniform contraction.

  • Strip localization: The strip-patch space is supported within a band of width O(h) around the interface.Each patch contains a cut cell, has diameter at most 2^d h, and therefore lies within distance C0h of Γ.
  • Strip localization: A Friedrichs inequality controls the full strip-band norm with a constant independent of level and cut configuration.The proof combines physical and fictitious parts using a normal collar, discrete extension, and ghost-penalty chain estimates.
  • Stable decomposition: The strip-patch spaces admit a stable decomposition with constants uniform in the level and cut.Nodal splitting assigns each relevant degree of freedom to a strip patch and uses the full-band estimate and norm equivalence.
  • Contraction: The multiplicative Schwarz estimate applied to the strip patches yields a contraction bound for every fixed sweep ordering.

B The two-cell threshold in the vanishing-cut limit

The two-cell reduction converts the vanishing-cut stability question into a polynomial quadratic-form problem. It yields an explicit semidefiniteness threshold when γD > p^2 and proves instability below it, while the full cut-uniform sharpness remains unresolved.

  • Model reduction: The two-cell model reduces the tensor-product problem to pairs of univariate polynomials on neighboring intervals, coupled by continuity across the interface face.Normal derivative jumps, interface traces, and ghost terms determine the reduced form.
  • Threshold definition: The threshold γ∗(κ) is the infimum of positive ghost weights making the reduced form semidefinite, and monotonicity makes admissible weights an interval.
  • Threshold result: γD > p^2 yields vanishing-cut semidefiniteness exactly when γ ≥ 1/(γD − p^2).This is the necessary and sufficient condition for the limiting form a0,γ.
  • Threshold result: γ(γD − p^2) < 1 produces an indefinite limiting form, while γD ≤ p^2 makes instability possible for every positive ghost weight.The witness construction realizes the negative quadratic-form direction and extends it to sufficiently small cut fractions.
  • Interpretation and scope: The endpoint constant p^-2 comes from the sharp polynomial evaluation bound and reduces the effective Nitsche weight from γ to γ/(1 + γp^2).The resulting lower bound for the cut-uniform threshold is supported by sampled eigenvalue computations, but equality for all cut fractions remains unproved.
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