V-cycle convergence for non-inherited CutFEM forms

Establish a mesh- and cut-uniform V-cycle convergence bound for geometric multigrid with vertex-patch smoothing applied to CutFEM discretizations whose level forms are non-inherited, using BPX-type machinery.

Background

The paper establishes a two-level convergence result and a conditional W-cycle result under the smallness assumption that the two-level rate is at most one quarter. The numerical experiments use the stronger V-cycle, but the analysis does not cover it because rediscretized, non-inherited level forms require a different multilevel framework. The authors explicitly identify a V-cycle bound as unresolved.

References

The numerical study intentionally uses the stronger V-cycle, whereas a V-cycle bound for the non-inherited forms requires BPX-type machinery.

— A Multigrid Method for CutFEM and its Convergence  (2609.04067 - Wichrowski, 3 Sep 2026) in Introduction; Section 4, item (b) of “The scope and limitations of the theorem”; Conclusion

The analysis of the proposed multigrid methods is not addressed here, and is left for future work. Due to the non-nested nature of the hierarchies, such analysis requires a substantially different approach from the body-fitted case and thus warrants its own paper.

— Geometric multigrid preconditioners for unfitted discretisations of the de Rham complex  (2610.22865 - Badia et al., 16 Oct 2026) in Section 1, Introduction

The analysis of the V-cycle is, however, left open.

— Geometric multigrid preconditioners for unfitted discretisations of the de Rham complex  (2610.22865 - Badia et al., 16 Oct 2026) in Section 1, Introduction