A Multigrid Method for CutFEM and its Convergence
Abstract
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.
Related
- Data-driven accelerated multigrid method for Riesz space fractional diffusion equations
- A Parallel Algebraic Multigrid Preconditioned Iterative Method Based on an MPI Process Grouping Strategy for Asynchronous Concurrent Execution of the Setup and Solve Phases
- Hybrid CutFEM-IFEM model for embedded and immersed interfaces with triply-cut cells for the 2D Poisson problem
- The geometric convergence of a parallel domain decomposition method on manifolds
- Sharp convergence rate of the estimation error of LASSO via the Feature Space Decomposition method
BibTeX
@misc{wichrowski2026multigrid,
title = {{A Multigrid Method for CutFEM and its Convergence}},
author = {Michal Wichrowski},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.04067},
eprint = {2609.04067},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.04067},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - A Multigrid Method for CutFEM and its Convergence
AU - Michal Wichrowski
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.04067
UR - https://arxiv.org/abs/2609.04067
AB - 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.
ER -