A Pressure-Free Virtual Element Method for the Surface Stokes Problem
Abstract
We develop a pressure-free virtual element method for the surface Stokes problem on polygonal approximations of closed surfaces of arbitrary genus. The method extends the planar VEM Stokes complex to discrete surfaces, yielding a nonconforming complex with an exactly tangential, H(div)-conforming velocity space, commuting interpolation, and the correct discrete cohomology. To obtain the weak consistency needed for optimal convergence, we introduce an edge correction that preserves the complex without adding degrees of freedom; the resulting construction reduces to the original complex in the planar setting. The virtual formulation requires no penalties, and a local divergence-preserving reconstruction provides a computable pressure-robust load. We also construct explicit macroelement representatives whose energy and load induce an alternative formulation on the VEM space, algebraically equivalent to a direct macroelement Galerkin method. The original virtual formulation and the macroelement method are stable and attain optimal first-order convergence in the broken H1 norm and second-order convergence in the L2 norm for the velocity. Numerical experiments illustrate these convergence rates, the role of the edge correction, and pressure robustness.
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BibTeX
@misc{hu2026pressurefree,
title = {{A Pressure-Free Virtual Element Method for the Surface Stokes Problem}},
author = {Jun Hu and Shengyang Xu and Hao Zhou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.39996},
eprint = {2609.39996},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39996},
}
RIS
TY - UNPB
TI - A Pressure-Free Virtual Element Method for the Surface Stokes Problem
AU - Jun Hu
AU - Shengyang Xu
AU - Hao Zhou
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2609.39996
UR - https://arxiv.org/abs/2609.39996
AB - We develop a pressure-free virtual element method for the surface Stokes problem on polygonal approximations of closed surfaces of arbitrary genus. The method extends the planar VEM Stokes complex to discrete surfaces, yielding a nonconforming complex with an exactly tangential, H(div)-conforming velocity space, commuting interpolation, and the correct discrete cohomology. To obtain the weak consistency needed for optimal convergence, we introduce an edge correction that preserves the complex without adding degrees of freedom; the resulting construction reduces to the original complex in the planar setting. The virtual formulation requires no penalties, and a local divergence-preserving reconstruction provides a computable pressure-robust load. We also construct explicit macroelement representatives whose energy and load induce an alternative formulation on the VEM space, algebraically equivalent to a direct macroelement Galerkin method. The original virtual formulation and the macroelement method are stable and attain optimal first-order convergence in the broken H^1 norm and second-order convergence in the L^2 norm for the velocity. Numerical experiments illustrate these convergence rates, the role of the edge correction, and pressure robustness.
ER -