Error analysis of penalized H(div)-conforming finite element methods for surface Stokes problems
Abstract
In recent years a number of finite element methods have been developed for the surface Stokes equations. Constructing standard conforming surface finite element methods for the velocity-pressure formulation of this problem would require simultaneously enforcing H1 conformity and tangentiality of the velocity field, which is not possible on the generalized polyhedral approximating surfaces on which surface FEM are typically posed. Thus methods for this problem weakly enforce either tangentiality or H1 conformity of the velocity field. In this paper we provide an error analysis for an H(div)-conforming method based on classical Brezzi-Douglas-Marini elements. In this method continuity of the velocity field is weakly enforced using an interior penalty-type formulation. Advantages of this formulation include exact enforcement of the incompressibility constraint and its construction based on a standard mixed finite element pair. Error analysis of this method in previous work was only valid for lowest-order elements and did not account for ``geometric errors'', i.e., variational crimes due to approximation of the continuous surface on which the PDE is posed by an approximating surface on which the finite element method is posed. Here we provide a full error analysis in a natural discontinuous Galerkin energy norm, with geometric errors accounted for. Optimal estimates for the L2 velocity error are also presented. In the case of lowest-order (polyhedral) surface approximations, our proof of optimal L2 estimates requires a stricter than usual condition on the placement of mesh nodes relative to the surface. Numerical experiments are used to illustrate our results.
Related
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BibTeX
@misc{demlow2026error,
title = {{Error analysis of penalized $H(\rm div)$-conforming finite element methods for surface Stokes problems}},
author = {Alan Demlow and Orsan Kilicer},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28729},
eprint = {2609.28729},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28729},
}
RIS
TY - UNPB
TI - Error analysis of penalized H(div)-conforming finite element methods for surface Stokes problems
AU - Alan Demlow
AU - Orsan Kilicer
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28729
UR - https://arxiv.org/abs/2609.28729
AB - In recent years a number of finite element methods have been developed for the surface Stokes equations. Constructing standard conforming surface finite element methods for the velocity-pressure formulation of this problem would require simultaneously enforcing H^1 conformity and tangentiality of the velocity field, which is not possible on the generalized polyhedral approximating surfaces on which surface FEM are typically posed. Thus methods for this problem weakly enforce either tangentiality or H^1 conformity of the velocity field. In this paper we provide an error analysis for an H(div)-conforming method based on classical Brezzi-Douglas-Marini elements. In this method continuity of the velocity field is weakly enforced using an interior penalty-type formulation. Advantages of this formulation include exact enforcement of the incompressibility constraint and its construction based on a standard mixed finite element pair. Error analysis of this method in previous work was only valid for lowest-order elements and did not account for ``geometric errors'', i.e., variational crimes due to approximation of the continuous surface on which the PDE is posed by an approximating surface on which the finite element method is posed. Here we provide a full error analysis in a natural discontinuous Galerkin energy norm, with geometric errors accounted for. Optimal estimates for the L_2 velocity error are also presented. In the case of lowest-order (polyhedral) surface approximations, our proof of optimal L_2 estimates requires a stricter than usual condition on the placement of mesh nodes relative to the surface. Numerical experiments are used to illustrate our results.
ER -