Journal of Scientific Computing

, Volume 61, Issue 3, pp 604–628

# A Stabilized Nitsche Fictitious Domain Method for the Stokes Problem

• André Massing
• Mats G. Larson
• Anders Logg
• Marie E. Rognes
Article

## Abstract

We present a novel finite element method for the Stokes problem on fictitious domains. We prove inf-sup stability, optimal order convergence and uniform boundedness of the condition number of the discrete system. The finite element formulation is based on a stabilized Nitsche method with ghost penalties for the velocity and pressure to obtain stability in the presence of small cut elements. We demonstrate for the first time the applicability of the Nitsche fictitious domain method to three-dimensional Stokes problems. We further discuss a general, flexible and freely available implementation of the method and present numerical examples supporting the theoretical results.

## Keywords

Fictitious domain Stokes problem Stabilized finite element methods Nitsche’s method

## Mathematics Subject Classification

65N12 65N30 76D07

## Notes

### Acknowledgments

The authors wish to thank Sebastian Warmbrunn for providing the surface geometry used in Sect. 8.4 and Kent-Andre Mardal for insightful discussion on preconditioning. This work is supported by an Outstanding Young Investigator grant from the Research Council of Norway, NFR 180450. This work is also supported by a Center of Excellence grant from the Research Council of Norway to the Center for Biomedical Computing at Simula Research Laboratory.

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## Authors and Affiliations

• André Massing
• 1
Email author
• Mats G. Larson
• 2
• Anders Logg
• 3
• Marie E. Rognes
• 1
1. 1.Simula Research LaboratoryOsloNorway
2. 2.Department of MathematicsUmeå UniversityUmeåSweden
3. 3.Mathematical SciencesChalmers University of TechnologyGothenburgSweden