Stability and Optimal Convergence of Unfitted Extended Finite Element Methods with Lagrange Multipliers for the Stokes Equations

We study a fictitious domain approach with Lagrange multipliers to discretize Stokes equations on a mesh that does not fit the boundaries. A mixed finite element method is used for fluid flow. Several stabilization terms are added to improve the approximation of the normal trace of the stress tensor and to avoid the inf-sup conditions between the spaces of the velocity and the Lagrange multipliers. We generalize first an approach based on eXtended Finite Element Method due to Haslinger-Renard (SIAM J Numer Anal 47(2):1474–1499, 2009) involving a Barbosa-Hughes stabilization and a robust reconstruction on the badly cut elements. Secondly, we adapt the approach due to Burman-Hansbo (Comput Methods Appl Mech Eng 199(41–44):2680–2686, 2010) involving a stabilization only on the Lagrange multiplier. Multiple choices for the finite elements for velocity, pressure and multiplier are considered. Additional stabilization on pressure (Brezzi-Pitkaranta, Interior Penalty) is added, if needed. We prove the stability and the optimal convergence of several variants of these methods under appropriate assumptions. Finally, we perform numerical tests to illustrate the capabilities of the methods.

[1]  F. Brezzi,et al.  On the Stabilization of Finite Element Approximations of the Stokes Equations , 1984 .

[2]  Vivette Girault,et al.  Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms , 1986, Springer Series in Computational Mathematics.

[3]  Helio J. C. Barbosa,et al.  The finite element method with Lagrange multiplier on the boundary: circumventing the Babuscka-Brezzi condition , 1991 .

[4]  Patrick Amestoy,et al.  A Fully Asynchronous Multifrontal Solver Using Distributed Dynamic Scheduling , 2001, SIAM J. Matrix Anal. Appl..

[5]  J. Guermond,et al.  Theory and practice of finite elements , 2004 .

[6]  M. Fortin,et al.  Finite Elements for the Stokes Problem , 2008 .

[7]  Michel Fortin,et al.  Mixed Finite Elements, Compatibility Conditions, and Applications , 2008 .

[8]  Jaroslav Haslinger,et al.  A New Fictitious Domain Approach Inspired by the Extended Finite Element Method , 2009, SIAM J. Numer. Anal..

[9]  Peter Hansbo,et al.  Interior-penalty-stabilized Lagrange multiplier methods for the finite-element solution of elliptic interface problems , 2010 .

[10]  Peter Hansbo,et al.  Fictitious domain finite element methods using cut elements: I. A stabilized Lagrange multiplier method , 2010 .

[11]  P. Hansbo,et al.  Fictitious domain finite element methods using cut elements , 2012 .

[12]  Tomas Bengtsson,et al.  Fictitious domain methods using cut elements : III . A stabilized Nitsche method for Stokes ’ problem , 2012 .

[13]  Alexei Lozinski,et al.  A fictitious domain approach for the Stokes problem based on the extended finite element method , 2013, 1303.6850.

[14]  S'ebastien Court,et al.  A fictitious domain finite element method for simulations of fluid-structure interactions: The Navier-Stokes equations coupled with a moving solid , 2015, 1502.03953.

[15]  Arnold Reusken,et al.  Analysis of an XFEM Discretization for Stokes Interface Problems , 2016, SIAM J. Sci. Comput..

[16]  Maxim A. Olshanskii,et al.  Numerical Analysis and Scientific Computing Preprint Seria Inf-sup stability of geometrically unfitted Stokes finite elements , 2016 .