A stabilized Nitsche cut finite element method for the Oseen problem

Abstract We provide the numerical analysis for a Nitsche-based cut finite element formulation for the Oseen problem, which has been originally presented for the incompressible Navier–Stokes equations by Schott and Wall (2014) and allows the boundary of the domain to cut through the elements of an easy-to-generate background mesh. The formulation is based on the continuous interior penalty (CIP) method of Burman et al. (2006) which penalizes jumps of velocity and pressure gradients over inter-element faces to counteract instabilities arising for high local Reynolds numbers and the use of equal order interpolation spaces for the velocity and pressure. Since the mesh does not fit the boundary, Dirichlet boundary conditions are imposed weakly by a stabilized Nitsche-type approach. The addition of CIP-like ghost-penalties in the boundary zone allows to prove that our method is inf–supstable and to derive optimal order a priori  error estimates in an energy-type norm, irrespective of how the boundary cuts the underlying mesh. All applied stabilization techniques are developed with particular emphasis on low and high Reynolds numbers. Two- and three-dimensional numerical examples corroborate the theoretical findings. Finally, the proposed method is applied to solve the transient incompressible Navier–Stokes equations on a complex geometry.

[1]  Carl E. Pearson,et al.  A computational method for viscous flow problems , 1965, Journal of Fluid Mechanics.

[2]  Miguel A. Fernández,et al.  Galerkin Finite Element Methods with Symmetric Pressure Stabilization for the Transient Stokes Equations: Stability and Convergence Analysis , 2008, SIAM J. Numer. Anal..

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

[4]  T. Belytschko,et al.  An Extended Finite Element Method for Two-Phase Fluids , 2003 .

[5]  P. Hansbo,et al.  A cut finite element method for a Stokes interface problem , 2012, 1205.5684.

[6]  Mats G. Larson,et al.  A Nitsche-Based Cut Finite Element Method for a Fluid--Structure Interaction Problem , 2013, 1311.2431.

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

[8]  W. Wall,et al.  A face‐oriented stabilized Nitsche‐type extended variational multiscale method for incompressible two‐phase flow , 2015 .

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

[10]  C. Y. Wang,et al.  On the low-Reynolds-number flow in a helical pipe , 1981, Journal of Fluid Mechanics.

[11]  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.

[12]  Li-Bin Liu,et al.  A Robust Adaptive Grid Method for a System of Two Singularly Perturbed Convection-Diffusion Equations with Weak Coupling , 2014, J. Sci. Comput..

[13]  André Massing,et al.  A Stabilized Nitsche Fictitious Domain Method for the Stokes Problem , 2012, J. Sci. Comput..

[14]  Alexandre Ern,et al.  Continuous interior penalty hp-finite element methods for advection and advection-diffusion equations , 2007, Math. Comput..

[15]  Peter Hansbo,et al.  CutFEM: Discretizing geometry and partial differential equations , 2015 .

[16]  Wolfgang A. Wall,et al.  An embedded Dirichlet formulation for 3D continua , 2010 .

[17]  Peter Hansbo,et al.  A stabilized cut finite element method for partial differential equations on surfaces: The Laplace–Beltrami operator , 2013, 1312.1097.

[18]  A. Ern,et al.  A continuous finite element method with face penalty to approximate Friedrichs' systems , 2007 .

[19]  J. Nitsche.  Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind , 1971 .

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

[21]  P. Moin,et al.  Application of a Fractional-Step Method to Incompressible Navier-Stokes Equations , 1984 .

[22]  Peter Hansbo,et al.  A cut discontinuous Galerkin method for the Laplace-Beltrami operator , 2015, 1507.05835.

[23]  Petr Knobloch,et al.  Improved stability and error analysis for a class of local projection stabilizations applied to the Oseen problem , 2013 .

[24]  Gert Lube,et al.  Some Remarks on Residual-based Stabilisation of Inf-sup Stable Discretisations of the Generalised Oseen Problem , 2009, Comput. Methods Appl. Math..

[25]  A. Chorin.  Numerical solution of the Navier-Stokes equations , 1968 .

[26]  P. Hansbo,et al.  Full gradient stabilized cut finite element methods for surface partial differential equations , 2016, 1602.01512.

[27]  P. Hansbo,et al.  An unfitted finite element method, based on Nitsche's method, for elliptic interface problems , 2002 .

[28]  Benedikt Schott,et al.  A new face-oriented stabilized XFEM approach for 2D and 3D incompressible Navier–Stokes equations , 2014 .

[29]  C. Ross Ethier,et al.  Exact fully 3D Navier–Stokes solutions for benchmarking , 1994 .

[30]  Erik Burman,et al.  Interior penalty variational multiscale method for the incompressible Navier-Stokes equation: Monitoring artificial dissipation , 2007 .

[31]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[32]  Erik Burman,et al.  Stabilized finite element methods for the generalized Oseen problem , 2007 .

[33]  L. Zabielski,et al.  Steady flow in a helically symmetric pipe , 1998, Journal of Fluid Mechanics.

[34]  Marcus Sarkis,et al.  Robust flux error estimation of an unfitted Nitsche method for high-contrast interface problems , 2016, 1602.00603.

[35]  Leonhard Kleiser,et al.  Subgrid‐scale energy transfer in the near‐wall region of turbulent flows , 1994 .

[36]  R. Codina.  Analysis of a stabilized finite element approximation of the Oseen equations using orthogonal subscales , 2008 .

[37]  Thomas J. R. Hughes,et al.  Weak imposition of Dirichlet boundary conditions in fluid mechanics , 2007 .

[38]  Peter Hansbo,et al.  A Cut Finite Element Method with Boundary Value Correction for the Incompressible Stokes Equations , 2019, Lecture Notes in Computational Science and Engineering.

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

[40]  André Massing,et al.  A stabilized Nitsche overlapping mesh method for the Stokes problem , 2012, Numerische Mathematik.

[41]  J. Cooper.  SINGULAR INTEGRALS AND DIFFERENTIABILITY PROPERTIES OF FUNCTIONS , 1973 .

[42]  Gert Lube,et al.  RESIDUAL-BASED STABILIZED HIGHER-ORDER FEM FOR A GENERALIZED OSEEN PROBLEM , 2006 .

[43]  Arnold Reusken,et al.  An extended pressure finite element space for two-phase incompressible flows with surface tension , 2007, J. Comput. Phys..

[44]  Miguel A. Fernández,et al.  An unfitted Nitsche method for incompressible fluid–structure interaction using overlapping meshes , 2014 .

[45]  Peter Hansbo,et al.  A cut finite element method for coupled bulk-surface problems on time-dependent domains , 2015, 1502.07142.

[46]  Erik Burman,et al.  A Stabilized Cut Finite Element Method for the Three Field Stokes Problem , 2014, SIAM J. Sci. Comput..

[47]  T. Belytschko,et al.  An Eulerian–Lagrangian method for fluid–structure interaction based on level sets , 2006 .

[48]  Wolfgang A. Wall,et al.  3D fluid–structure-contact interaction based on a combined XFEM FSI and dual mortar contact approach , 2010 .

[49]  Miguel A. Fernández,et al.  Continuous Interior Penalty Finite Element Method for Oseen's Equations , 2006, SIAM J. Numer. Anal..

[50]  Wolfgang A. Wall,et al.  An XFEM‐based embedding mesh technique for incompressible viscous flows , 2011 .

[51]  Peter Hansbo,et al.  Cut finite element methods for partial differential equations on embedded manifolds of arbitrary codimensions , 2016, ESAIM: Mathematical Modelling and Numerical Analysis.

[52]  Peter Hansbo,et al.  Fictitious domain methods using cut elements: III. A stabilized Nitsche method for Stokes’ problem , 2014 .

[53]  Erik Burman,et al.  A Unified Analysis for Conforming and Nonconforming Stabilized Finite Element Methods Using Interior Penalty , 2005, SIAM J. Numer. Anal..

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

[55]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[56]  Peter Hansbo,et al.  A cut finite element method with boundary value correction , 2015, Math. Comput..

[57]  M. Germano,et al.  On the effect of torsion on a helical pipe flow , 1982, Journal of Fluid Mechanics.

[58]  Benedikt Schott,et al.  A stabilized Nitsche‐type extended embedding mesh approach for 3D low‐ and high‐Reynolds‐number flows , 2016 .

[59]  P. Hansbo,et al.  A FINITE ELEMENT METHOD ON COMPOSITE GRIDS BASED ON NITSCHE'S METHOD , 2003 .

[60]  Mats G. Larson,et al.  Efficient implementation of finite element methods on non-matching and overlapping meshes in 3D , 2012, 1210.7076.

[61]  W. Wall,et al.  An eXtended Finite Element Method/Lagrange multiplier based approach for fluid-structure interaction , 2008 .

[62]  ANDRÉ MASSING,et al.  Efficient Implementation of Finite Element Methods on Nonmatching and Overlapping Meshes in Three Dimensions , 2013, SIAM J. Sci. Comput..