Inf-sup stability of the trace P2-P1 Taylor-Hood elements for surface PDEs

The paper studies a geometrically unfitted finite element method (FEM), known as trace FEM or cut FEM, for the numerical solution of the Stokes system posed on a closed smooth surface. A trace FEM based on standard Taylor-Hood (continuous P2-P1) bulk elements is proposed. A so-called volume normal derivative stabilization, known from the literature on trace FEM, is an essential ingredient of this method. The key result proved in the paper is an inf-sup stability of the trace P2-P1 finite element pair, with the stability constant uniformly bounded with respect to the discretization parameter and the position of the surface in the bulk mesh. Optimal order convergence of a consistent variant of the finite element method follows from this new stability result and interpolation properties of the trace FEM. Properties of the method are illustrated with numerical examples.

[1]  C. B. Morrey.  Multiple Integrals in the Calculus of Variations , 1966 .

[2]  Rüdiger Verfürth,et al.  Error estimates for a mixed finite element approximation of the Stokes equations , 1984 .

[3]  Maxim A. Olshanskii,et al.  Iterative Methods for Linear Systems - Theory and Applications , 2014 .

[4]  Thomas-Peter Fries,et al.  Higher‐order surface FEM for incompressible Navier‐Stokes flows on manifolds , 2017, ArXiv.

[5]  Michael E. Taylor,et al.  Analysis on Morrey Spaces and Applications to Navier-Stokes and Other Evolution Equations , 1992 .

[6]  I. Nitschke,et al.  A finite element approach to incompressible two-phase flow on manifolds , 2012, Journal of Fluid Mechanics.

[7]  Arnold Reusken,et al.  Stream function formulation of surface Stokes equations , 2018, IMA Journal of Numerical Analysis.

[8]  Maxim A. Olshanskii,et al.  An Adaptive Surface Finite Element Method Based on Volume Meshes , 2012, SIAM J. Numer. Anal..

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

[10]  J. Marsden,et al.  Groups of diffeomorphisms and the motion of an incompressible fluid , 1970 .

[11]  Axel Voigt,et al.  Solving the incompressible surface Navier-Stokes equation by surface finite elements , 2017, 1709.02803.

[12]  Peter Hansbo,et al.  A stabilized cut finite element method for the Darcy problem on surfaces , 2015, 1511.03747.

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

[14]  Axel Voigt,et al.  The Interplay of Curvature and Vortices in Flow on Curved Surfaces , 2014, Multiscale Model. Simul..

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

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

[17]  Maxim A. Olshanskii,et al.  A Finite Element Method for the Surface Stokes Problem , 2018, SIAM J. Sci. Comput..

[18]  Arnold Reusken,et al.  Trace finite element methods for surface vector-Laplace equations , 2019 .

[19]  A. Bonito,et al.  A divergence-conforming finite element method for the surface Stokes equation , 2019, SIAM J. Numer. Anal..

[20]  Kenneth Steiglitz,et al.  Operations on Images Using Quad Trees , 1979, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[21]  Harald Garcke,et al.  A stable numerical method for the dynamics of fluidic membranes , 2016, Numerische Mathematik.

[22]  Arnold Reusken,et al.  Error analysis of higher order Trace Finite Element Methods for the surface Stokes equation , 2019, J. Num. Math..

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

[24]  Maxim A. Olshanskii,et al.  Incompressible fluid problems on embedded surfaces: Modeling and variational formulations , 2017, Interfaces and Free Boundaries.

[25]  Morton E. Gurtin,et al.  A continuum theory of elastic material surfaces , 1975 .

[26]  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.

[27]  Marius Mitrea,et al.  Navier-Stokes equations on Lipschitz domains in Riemannian manifolds , 2001 .

[28]  Arnold Reusken,et al.  Analysis of trace finite element methods for surface partial differential equations , 2015 .

[29]  Yoshikazu Giga,et al.  Energetic variational approaches for incompressible fluid systems on an evolving surface , 2016 .

[30]  I. Holopainen.  Riemannian Geometry , 1927, Nature.

[31]  Volker John,et al.  Finite Element Methods for Incompressible Flow Problems , 2016 .

[32]  Peter Hansbo,et al.  Analysis of finite element methods for vector Laplacians on surfaces , 2016, IMA Journal of Numerical Analysis.

[33]  Maxim A. Olshanskii,et al.  Trace Finite Element Methods for PDEs on Surfaces , 2016, 1612.00054.

[34]  R. Temam.  Infinite Dimensional Dynamical Systems in Mechanics and Physics Springer Verlag , 1993 .

[35]  Maxim A. Olshanskii,et al.  A Penalty Finite Element Method for a Fluid System Posed on Embedded Surface , 2018, Journal of Mathematical Fluid Mechanics.

[36]  Arnold Reusken,et al.  A Higher Order Finite Element Method for Partial Differential Equations on Surfaces , 2016, SIAM J. Numer. Anal..

[37]  Christoph Lehrenfeld,et al.  Analysis of a High-Order Trace Finite Element Method for PDEs on Level Set Surfaces , 2016, SIAM J. Numer. Anal..

[38]  Marino Arroyo,et al.  Relaxation dynamics of fluid membranes. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[39]  Jean E. Roberts,et al.  Mixed and hybrid finite element methods , 1987 .

[40]  V. Arnold,et al.  Topological methods in hydrodynamics , 1998 .

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

[42]  Maxim A. Olshanskii,et al.  A Finite Element Method for Elliptic Equations on Surfaces , 2009, SIAM J. Numer. Anal..