The MINI mixed finite element for the Stokes problem: An experimental investigation

Abstract O h 3 ∕ 2 superconvergence in pressure and velocity has been experimentally investigated for the two-dimensional Stokes problem discretized with the MINI mixed finite element. Even though the classic mixed finite element theory for the MINI element guarantees linear convergence for the total error, recent theoretical results indicate that superconvergence of order O h 3 ∕ 2 in pressure and of the linear part of the computed velocity to the piecewise-linear nodal interpolation of the exact velocity is in fact possible with structured, three-directional triangular meshes. The numerical experiments presented here suggest a more general validity of O h 3 ∕ 2 superconvergence, possibly to automatically generated and unstructured triangulations. In addition, the approximating properties of the complete computed velocity have been compared with the approximating properties of the piecewise-linear part of the computed velocity, finding that the former is generally closer to the exact velocity, whereas the latter conserves mass better.

[1]  Jiten C. Kalita,et al.  Moffatt eddies in the driven cavity: A quantification study by an HOC approach , 2018, Comput. Math. Appl..

[2]  D. A. Field.  Qualitative measures for initial meshes , 2000 .

[3]  Volker John,et al.  On the Divergence Constraint in Mixed Finite Element Methods for Incompressible Flows , 2015, SIAM Rev..

[4]  M. Benzi.  Preconditioning techniques for large linear systems: a survey , 2002 .

[5]  G. Tallini,et al.  ON THE EXISTENCE OF , 1996 .

[6]  S. Repin,et al.  ON THE FUNCTIONAL TYPE A POSTERIORI ERROR ESTIMATES FOR THE STOKES PROBLEM. , 2004 .

[7]  R. Panton.  Incompressible Flow: Panton/Incompressible Flow 4E , 2013 .

[8]  A. J. Wathen,et al.  Preconditioning , 2015, Acta Numerica.

[9]  I. Babuska.  The finite element method with Lagrangian multipliers , 1973 .

[10]  Sungyun Lee,et al.  Modified Mini finite element for the Stokes problem in ℝ2 or ℝ3 , 2000, Adv. Comput. Math..

[11]  F. Brezzi.  On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers , 1974 .

[12]  Alessandro Russo,et al.  A posteriori error estimators for the Stokes problem , 1995 .

[13]  T. Shih,et al.  Effects of grid staggering on numerical schemes , 1989 .

[14]  R. Verfürth.  A posteriori error estimators for the Stokes equations , 1989 .

[15]  M. Fortin,et al.  Mixed Finite Element Methods and Applications , 2013 .

[16]  Randolph E. Bank,et al.  A posteriori error estimates for the Stokes equations: a comparison , 1990 .

[17]  Barry Lee,et al.  Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics , 2006, Math. Comput..

[18]  Victor M. Calo,et al.  Performance evaluation of block-diagonal preconditioners for the divergence-conforming B-spline discretization of the Stokes system , 2015, J. Comput. Sci..

[19]  M. Fortin,et al.  A stable finite element for the stokes equations , 1984 .

[20]  D. A. Dunavant.  High degree efficient symmetrical Gaussian quadrature rules for the triangle , 1985 .

[21]  Hehu Xie,et al.  Supercloseness and superconvergence of stabilized low-order finite element discretizations of the Stokes Problem , 2011, Math. Comput..

[22]  Per-Olof Persson,et al.  A Simple Mesh Generator in MATLAB , 2004, SIAM Rev..