DUAL-MIXED FINITE ELEMENT METHODS FOR THE NAVIER-STOKES EQUATIONS

A mixed finite element method for the Navier-Stokes equations is introduced in which the stress is a primary variable. The variational formulation retains the mathematical structure of the Navier-Stokes equations and the classical theory extends naturally to this setting. Finite element spaces satisfying the associated inf-sup conditions are developed.

[1]  P. Raviart,et al.  Finite Element Approximation of the Navier-Stokes Equations , 1979 .

[2]  Zhiqiang Cai,et al.  Pseudostress–velocity formulation for incompressible Navier–Stokes equations , 2010 .

[3]  Jacques Rappaz,et al.  Finite Dimensional Approximation of Non-Linear Problems .1. Branches of Nonsingular Solutions , 1980 .

[4]  Douglas N. Arnold,et al.  Mixed finite element methods for linear elasticity with weakly imposed symmetry , 2007, Math. Comput..

[5]  Jay Gopalakrishnan,et al.  A Second Elasticity Element Using the Matrix Bubble , 2012 .

[6]  P. Clément.  Approximation by finite element functions using local regularization , 1975 .

[7]  O. C. Zienkiewicz,et al.  The superconvergent patch recovery (SPR) and adaptive finite element refinement , 1992 .

[8]  Shun Zhang,et al.  Mixed Finite Element Methods for Incompressible Flow: Stationary Navier-Stokes Equations , 2010, SIAM J. Numer. Anal..

[9]  J. Z. Zhu,et al.  The superconvergent patch recovery and a posteriori error estimates. Part 2: Error estimates and adaptivity , 1992 .

[10]  Alan Shapiro.  The Use of an Exact Solution of the Navier–Stokes Equations in a Validation Test of a Three-Dimensional Nonhydrostatic Numerical Model , 1993 .

[11]  Jason S. Howell,et al.  Inf–sup conditions for twofold saddle point problems , 2011, Numerische Mathematik.

[12]  D. Boffi,et al.  Mixed finite elements, compatibility conditions, and applications : lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 26-July 1, 2006 , 2008 .

[13]  L. R. Scott,et al.  Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials , 1985 .

[14]  Bernardo Cockburn,et al.  A new elasticity element made for enforcing weak stress symmetry , 2010, Math. Comput..

[15]  Mohamed Farhloul,et al.  Analysis of non-singular solutions of a mixed Navier-Stokes formulation , 1996 .

[16]  O. C. Zienkiewicz,et al.  Reduced integration technique in general analysis of plates and shells , 1971 .

[17]  S. Nicaise,et al.  A refined mixed finite-element method for the stationary Navier–Stokes equations with mixed boundary conditions , 2007 .

[18]  F. Brezzi,et al.  Finite dimensional approximation of nonlinear problems , 1981 .

[19]  S. Nicaise,et al.  A priori and a posteriori error estimations for the dual mixed finite element method of the Navier‐Stokes problem , 2009 .

[20]  Zhimin Zhang.  Ultraconvergence of the patch recovery technique , 1996, Math. Comput..

[21]  L. D. Marini,et al.  Two families of mixed finite elements for second order elliptic problems , 1985 .

[22]  J. Z. Zhu,et al.  The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique , 1992 .

[23]  R. Stenberg.  Analysis of mixed finite elements methods for the Stokes problem: a unified approach , 1984 .

[24]  Shangyou Zhang,et al.  A new family of stable mixed finite elements for the 3D Stokes equations , 2004, Math. Comput..

[25]  M. Fortin,et al.  Reduced symmetry elements in linear elasticity , 2008 .

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

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

[28]  R. Showalter.  Monotone operators in Banach space and nonlinear partial differential equations , 1996 .

[29]  C. P. Gupta,et al.  A family of higher order mixed finite element methods for plane elasticity , 1984 .

[30]  Numedsche,et al.  A Family of Mixed Finite Elements for the Elasticity Problem , 2022 .

[31]  J. Douglas,et al.  PEERS: A new mixed finite element for plane elasticity , 1984 .

[32]  J. Barlow,et al.  Optimal stress locations in finite element models , 1976 .

[33]  William Layton,et al.  Introduction to the Numerical Analysis of Incompressible Viscous Flows , 2008 .