NONCONFORMING MIXED FINITE ELEMENT METHOD FOR THE STATIONARY CONDUCTION-CONVECTION PROBLEM

In this paper, a new stable nonconforming mixed flnite element scheme is proposed for the stationary conduction-convection problem, in which a new low order Crouzeix-Raviart type nonconforming rectangular element is taken as approximation space for the velocity, the piecewise constant element for the pressure and the bilinear element for the temperature, respectively. The convergence analysis is presented and the optimal error estimates in a broken H1-norm for the velocity, L2-norm for the pressure and H1-seminorm for the temperature are derived.

[1]  Dongwoo Sheen,et al.  Nonconforming Galerkin methods based on quadrilateral elements for second order elliptic problems , 1999 .

[2]  J. Tinsley Oden,et al.  Finite Elements: Fluid Mechanics. , 1989 .

[3]  Eberhard Zeidler,et al.  Applications to mathematical physics , 1988 .

[4]  E. Zeidler.  Nonlinear functional analysis and its applications , 1988 .

[5]  Dongwoo Sheen,et al.  A new quadratic nonconforming finite element on rectangles , 2006 .

[6]  T. Zhou,et al.  A least squares Petrov-Galerkin finite element method for the stationary Navier-Stokes equations , 1993 .

[7]  Jincheng Ren,et al.  Nonconforming mixed finite element approximation to the stationary Navier–Stokes equations on anisotropic meshes , 2009 .

[8]  M. Marion,et al.  Nonlinear Galerkin methods and mixed finite elements: two-grid algorithms for the Navier-Stokes equations , 1994 .

[9]  Jim Douglas,et al.  An absolutely stabilized finite element method for the stokes problem , 1989 .

[10]  P. Raviart.  Finite element methods and Navier-Stokes equations , 1979 .

[11]  S. Nicaise,et al.  A non-conforming finite volume element method of weighted upstream type for the two-dimensional stationary Navier--Stokes system , 2008 .

[12]  Louis B. Rall,et al.  Nonlinear Functional Analysis and Applications , 1971 .

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

[14]  C. Bernardi,et al.  Analysis of some finite elements for the Stokes problem , 1985 .

[15]  Q. Lin,et al.  Superconvergence and extrapolation of non-conforming low order finite elements applied to the Poisson equation , 2005 .

[16]  M. O'Leary.  Conduction-Convection Problems with Change of Phase. , 1995 .

[17]  Thomas J. R. Hughes,et al.  The Stokes problem with various well-posed boundary conditions - Symmetric formulations that converge for all velocity/pressure spaces , 1987 .

[18]  R. Shvydkoy.  LECTURES ON FUNCTIONAL ANALYSIS , 2010 .

[19]  T. Hughes,et al.  A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuscka-Brezzi condition: A stable Petrov-Galerkin formulation of , 1986 .

[20]  R. L. Sani,et al.  Solution of the time-dependent incompressible Navier-Stokes and Boussinesq equations using the Galerkin finite element method , 1980 .

[21]  J. Douglas,et al.  Stabilized mixed methods for the Stokes problem , 1988 .

[22]  Shao-chunChen,et al.  AN ANISOTROPIC NONCONFORMING FINITE ELEMENT WITH SOME SUPERCONVERGENCE RESULTS , 2005 .

[23]  Lutz Tobiska,et al.  The streamline–diffusion method for nonconforming Qrot1 elements on rectangular tensor–product meshes , 2001 .

[24]  R. Rannacher,et al.  Simple nonconforming quadrilateral Stokes element , 1992 .

[25]  R. Temam.  Navier-Stokes Equations , 1977 .

[26]  Philippe G. Ciarlet,et al.  Handbook of Numerical Analysis , 1976 .

[27]  T. Hughes,et al.  Two classes of mixed finite element methods , 1988 .

[28]  Jukka Saranen,et al.  Streamline diffusion methods for the incompressible Euler and Navier-Stokes equations , 1986 .

[29]  Thomas J. R. Hughes,et al.  Finite element methods for first-order hyperbolic systems with particular emphasis on the compressible Euler equations , 1984 .