New one- and two-level Newton iterative mixed finite element methods for stationary conduction-convection problems

In this article, new one- and two-level Newton iterative mixed finite element methods (MFEM) for two-dimensional stationary conduction-convection problems are presented. In both of the methods, we decouple the conduction-convection problem into two systems: one is for the temperature and the other is for the velocity and pressure. Thus much computational time can be saved. In the new Newton MFEM, we first solve the equation for the temperature, then solve a Navier-Stokes problem by Newton iterative method. While in the two-level method, we solve the decoupled conduction-convection problem as same as in the one-level method using a coarse grid firstly, then seek a fine grid solution by solving a linearized problem on a fine grid. Stability analysis is performed and error estimates of the methods are derived, which show that our methods are stable and have a good precision. Numerical experiments are also given, which confirm the theoretical analysis and demonstrate the efficiency of the new methods.

[1]  Jincheng Ren,et al.  NONCONFORMING MIXED FINITE ELEMENT METHOD FOR THE STATIONARY CONDUCTION-CONVECTION PROBLEM , 2009 .

[2]  Weiwei Sun,et al.  Stability and Convergence of the Crank-Nicolson/Adams-Bashforth scheme for the Time-Dependent Navier-Stokes Equations , 2007, SIAM J. Numer. Anal..

[3]  On identical problems of natural convection in enclosures and applications of the identity character , 1993 .

[4]  Maxim A. Olshanskii,et al.  Two-level method and some a priori estimates in unsteady Navier—Stokes calculations , 1999 .

[5]  R. M. Cotta,et al.  Hybrid formulation and solution for transient conjugated conduction-external convection , 2009 .

[6]  Jinchao Xu.  Two-grid Discretization Techniques for Linear and Nonlinear PDEs , 1996 .

[7]  Yinnian He,et al.  Convergence of three iterative methods based on the finite element discretization for the stationary Navier–Stokes equations☆ , 2009 .

[8]  Charbel Farhat,et al.  A discontinuous enrichment method for the finite element solution of high Péclet advection-diffusion problems , 2009 .

[9]  S. P. Oliveira,et al.  Streamline design of stability parameters for advection—diffusion problems , 2001 .

[10]  Jing Chen,et al.  An optimizing reduced PLSMFE formulation for non‐stationary conduction–convection problems , 2009 .

[11]  Yinnian He,et al.  Two-Level Method Based on Finite Element and Crank-Nicolson Extrapolation for the Time-Dependent Navier-Stokes Equations , 2003, SIAM J. Numer. Anal..

[12]  Lutz Tobiska,et al.  A Two-Level Method with Backtracking for the Navier--Stokes Equations , 1998 .

[13]  Yinnian He,et al.  A simplified two-level method for the steady Navier–Stokes equations , 2008 .

[14]  Vivette Girault,et al.  Two-grid finite-element schemes for the transient Navier-Stokes problem , 2001 .

[15]  Jie Shen,et al.  An overview of projection methods for incompressible flows , 2006 .

[16]  William Layton,et al.  Error analysis for finite element methods for steady natural convection problems , 1990 .

[17]  Juan Andrés Martín García,et al.  Two-dimensional non-linear inverse heat conduction problem based on the singular value decomposition , 2009 .

[18]  Mo Yang,et al.  Natural convection in a square enclosure with an internal isolated vertical plate , 1994 .

[19]  Maximilian S. Mesquita,et al.  Optimal multigrid solutions of two-dimensional convection-conduction problems , 2004, Appl. Math. Comput..

[20]  Yinnian He,et al.  TWO-LEVEL PENALIZED FINITE ELEMENT METHODS FOR THE STATIONARY NAVIER-STOKE EQUATIONS , 2005 .

[21]  Endre Süli,et al.  Approximation of the global attractor for the incompressible Navier–Stokes equations , 2000 .

[22]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[23]  Yin-nianHe,et al.  A TWO-LEVEL FINITE ELEMENT GALERKIN METHOD FOR THE NONSTATIONARY NAVIER-STOKES EQUATIONS I： SPATIAL DISCRETIZATION , 2004 .

[24]  V. Girault,et al.  Two-grid finite-element schemes for the steady Navier-Stokes problem in polyhedra. , 2001 .

[25]  Kikuji Chida.  Surface temperature of a flat plate of finite thickness under conjugate laminar forced convection heat transfer condition , 2000 .

[26]  Yinnian He,et al.  Two-level Stabilized Finite Element Methods for the Steady Navier–Stokes Problem , 2005, Computing.

[27]  H. K. Lee,et al.  Numerical Solution of the Stationary Navier-Stokes Equations Using a Multilevel Finite Element Method , 1998, SIAM J. Sci. Comput..

[28]  Young Don Choi,et al.  Analysis of conduction – natural convection conjugate heat transfer in the gap between concentric cylinders under solar irradiation , 2009 .

[29]  H. Lenferink,et al.  An accurate solution procedure for fluid flow with natural convection , 1994 .

[30]  Jinchao Xu,et al.  A Novel Two-Grid Method for Semilinear Elliptic Equations , 1994, SIAM J. Sci. Comput..

[31]  Vivette Girault,et al.  An analysis of a mixed finite element method for the Navier-Stokes equations , 1979 .