##
**A coupled Newton iterative mixed finite element method for stationary conduction-convection problems.**
*(English)*
Zbl 1206.65249

The authors give a coupled Newton iterative mixed finite element method (MFEM) for solving stationary conduction-convection problems in two dimension. For solving the equations of conduction-convection problems, they use the Newton iterative MFEM. They derive a stability and convergence analysis in the \(H ^{1}\)-norm of \({u_h^n, T_h^n}\) and the \(L ^{2}\)-norm of \({p_h^n}.\) They prove that their method is stable and has a good precision. They also give some numerical results, which prove that the coupled Newton iterative MFEM is highly efficient for stationary conduction-convection problems.

Reviewer: Yaşar Sözen (Istanbul)

### MSC:

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |

35J65 | Nonlinear boundary value problems for linear elliptic equations |

65N15 | Error bounds for boundary value problems involving PDEs |

### Keywords:

Boussinesq approximation; mixed finite element method; coupled Newton iterative; stability analysis; error estimates; stationary conduction-convection problems
PDF
BibTeX
XML
Cite

\textit{Z. Si} and \textit{Y. He}, Computing 89, No. 1--2, 1--25 (2010; Zbl 1206.65249)

Full Text:
DOI

### References:

[1] | Adams RA (1975) Sobolev space, pure and applied mathematics, vol 65. Academic Press, New York |

[2] | Ciarlet PG (1978) The finite element method for elliptic problems. North-Holland, Amsterdam · Zbl 0383.65058 |

[3] | Garcia J, Cabeza J, Rodriguez A (2009) Two-dimensional non-linear inverse heat conduction problem based on the singular value decomposition. Int J Thermal Sci 48: 1081–1093 |

[4] | Chen ZX (2005) Finite element methods and their applications. Springer, Berlin |

[5] | Girault V, Raviart PA (1987) Finite element method for Navier–Stokes equations: theory and algorithms. Springer, Berlin |

[6] | He YN (2003) Two-level method based on finite element and Crank-Nicolson extrapolation for the time-dependent Navier–Stokes equations. SIAM J Numer Anal 41(4): 1263–1285 · Zbl 1130.76365 |

[7] | He YN, Li J (2009) Convergence of three iterative methods based on the finite element discretization for the stationary Navier–Stokes equations. Comput Methods Appl Mech Eng 198: 1351–1359 · Zbl 1227.76031 |

[8] | He YN, Li KT (2005) Two-level stabilized finite element methods for the steady Navier–Stokes problem. Computing 74: 337–351 · Zbl 1099.65111 |

[9] | He YN, Sun WW (2007) Stability and convergence of the Crank-Nicolson/Adams-Bashforth scheme for the time-dependent Navier–Stokes equations. SIAM J Numer Anal 45(2): 837–869 · Zbl 1145.35318 |

[10] | He YN, Wang AW (2008) A simplified two-level method for the steady Navier–Stokes equations. Comput Methods Appl Mech Eng 197: 1568–1576 · Zbl 1194.76120 |

[11] | Hill AT, Sli E (2000) Approximation of the global attractor for the incompressible Navier–Stokes equations. IMA J Numer Anal 20: 633–667 · Zbl 0982.76022 |

[12] | Layton W, Lenferink HWJ (1996) A multilevel mesh independence principle for the Navier–Stokes equations. SIAM J Numer Anal 33: 17–30 · Zbl 0844.76053 |

[13] | Layton W, Leferink HWJ (1995) Two-level Picard and modified Picard methods for the Navier–Stokes equations. Appl Math Comput 69: 263–274 · Zbl 0828.76017 |

[14] | Li KT, Hou YR (2001) An AIM and one-step Newton method for the Navier–Stokes equations. Comput Methods Appl Mech Eng 190: 6141–6155 · Zbl 1011.76045 |

[15] | Luo ZD (2006) Mixed finite element foundation and its application. Science Press, Beijing (in Chinese) |

[16] | Luo ZD, Chen J, Navon IM, Zhu J (2009) An optimizing reduced PLSMFE formulation for non-stationary conduction–convection problems. Int J Numer Methods Fluids 60: 409–436 · Zbl 1161.76032 |

[17] | Luo ZD, Lu XM (2003) A least squares Galerkin/Petrov mixed finite element method for the stationary conduction–convection problems. Mathematica Numerica Sinica 25(2): 231–244 |

[18] | Luo ZD, Lu XM (2003) A nonlinear Galerkin/Petrov least squares mixed finite element method for the stationary conduction–convection problems. Mathematica Numerica Sinica 25(4): 447–462 |

[19] | Luo ZD, Wang LH (1998) Nonlinear Galerkin mixed element methods for the non stationary conduction–convection problems (I): the continuous-time case. C J Numer Math Appl 20(4): 71–94 |

[20] | Kim D, Choi Y (2009) Analysis of conduction-natural convection conjugate heat transfer in the gap between concentric cylinders under solar irradiation. Int J Thermal Sci 48: 1247–1258 |

[21] | Mesquita MS, de Lemos MJS (2004) Optimal multigrid solutions of two-dimensional convection–conduction problems. Appl Math Comput 152: 725–742 · Zbl 1077.65508 |

[22] | Naveira CP, Lachi M, Cotta RM, Padet J (2009) Hybrid formulation and solution for transient conjugated conduction-external convection. Int J Heat Mass Transf 52: 112–123 · Zbl 1156.80357 |

[23] | Reddy JN, Gartling DK (2000) The finite element method transfer and fluid dunamics (second edition). CRC Pess, Washington |

[24] | Tang L, Tsang T (1993) A least-squsres finite element method for time-depent incompressible flows with thermal convection. Int J Numer Methods Fluids 17: 271–289 · Zbl 0779.76045 |

[25] | Temam R (1984) Navier–Stokes equation: theory and numerical analysis (third edition). North- Holland, Amsterdam · Zbl 0568.35002 |

[26] | Wang QW, Yang M, Tao WQ (1994) Natural convection in a square enclosure with an internal isolated vertical plate. Warme-Stoffubertrag 29: 161–169 |

[27] | Yang M, Tao WQ, Wang QW, Lue SS (1993) On identical problems of natural convection in enclosure and applications of the identity character. J Thermal Sci 2: 116–125 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.