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Adaptive mixed least squares Galerkin/Petrov finite element method for stationary conduction convection problems. (English) Zbl 1237.76078
Summary: An adaptive mixed least squares Galerkin/Petrov finite element method (FEM) is developed for stationary conduction convection problems. The mixed least squares Galerkin/Petrov FEM is consistent and stable for any combination of discrete velocity and pressure spaces without requiring the Babuska-Brezzi stability condition. Using the general theory of Verfürth, the posteriori error estimates of the residual type are derived. Finally, numerical tests are presented to illustrate the effectiveness of the method.

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
65Z05 Applications to the sciences
Software:
FreeFem++
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References:
[1] Zhou, T. X. and Feng, M. F. A least squares Galerkin/Petrov mixed finite element method for the stationary Navier-Stokes equations. Mathematics of Computation, 60, 531–543 (1993) · Zbl 0778.65081
[2] Luo, Z. D., Zhu, J., and Wang, H. J. A nonlinear Galerkin/Petrov-least squares mixed element method for the stationary Navier-Stokes equation. Applied Mathematics and Mechanics (English Edition), 23(7), 783–793 (2002) DOI 10.1007/BF02439 · Zbl 1036.76032
[3] Luo, Z. D., Mao, Y. K., and Zhu, J. Galerkin-Petrov least squares mixed element method for stationary incompressible magnetohydrodynamics. Applied Mathematics and Mechanics (English Edition), 28(3), 395–404 (2007) DOI 10.1007/s10483-007-0312-x · Zbl 1231.65214
[4] Hughes, T. J. and Tezduyar, T. E. Finite element methods for first-order hyperbolic systems with particular emphasis on the compressible Euler equations. Computer Methods in Applied Mechanics and Engineering, 45, 217–284 (1984) · Zbl 0542.76093
[5] Johnson, C. and Saranen, J. Streamline diffusion methods for the incompressible Euler and Navier-Stokes equations. Mathematics of Computation, 47, 1–18 (1986) · Zbl 0609.76020
[6] Luo, Z. D. and Lu, X. M. A least squares Galerkin/Petrov mixed finite element method for the stationary conduction convection problems. Mathematic Numerica Sinica, 25(2), 231–244 (2003)
[7] Sun, P., Luo, Z. D., and Chen, J. A Petrov least squares mixed finite element method for the nonstationary conduction convection problems. Mathematica Numerica Sinica, 31(1), 87–98 (2009) · Zbl 1199.65342
[8] Luo, Z. D., Chen, J., Navon, I. M., and Zhu, J. An optimizing reduced PLSMFE formulation for non-stationary conduction-convection problems. International Journal for Numerical Methods in Fluids, 60, 409–436 (2009) · Zbl 1161.76032
[9] Mesquita, M. S. and de Lemos, M. J. S. Optimal multigrid solutions of two dimensional convection conduction problems. Applied Mathematics and Computation, 152(3), 725–742 (2004) · Zbl 1077.65508
[10] Wang, Q. W., Yang, M., and Tao, W. Q. Natural convection in a square enclosure with an internal isolated vertical plate. Warme-Stoffubertrag, 29, 161–169 (1994)
[11] Yang, M., Tao, W. Q., Wang, Q. W., and Lue, S. S. On identical problems of natural convection in enclosure and applications of the identity character. Journal of Thermal Science, 2(2), 116–125 (1993)
[12] Luo, Z. D. The Bases and Applications of Mixed Finite Element Methods (in Chinese), Chinese Science Press, Beijing (2006)
[13] Luo, Z. D., Zhu, J., Xie, Z. H., and Zhang, G. F. A difference scheme and numerical simulation based on mixed finite element method for natural convection problem. Applied Mathematics and Mechanics (English Edition), 24(9), 973–983 (2003) DOI 10.1007/BF02437642
[14] Si, Z. Y. and He, Y. N. A coupled Newton iterative mixed finite element method for stationary conduction convection problems. Computing, 89(1–2), 1–25 (2010) · Zbl 1206.65249
[15] Zhang, Y. Z., Hou, Y. R., and Zou, H. L. A posteriori error estimation and adaptive computation of conduction convection problems. Applied Mathematical Modelling, 35, 2336–2347 (2011) · Zbl 1217.76040
[16] Babuska, I. and Strouboulis, T. The Finite Element Method and Its Reliability, Oxford University Press, London (2001)
[17] Chen, Z. X. Finite Element Methods and Their Applications, Springer-Verlag, Heidelberg (2005) · Zbl 1082.65118
[18] Zheng, H. B., Hou, Y. R., and Shi, F. Adaptive variational multiscale methods for incompressible flow based on two local Gauss integrations. Journal of Computational Physics, 229(19), 7030–7041 (2010) · Zbl 1425.76067
[19] Zheng, H. B., Hou, Y. R., and Shi, F. A posteriori error estimates of stabilization of low-order mixed finite elements for incompressible flow. SIAM Journal on Scientific Computing, 32, 1346–1361 (2010) · Zbl 1410.76206
[20] Luo, Z. D. and Zhu, J. A nonlinear Galerkin mixed element method and a posteriori error estimator for the stationary Navier-Stokes equations. Applied Mathematics and Mechanics (English Edition), 23(10), 1194–1206 (2002) DOI 10.1007/BF02437668 · Zbl 1143.76481
[21] Zhang, Y. Z. and Hou, Y. R. Posteriori analysis of unsteady Navier-Stokes equations with the coriolis force. Dynamics of Continuous, Discrete and Impulsive Systems, Series B, 18(2), 229–244 (2011)
[22] Ervin, V. J., Layton, W. J., and Maubach, J. M. An adaptive defect correction method for viscous incompressible flow problems. SIAM Journal on Numerical Analysis, 37, 1165–1185 (2000) · Zbl 1049.76038
[23] Verfürth, R. A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley Teubner, New York (1996) · Zbl 0853.65108
[24] Berrone, S. Adaptive discretization of stationary and incompressible Navier-Stokes equations by stabilized finite element methods. Computer Methods in Applied Mechanics and Engineering, 190, 4435–4455 (2001)
[25] Verfürth, R. A posteriori error estimates for nonlinear problems, finite element discretizations of elliptic equations. Mathematics of Computation, 62, 445–475 (1994) · Zbl 0799.65112
[26] Ervin, V. J. and Louis, N. N. A posteriori error estimation and adaptive computation of viscoelastic fluid flows. Numerical Methods for Partial Differential Equations, 21, 297–322 (2005) · Zbl 1141.76434
[27] Du, Q. and Zhang, J. Adaptive finite element method for a phase field bending elasticity model of vesicle membrane deformations. SIAM Journal on Scientific Computing, 30(3), 1634–1657 (2008) · Zbl 1162.74042
[28] Verfürth, R. A posteriori error estimators for the Stokes equations. Numerische Mathematik, 55, 309–325 (1989) · Zbl 0674.65092
[29] Adams, R. Sobolev Space, Pure and Applied Mathematics, Vol. 65, Academic Press, New York (1975)
[30] Hou, Y. R. and Mei, L. Q. Full discrete two-level correction scheme for Navier-Stokes equations. Journal of Computational Mathematics, 26(2), 209–226 (2008) · Zbl 1174.76009
[31] He, Y. N. and Li, J. Convergence of three iterative methods based on the finite element discretization for the stationary Navier-Stokes equations. Computer Methods in Applied Mechanics and Engineering, 198, 1351–1359 (2009) · Zbl 1227.76031
[32] Temam, R. Navier-Stokes Equation: Theory and Numerical Analysis, 3rd ed., North-Holland Publishing Co., Amsterdam/New York (1984) · Zbl 0568.35002
[33] Clément, P. Approximation by finite element functions using local regularization. RAIRO Analyse Numérique, 9(2), 77–84 (1975)
[34] Hecht, F., Pironneau, O., Hyaric, A. L., and Ohtsuka, K. FreeFem++, Preprint at < http://www.freefem.org/ff++ > (2008)
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