zbMATH — the first resource for mathematics

A posteriori error estimation and adaptive computation of conduction convection problems. (English) Zbl 1217.76040
Summary: An adaptive finite element method is developed for stationary conduction convection problems. Using a mixed finite element formulation, residual type a posteriori error estimates are derived by means of the general framework of R. Verfürth. The effectiveness of the adaptive method is further demonstrated through two numerical examples. The first example is problem with known solution and the second example is a physical model of square cavity stationary flow.

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
76R10 Free convection
80A20 Heat and mass transfer, heat flow (MSC2010)
Full Text: DOI
[1] Luo, Z.; Lu, X., A least squares Galerkin/Petrov mixed finite element method for the stationary conduction – convection problems, Math. numer. sin., 25, 31-244, (2003)
[2] Luo, Z.; Lu, X., A nonlinear Galerkin/Petrov least squares mixed finite element method for the stationary conduction – convection problems, Math. numer. sin., 25, 447-462, (2003)
[3] Mesquita, M.S.; de Lemos, M.J.S., Optimal multigrid solutions of two-dimensional convection – conduction problems, Appl. math. comput., 152, 725-742, (2004) · Zbl 1077.65508
[4] Wang, Q.W.; Yang, M.; Tao, W.Q., Natural convection in a square enclosure with an internal isolated vertical plate, Warme-stoffubertrag, 29, 161-169, (1994)
[5] Yang, M.; Tao, W.Q.; Wang, Q.W.; Lue, S.S., On identical problems of natural convection in enclosure and applications of the identity character, J. thermal sci., 2, 116-125, (1993)
[6] Luo, Z., Mixed finite element foundation and its application, (2006), Science Press Beijing, in Chinese
[7] Luo, Z.; Chen, J.; Navon, I.M.; Zhu, J., An optimizing reduced PLSMFE formulation for non-stationary conduction – convection problems, Int. J. numer. meth. fluids, 60, 409-436, (2009) · Zbl 1161.76032
[8] Luo, Z.; Wang, L., 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, (1998)
[9] Reddy, J.N.; Gartling, D.K., The finite element method in heat transfer and fluid dynamics, (2001), CRC Pess Washington · Zbl 0855.76002
[10] Eriksson, K.; Johnson, C., Adaptive streamline diffusion finite element methods for stationary convection – diffusion problems, Math. comput., 60, 167-188, (1993) · Zbl 0795.65074
[11] Zheng, H.; Hou, Y.; Shi, F., Adaptive variational multiscale methods for incompressible flow based on two local Gauss integrations, J. comput. phys., 229, 7030-7041, (2010) · Zbl 1425.76067
[12] Zheng, H.; Hou, Y.; Shi, F., A posteriori error estimates of stabilization of low-order mixed finite elements for incompressible flow, SIAM J. sci. comp., 32, 1346-1361, (2010) · Zbl 1410.76206
[13] Ervin, V.J.; Layton, W.J.; Maubach, J.M., An adaptive defect correction method for viscous incompressible flow problems, SIAM J. numer. anal., 37, 1165-1185, (2000) · Zbl 1049.76038
[14] Verfürth, R., A review of a posteriori error estimation and adaptive mesh-refinement techniques, (1996), Wiley Teubner · Zbl 0853.65108
[15] Ainsworth, M.; Oden, J.T., A posteriori error estimation in finite element analysis, (2000), Wiley Interscience New York · Zbl 1008.65076
[16] Berrone, S., Adaptive discretization of stationary and incompressible navier – stokes equations by stabilized finite element methods, comput, Meth. appl. mech. eng., 190, 4435-4455, (2001)
[17] Verfürth, R., A posteriori error estimates for nonlinear problems, Finite elem. discretizations elliptic equ. math. comput., 62, 445-475, (1994) · Zbl 0799.65112
[18] Ervin, V.J.; Ntasin, Louis N., A posteriori error estimation and adaptive computation of viscoelastic fluid flows, Numer. methods partial differential eq., 21, 297-322, (2005) · Zbl 1141.76434
[19] Du, Q.; Zhang, J., Adaptive finite element method for a phase field bending elasticity model of vesicle membrane deformations, SIAM J. sci. comput., 30, 3, 1634-1657, (2008) · Zbl 1162.74042
[20] Adams, R., Sobolev space, Pure and applied mathematics, vol. 65, (1975), Academic press New York
[21] He, Y.; Li, J., 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, (2009) · Zbl 1227.76031
[22] Temam, R., Navier – stokes equation: theory and numerical analysis, (1984), North-Holland Amsterdam, New York, Oxford
[23] Clément, P., Approximation by finite elements using local regularization, RAIRO anal. numér., 2, 77-84, (1975) · Zbl 0368.65008
[24] F.Hecht, O.Pironneau, A. Le Hyaric, K. Ohtsuka, FreeFem++, 2008. <http://www.freefem.org/ff>++.
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.