zbMATH — the first resource for mathematics

A posteriori error estimation for a defect correction method applied to conduction convection problems. (English) Zbl 1364.76104
Summary: We present a posteriori error estimate for a defect correction method for approximating solutions of the stationary conduction convection problems in two dimension. The defect correction method is aiming at small viscosity \(\nu\). A reliable a posteriori error estimation is derived for the defect correction method. Finally, two numerical examples validate our theoretical results. The first example is a 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
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
Full Text: DOI
[1] Luo, A least squares Galerkin/Petrov mixed finite element method for the stationary conduction-convection problems, Math Numer Sin 25 pp 231– (2003)
[2] Luo, A nonlinear Galerkin/Petrov least squares mixed finite element method for the stationary conduction-convection problems, Math Numer Sin 25 pp 447– (2003)
[3] Wang, Natural convection in a square enclosure with an internal isolated vertical plate, Warme-Stoffubertrag 29 pp 161– (1994)
[4] Yang, On identical problems of natural convection in enclosure and applications of the identity character, J Thermal Sci 2 pp 116– (1993)
[5] Luo, Mixed finite element foundation and its application (2006)
[6] Si, A defect-correction mixed finite element method for stationary conduction-convection problems, Math Problems Eng 2011 pp 28– (2011) · Zbl 1204.76021
[7] Luo, An optimizing reduced PLSMFE formulation for non-stationary conduction-convection problems, Int J Numer Methods Fluids 60 pp 409– (2009) · Zbl 1161.76032
[8] Si, A defect-correction method for unsteady conduction convection problems I: spatial discretization, Sci China Math 54 pp 185– (2011) · Zbl 1227.76040
[9] Reddy, The finite element method in heat transfer and fluid dynamics (2001)
[10] Ervin, A study of defect correction finite difference methods for convection dominated convection diffusion equations, SIAM J Numer Anal 26 pp 169– (1989) · Zbl 0672.65063
[11] Eriksson, Adaptive streamline diffusion finite element methods for stationary convection-diffusion problems, Math Comput 60 pp 167– (1993) · Zbl 0795.65074
[12] Cawood, Adaptive defect correction methods for convection dominated, convection diffusion problems, J Comput Appl Math 116 pp 1– (2000) · Zbl 0979.65096
[13] Ervin, An adaptive defect correction method for viscous incompressible flow problems, SIAM J Numer Anal 37 pp 1165– (2000) · Zbl 1049.76038
[14] Linss, A posteriori error estimation for a defect-correction method applied to convection-diffusion problems, Int J Num Anal Model 7 pp 718– (2010) · Zbl 1197.65100
[15] Zhang, Posteriori analysis of unsteady Navier-Stokes equations with the coriolis force, Dyn Continuous Discrete Impulsive Syst, Ser B 18 pp 229– (2011)
[16] Verfürth, A posteriori error estimates for nonlinear problems, finite element discretizations of elliptic equations, Math Comput 62 pp 445– (1994) · Zbl 0799.65112
[17] Verfürth, A review of a posteriori error estimation and adaptive mesh-refinement techniques (1996) · Zbl 0853.65108
[18] Ainsworth, A posteriori error estimation in finite element analysis (2000) · Zbl 1008.65076
[19] Ervin, A posteriori error estimation and adaptive computation of viscoelastic fluid flows, Numer Methods Partial Differential Equations 21 pp 297– (2005) · Zbl 1141.76434
[20] Du, Adaptive finite element method for a phase field bending elasticity model of vesicle membrane deformations, SIAM J Sci Comput 30 pp 1634– (2008) · Zbl 1162.74042
[21] Zheng, Adaptive variational multiscale methods for incompressible flow based on two local Gauss integrations, J Comput Phys 229 pp 7030– (2010) · Zbl 1425.76067
[22] Zheng, A posteriori error estimates of stabilization of low-order mixed finite elements for incompressible flow, SIAM J Sci Comput 32 pp 1346– (2010) · Zbl 1410.76206
[23] Zhang, A posteriori error estimation and adaptive computation of conduction convection problems, Appl Math Model 35 pp 2336– (2011) · Zbl 1217.76040
[24] Zhang, Adaptive least squares Galerkin/Petrov finite element method for the stationary conduction convection problems, Appl Math Mech 32 pp 1269– (2011) · Zbl 1237.76078
[25] Lee, Analysis of a defect correction method for viscoelastic fluid flow, Comput Math Appl 48 pp 1213– (2004) · Zbl 1063.76056
[26] Ervin, Defect correction method for viscoelastic flows at high Weissenberg number, Numer Methods Partial Differential Equations 22 pp 145– (2006) · Zbl 1080.76037
[27] Ervin, A two-parameter defect-correction method for computation of steady-state viscoelastic fluid flow, Appl Math Comp 196 pp 818– (2008) · Zbl 1132.76033
[28] Zhang, Defect correction method for time dependent viscoelastic fluid flow, Int J Comput Math 88 pp 1546– (2011) · Zbl 1331.76076
[29] Zhang, A defect-correction method for time-dependent viscoelastic fluid flow based on SUPG formulation, Discrete Dyn Nat Soc, 2011 pp 25– (2011) · Zbl 1284.76260
[30] Frank, The application of iterated defect correction to variational methods for elliptic boundary value problems, Computing 30 pp 121– (1983) · Zbl 0498.65051
[31] Gracia, A defect-correction parameter-uniform numerical method for a singularly perturbed convection diffusion problem in one dimension, Numer Algorithms 41 pp 359– (2006) · Zbl 1098.65083
[32] Labovschii, A defect correction method for the time-dependent Navier-Stokes equations, Numer Methods Partial Differential Equations 25 pp 1– (2009) · Zbl 1394.76087
[33] Clément, Approximation by finite elements using local regularization, RAIRO Anal NumJW-NUMT1200160x.png@Picture{acute} r 2 pp 77– (1975)
[34] Adams, Sobolev space (1975)
[35] Temam, Navier-Stokes equation, theory and numerical analysis (1983) · Zbl 0522.35002
[36] Girault, Finite element method for Navier-Stokes equations, theory and algorithms (1987)
[37] F. Hecht O. Pironneau A. LeHyaric K. Ohtsuka FreeFem ++ 2008 http://www.freefem.org/ff++.
[38] Verfurth, A posteriori error estimators for the Stokes equations, Numer Math 55 pp 309– (1989) · Zbl 0674.65092
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.