J. Korean Math. Soc. 2013; 50(5): 1129-1163
Printed September 1, 2013
https://doi.org/10.4134/JKMS.2013.50.5.1129
Copyright © The Korean Mathematical Society.
Yueqiang Shang, Do Wan Kim, and Tae-Chang Jo
Southwest University, Inha University, Inha University
Based on finite element discretization, two linearization approaches to the defect-correction method for the steady incompressible Navier-Stokes equations are discussed and investigated. By applying $m$ times of Newton and Picard iterations to solve an artificial viscosity stabilized nonlinear Navier-Stokes problem, respectively, and then correcting the solution by solving a linear problem, two linearized defect-correction algorithms are proposed and analyzed. Error estimates with respect to the mesh size $h$, the kinematic viscosity $\nu$, the stability factor $\alpha$ and the number of nonlinear iterations $m$ for the discrete solution are derived for the linearized one-step defect-correction algorithms. Efficient stopping criteria for the nonlinear iterations are derived. The influence of the linearizations on the accuracy of the approximate solutions are also investigated. Finally, numerical experiments on a problem with known analytical solution, the lid-driven cavity flow, and the flow over a backward-facing step are performed to verify the theoretical results and demonstrate the effectiveness of the proposed defect-correction algorithms.
Keywords: Navier-Stokes equations, finite element, defect-correction meth\-od, Newton iteration, Picard iteration
MSC numbers: 65N15, 65N30, 76D05, 76M10
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