A defect correction method for the time‐dependent Navier‐Stokes equations

Abstract

A method for solving the time dependent Navier-Stokes equations, aiming at higher Reynolds' number, is presented. The direct numerical simulation of flows with high Reynolds' number is computationally expensive. The method presented is unconditionally stable, computationally cheap, and gives an accurate approximation to the quantities sought. In the defect step, the artificial viscosity parameter is added to the inverse Reynolds number as a stability factor, and the system is antidiffused in the correction step. Stability of the method is proven, and the error estimations for velocity and pressure are derived for the one-and two-step defect-correction methods. The spacial error is O(h) for the one-step defect-correction method, and O(h 2 ) for the two-step method, where h is the diameter of the mesh. The method is compared to an alternative approach, and both methods are applied to a singularly perturbed convection-diffusion problem. The numerical results are given, which demonstrate the advantage (stability, no oscillations) of the method presented.

Error analysis of a fully discrete finite element variational multiscale method for time‐dependent incompressible Navier–Stokes equations

“…As an improvement of the artificial viscosity method, the defect-correction methods in general consist of an initial defect step followed by several correction steps. In the defect step, an artificial viscosity stabilized Navier-Stokes problem is first solved, and the system is then antidiffused in the correction steps where a linear problem is solved at each step; see, for example, [10,12] for the steady Navier-Stokes equations and [14] for the unsteady Navier-Stokes equations. The subgrid stabilization methods are generally based on the notion of scale separation, which assumes that there exist large scales and small scales of the flow.…”

“…There are many stabilized methods for the simulation of high Reynolds number flows in literature, for example, the artificial viscosity method [9], the defect-correction methods [10][11][12][13][14], the subgrid stabilization methods [15][16][17][18][19][20][21][22], and the variational multiscale methods [23][24][25][26][27], among others. The artificial viscosity method adds an artificial viscosity to the inverse Reynolds number as a stability factor.…”

Second order modified method of characteristics mixed defect-correction finite element method for time dependent Navier–Stokes problems

“…Because it is highly efficient, there are many works devoted to this method, e.g. the convection-diffusion equation [17], adaptive refinement for the convection-diffusion problems [2], adaptive defect correction methods for the viscous incompressible flow [18], two-parameter defectcorrection method for computation of the steady-state viscoelastic fluid flow [20], variational methods for the elliptic boundary value problems [21], defectcorrection parameter-uniform numerical method for a singularly perturbed convection-diffusion problem [28], the convection-dominated flow [33], finite volume local defect correction method for solving the transport equation [34], the singular initial value problems [35], the time-dependent Navier-Stokes equations [36,37,41], the stationary Navier-Stokes equation [39], second order defect correction scheme [42], finite element eigenvalues with applications to quantum chemistry [46] and so on. In [21], Frank et al give a method which makes it possible to apply the idea of iterated defect-correction to finite element methods.…”

On the Linearization of Defect-Correction Method for the Steady Navier-Stokes Equations

“…Subsequently, the method was combined with the adaptive technique [5], the subgrid stabilization method [16], and the twolevel method [21]. It was also applied to the viscoelastic fluid flow problems [20], the time-dependent Navier-Stokes equations [17], and the time-dependent conduction-convection problem [22,23]. We refer to [16] and [22] for a literature review for the defect-correction method.…”

“…We analyze the stability and convergence of the developed algorithms, and provide error estimates with respect to the mesh size h, the kinematic viscosity ν, the stability factor α and the number of nonlinear iterations m for the discrete velocity and pressure for onestep defect-correction algorithms. Moreover, it was commonly believed that s-step corrections in the defect-correction method are enough from the optimally asymptotical error point of view (see, e.g., [16,17]), where s is the local polynomial degree of finite element subspace for the velocity. However, our present study shows that in small λ cases (see (2.5) for the definition of λ), a little more than s steps corrections are necessary for a good approximate solution (see Remark 4.5 and the numerical results in Section 5.3).…”