An efficient two-step algorithm for the incompressible flow problem

Abstract

A new two-step stabilized finite element method for the 2D/3D stationary Navier-Stokes equations based on local Gauss integration is introduced and analyzed in this paper. The method consists of solving one Navier-Stokes problem based on the P 1 − P 1 finite element pair and then solving a general Stokes problem based on the P 2 − P 2 finite element pair, i.e., computes a lower order predictor and a higher order corrector. Moreover, the stability and convergence of the present method are deduced, which show that the new method provides an approximate solution with the convergence rate of the same order as the P 2 − P 2 stabilized finite element solution solving the Navier-Stokes equations on the same mesh width. However, our method can save a large amount of computational time. Finally, numerical tests confirm the theoretical results of the method.

Stabilized mixed finite element methods for the Navier‐Stokes equations with damping

“…Two‐level method has also been applied to many other nonlinear equations, such as the Navier‐Stokes equations, the natural convection equations, the reaction diffusion equations, the miscible displacement equations, the Cahn‐Hilliard equations, and the nonlinear parabolic equation . In Huang et al, a novel two‐step method was proposed for the Navier‐Stokes equations, which uses a lower‐order element to solve one nonlinear system as an initial approximation and then uses a higher‐order element to solve one linear system. In fact, the two‐step method can be cast in the framework of the two‐level method, but compared with the two‐level method, the two‐step method only needs one mesh size, which can avoid the discussion on the relation of the coarse and fine meshes.…”

Stability and convergence of two‐grid Crank‐Nicolson extrapolation scheme for the time‐dependent natural convection equations

“…The main contributions can be list as follows. Compared with He, He and Sun, Ammi and Marion, Dai and Cheng, He, and Huang et al, our model not only inherits all the difficulties of the Navier‐Stokes equations but also increases the coupling among the variables, how to deal with the nonlinear terms and reduce the computational scale efficiently are some meaningful topics. Compared with Su et al and Zhang and Hou, we not only provide some novel stability results of the Crank‐Nicolson extrapolation scheme but also establish the theoretical analysis of the two‐grid Crank‐Nicolson extrapolation scheme for the time‐dependent natural convection problem. Optimal error estimates of the numerical solutions are established both in one grid and two‐grid Crank‐Nicolson extrapolation schemes for the 2D/3D cases. …”

“…This means that solving a nonlinear equation is not much more difficult than solving a linear equation because , the work for solving u H is relatively very small. Later on, the two‐grid method was considered by many authors, we just refer to the eigenvalue problems, the nonlinear elliptic problems, the nonlinear parabolic equations, the Navier‐Stokes equations as the examples. Now, the two‐grid method has been shown to be an efficient scheme for solving the nonlinear problems of various types.…”

Two‐grid stabilized algorithms for the steady Navier–Stokes equations with damping

“…Recently, many investigations have focused on stabilization of the lowest equal-order elements pair using the projection of the pressure onto the piecewise constant space (see, e.g., other works [27][28][29][30][31][32][33] ) and of the quadratic equal-order elements pair using the projection of the gradient of pressure onto the piecewise constant space (cf. other studies [34][35][36][37][38] ). Most attractive features of these stabilized methods are free of parameter, avoiding higher-order derivatives or edge-based data structures, and stabilization being completely local at the element level.…”