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We present and study a parallel subgrid stabilized algorithm for simulating the incompressible Navier-Stokes equations with nonlinear slip boundary conditions. The algorithm uses finite element discretizations and an approach of completely overlapping region division for the parallelization, where an elliptic projection operator is applied to define the stabilization term. It has the following appealing features: 1) each subproblem used to calculate a local solution in a subdomain is actually a global problem defined on a global mesh that is locally refined around the subdomain; 2) it can re-use existing sequential software in coding, where both parallelization and stabilization are easy to implement based on existing sequential codes without extensive effort; 3) it inherits the advantages of the subgrid stabilization method and the completely overlapping region division approach; 4) with suitable algorithmic parameters, it is able to yield an optimal convergence rate for the approximate solutions with a comparable accuracy to that of the solutions from the global subgrid stabilized method, with considerable reduction in computational time. With the help of a local a priori estimate, we estimate error bounds of the obtained solutions from the algorithm, and perform some numerical tests to validate the theoretical prediction and illustrate the applicability of the proposed algorithm. [ABSTRACT FROM AUTHOR]

This paper introduces a three‐step Oseen‐linearized finite element method for the 2D/3D steady incompressible Navier–Stokes equations with nonlinear damping term. Within this method, we solve a nonlinear problem over a coarse grid followed by solving two Oseen‐linearized problems over a fine grid, which possess the same stiffness matrices with only various right‐hand sides. We theoretically analyze the stability of the present method, and derive optimal error estimates of the finite element solutions. We conduct a series of numerical experiments which support the theoretical analysis and test the effectiveness of the proposed method. We demonstrate numerically that there is a significant improvement in the accuracy of the approximate solutions over those for the standard two‐level method. [ABSTRACT FROM AUTHOR]

This paper presents and studies three two‐grid stabilized quadratic equal‐order finite element algorithms based on two local Gauss integrations for the steady Navier–Stokes equations with damping. In these algorithms, we first solve a stabilized nonlinear problem on a coarse grid, and then pass the coarse grid solution to a fine grid and solve a stabilized linear problem. Using some nonlinear analysis techniques, we analyze stability of the algorithms and derive optimal order error estimates of the approximate solutions. Theoretical and numerical results show that, when the algorithmic parameters are chosen appropriately, the accuracy of the approximate solutions computed by our two‐grid stabilized algorithms is comparable to that of solving a fully stabilized nonlinear problem on the same fine grid; however, our two‐grid algorithms save a large amount of CPU time than the one‐grid stabilized algorithm. [ABSTRACT FROM AUTHOR]

This paper is concerned with a parallel stabilized quadratic equal-order finite element algorithm for the steady incompressible Navier–Stokes equations where a fully overlapping domain decomposition is used for parallelization. The basic idea of the studied algorithm is that each processor independently calculates a local stabilized solution in an interesting subdomain on a multiscale mesh that is fine around the subdomain and coarse on the rest of subdomain. The present algorithm can be solved simultaneously by the existing Navier–Stokes codes without substantial recoding. In the light of local a priori estimate for stabilized finite element solution, we analyse the error bound of the stabilized solution. Moreover, several parallel iterative stabilized quadratic equal-order finite element algorithms are presented, and three numerical examples which confirm the high efficiency of the parallel stabilized algorithms are provided. [ABSTRACT FROM AUTHOR]