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A mixed continuous/discontinuous finite element discretization of the incompressible NS equations

机译：不可压缩NS方程的混合连续/不连续有限元离散化

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摘要

A projection scheme for the numerical solution of the incompressible Navier-Stokes equations is presented. Finite element discontinuous Galerkin (dG) discretization for the velocity in the momentum equations is employed. The incompressibility constraint is enforced by numerically solving the Poisson equation for the pressure by using a continuous Galerkin (cG) discretization. The main advantage of the method is that it does not require the velocity and pressure approximation spaces to satisfy the usual inf-sup condition, thus equal order finite element approximations for both velocity and pressure can be used. Furthermore, by using cG discretization for the Poisson equation, no auxiliary equations are needed as it is required for dG approximations of second order derivatives. In order to enable large time steps for time marching to steady-state and time evolving problems, implicit schemes are used in connection with high order implicit RK methods. Numerical tests demonstrate that the overall scheme is accurate and computationally efficient.
机译：提出了不可压缩的Navier-Stokes方程数值解的投影方案。动量方程中的速度采用了有限元不连续Galerkin（dG）离散化方法。通过使用连续Galerkin（cG）离散化对压力的泊松方程进行数值求解，可以增强不可压缩性约束。该方法的主要优点是它不需要速度和压力近似空间来满足通常的注入条件，因此可以使用速度和压力的等阶有限元近似。此外，通过对泊松方程使用cG离散化，不需要辅助方程，因为它对于二阶导数的dG逼近是必需的。为了使较大的时间步长可以进入到稳态和时间演变问题，将隐式方案与高阶隐式RK方法结合使用。数值测试表明，该总体方案是准确的，并且计算效率高。

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