This paper studies fully discrete finite element approximations to the Navier-Stokes equations using inf-sup stable elements and grad-div stabilization. For the time integration two implicit-explicit second order backward differentiation formulae (BDF2) schemes are applied. In both the laplacian is implicit while the nonlinear term is explicit, in the first one, and semi-implicit, in the second one. The grad-div stabilization allow us to prove error bounds in which the constants are independent of inverse powers of the viscosity. Error bounds of order $r$ in space are obtained for the $L^2$ error of the velocity using piecewise polynomials of degree $r$ to approximate the velocity together with second order bounds in time, both for fixed time step methods and for methods with variable time steps. A CFL-type condition is needed for the method in which the nonlinear term is explicit relating time step and spatial mesh sizes parameters.
翻译:本文使用 Inf- sup 稳定元素和 grad- div 稳定化 来研究与 Navier- Stokes 等式完全离散的有限元素近似值。 对于时间整合,应用了两个隐含的第二顺序后向偏差公式(BDF2) 。 在 lapacian 中, 隐含了非线性词, 而第二个则隐含了非线性词, 而第二个则隐含了。 分级稳定化让我们能够证明, 常数与粘度反动力不相容的错误界限 。 空间顺序值$( $) 的错误界限是: 速度误差, 速度误差是使用小巧的多级多级方块值$($) 来接近速度, 与第二顺序相近, 时间相近于固定时间步法和可变时间步法, 需要 CFLL- 类型条件, 用于非线性术语明确与时间步骤和空间网格参数有关的方法 。