A split-step finite-element method for incompressible Navier-Stokes equations with high-order accuracy up-to the boundary

Abstract

An efficient and accurate finite-element algorithm is described for the numerical solution of the incompressible Navier-Stokes (INS) equations. The new algorithm that solves the INS equations in a velocity-pressure reformulation is based on a split-step scheme in conjunction with the standard finite-element method. The split-step scheme employed for the temporal discretization of our algorithm completely separates the pressure updates from the solution of velocity variables. When the pressure equation is formed explicitly, the algorithm avoids solving a saddle-point problem; therefore, our algorithm has more flexibility in choosing finite-element spaces. In contrast, popular mixed finite-element methods that solve the INS equations in the primitive variables (or velocity-divergence formulation) lead to discrete saddle-point problems whose solution depends on the choice of finite-element spaces for velocity and pressure that is subject to the wellknown Ladyzenskaja-Babuška-Brezzi (LBB) (or inf-sup) condition. For efficiency and robustness, Lagrange (piecewise-polynomial) finite elements of equal order for both velocity and pressure are used. Accurate numerical boundary condition for the pressure equation is also investigated. Motivated by a post-processing technique that calculates derivatives of a finite element solution with super-convergent error estimates, an alternative numerical boundary condition is proposed for the pressure equation at the discrete level. The new numerical pressure boundary condition that can be regarded as a better implementation of the compatibility boundary condition improves the boundary-layer errors of the pressure solution. A normal-mode analysis is performed using a simplified model problem on a uniform mesh to demonstrate the numerical properties of our methods. Convergence studies using P 1 elements support the analytical results and demonstrates that our algorithm with the new numerical boundary condition achieves the optimal second-order accuracy for both velocity and pressure up-to the boundary. Benchmark problems are also computed and carefully compared with existing studies. Finally, as an example to illustrate that our approach can be easily adapted for higher-order finite elements, we solve the classical flow-past-a-cylinder problem using P n finite elements with n ≥ 1.

Pressure Stabilization Strategies for a LES Filtering Reduced Order Model

“…The reader interested in enforcing non-homogeneous Neumann conditions for the pressure fields is referred to [55,56]. We remark that systems ( 9) and ( 10) and ( 17) and ( 14) are not equivalent to systems ( 9) and ( 46) and ( 17) and ( 48) for steady flows [55][56][57].…”

A global residual‐based stabilization for equal‐order finite element approximations of incompressible flows

“…Although their proof requires ∇p − Δu ∈ [L 2 (Ω)] d , which is still more restrictive than standard weak formulations, there is substantial numerical evidence that such PPE-based methods provide accurate approximations even for problems with singularities. 23,24,[37][38][39][40][41][42]…”

“…A slightly less restrictive proof considering the weak problem was presented by Sani et al 36 for the “pure PPE” () and can be extended to the present case. Although their proof requires , which is still more restrictive than standard weak formulations, there is substantial numerical evidence that such PPE‐based methods provide accurate approximations even for problems with singularities 23,24,37‐42 …”

“…We remark, however, that our numerical experiments with highly nonuniform meshes ( C > 10) and even adaptive ones for different values of do not indicate any apparent signs of stability restrictions. In fact, “pure PPE” formulations such as that by Johnston and Liu 22 also have a “nonvanishing” divergence in the bilinear form and, although a stability proof for C 0 finite elements is also still open in their case, 39 several variations of their method have been used successfully in the past decade in the solution of challenging problems with general meshes, 23,39,42 corner singularities 37,38,40 and wide ranges of Reynolds numbers 24,41 —all providing strong numerical evidence towards good stability properties.…”

Efficient split-step schemes for fluid-structure interaction involving incompressible generalised Newtonian flows

“…There is an intermediate alternative available, though: As done by Liu [102] in the original scheme, we apply Leray projection on the past velocities only, effectively skipping the L 2 -projection step to obtain ǔ f , and thereby fulfill the Dirichlet boundary conditions on the velocity u f exactly. This technique is also referred to as divergence damping [134,135] and has been shown to effectively reduce mass conservation errors and improve overall stability, while being cheaper than standard Leray projection and preserving boundary conditions on the velocity. Then, we construct the time-discrete weak form of the split-step scheme using BDF-m schemes of the form…”