Stabilised mixed velocity-pressure formulations are one of the widely-used finite element schemes for computing the numerical solutions of laminar incompressible Navier-Stokes. In these formulations, the Newton-Raphson scheme is employed to solve the nonlinearity in the convection term. One fundamental issue with this approach is the computational cost incurred in the Newton-Raphson iterations at every load/time step. In this paper, we present an iteration-free mixed-stabilised finite element formulation for incompressible Navier-Stokes that preserves second-order temporal accuracy for both velocity and pressure fields. We prove the second-order temporal accuracy using an example with a manufactured solution. We then illustrate the accuracy and the computational benefits of the proposed scheme by studying the benchmark example of flow past a fixed circular cylinder and then using two benchmark examples in fluid-flexible structure interaction.

中文翻译：

---

层流不可压缩 Navier-Stokes 的精确无迭代混合稳定公式：流固耦合的应用

稳定混合速度-压力公式是计算层流不可压缩 Navier-Stokes 数值解的广泛使用的有限元方案之一。在这些公式中，采用 Newton-Raphson 方案来解决对流项中的非线性问题。这种方法的一个基本问题是在每个负载/时间步长的 Newton-Raphson 迭代中产生的计算成本。在本文中，我们提出了一种用于不可压缩 Navier-Stokes 的无迭代混合稳定有限元公式，该公式保持速度场和压力场的二阶时间精度。我们使用带有制造解决方案的示例来证明二阶时间精度。