The paper addresses an error analysis of an Eulerian finite element method used for solving a linearized Navier–Stokes problem in a time-dependent domain. In this study, the domain’s evolution is assumed to be known and independent of the solution to the problem at hand. The numerical method employed in the study combines a standard backward differentiation formula-type time-stepping procedure with a geometrically unfitted finite element discretization technique. Additionally, Nitsche’s method is utilized to enforce the boundary conditions. The paper presents a convergence estimate for several velocity–pressure elements that are inf-sup stable. The estimate demonstrates optimal order convergence in the energy norm for the velocity component and a scaled $L^{2}(H^{1})$-type norm for the pressure component.

中文翻译：

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用于演化域中线性化 Navier-Stokes 问题的欧拉有限元方法


本文讨论了欧拉有限元方法的误差分析，该方法用于求解瞬态域中的线性 Navier-Stokes 问题。在这项研究中，假设该领域的演变是已知的，并且与手头问题的解决方案无关。该研究中采用的数值方法将标准的向后微分公式类型的时间步进程序与几何未拟合的有限元离散化技术相结合。此外，Nitsche 的方法还用于强制执行边界条件。本文提出了几种 inf-sup 稳定的速度-压力元件的收敛估计。该估计证明了速度分量的能量模中最优阶收敛，以及压力分量的 $L^{2}（H^{1}）$ 型模。