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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演化域中线性化纳维-斯托克斯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纳维-斯托克斯问题。在本研究中，假设领域的演化是已知的并且独立于当前问题的解决方案。研究中采用的数值方法将标准后向微分公式型时间步进程序与几何不拟合的有限元离散技术相结合。此外，Nitsche 方法用于强制执行边界条件。本文提出了几个 inf-sup 稳定的速度-压力元素的收敛估计。该估计表明速度分量的能量范数和压力分量的缩放 $L^{2}(H^{1})$ 型范数的最优阶收敛。