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 (BDF)-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.

翻译：本文针对在随时间变化的域中求解线性化Navier-Stokes问题所采用的欧拉有限元方法进行了误差分析。在本研究中，假设域的演化已知且独立于待求解问题。研究所采用的数值方法将标准后向微分公式（BDF）型时间步进程序与几何非拟合有限元离散化技术相结合。此外，利用Nitsche方法施加边界条件。本文给出了若干inf-sup稳定的速度-压力单元的收敛性估计。该估计表明速度分量在能量范数下、压力分量在标度化$L^2(H^1)$型范数下均具有最优阶收敛性。