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：本文针对演化域中线性化Navier–Stokes问题所采用的欧拉有限元方法进行误差分析。研究中假定演化域的变化是已知的且与所求解问题无关。该数值方法结合了标准向后微分公式（BDF）类型时间步进过程与几何非匹配有限元离散技术，并利用Nitsche方法施加边界条件。文章针对满足inf-sup稳定性的多种速度-压力单元给出了收敛性估计。该估计表明，速度分量在能量范数下达到最优阶收敛，而压力分量在缩放后的L^2(H^1)型范数下亦呈现最优阶收敛性。