An Eulerian finite element method for the linearized Navier--Stokes problem in an evolving domain

TL;DR

This paper presents an error analysis of an Eulerian finite element method combining BDF time-stepping, unfitted discretization, and Nitsche's method for solving linearized Navier--Stokes equations in evolving domains, demonstrating optimal convergence.

Contribution

It provides the first convergence estimates for an Eulerian finite element approach applied to linearized Navier--Stokes problems in moving domains, including stability and optimal error bounds.

Findings

Optimal order convergence in energy norm for velocity

Optimal scaled $L^2(H^1)$-norm convergence for pressure

Effective enforcement of boundary conditions via Nitsche's method

Abstract

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.