Optimal convergence of arbitrary Lagrangian–Eulerian finite element methods for the Stokes equations in an evolving domain

Abstract The numerical solution of the Stokes equations in an evolving domain with a moving boundary is studied based on the arbitrary Lagrangian–Eulerian finite element method along the trajectories of the evolving mesh, where the Taylor–Hood finite elements of degree $r\ge 2$ is employed for spatial discretization. The error of the semidiscrete arbitrary Lagrangian–Eulerian method is proved to be $O(h^{r+1})$ for velocity approximation in $ L^{\infty }(0,T;L^{2}) $ norm and $ O (h^{r}) $ for pressure approximation in $ L^{2}(0,T;L^{2}) $ norm, respectively. The analyses are based on a Nitsche’s duality argument adapted to an evolving mesh, by proving that the material derivative and the Stokes–Ritz projection commute up to terms that have optimal-order convergence in the $L^{2}$ norm. Numerical examples are provided to support the theoretical analysis.