Convergence of Arbitrary Lagrangian-Eulerian Second-order Projection Method for the Stokes Equations on an Evolving Domain

Abstract

The numerical solution of the Stokes equations on an evolving domain with a moving boundary is studied based on the arbitrary Lagrangian-Eulerian finite element method and a second-order projection method along the trajectories of the evolving mesh for decoupling the unknown solutions of velocity and pressure. The error of the semidiscrete arbitrary Lagrangian-Eulerian method is shown to be $O(h^{r+1})$ for the Taylor--Hood finite elements of degree $r\ge 2$, using Nitsche's duality argument adapted to an evolving mesh, by proving that the material derivative and the Stokes--Ritz projection commute up to terms which have optimal-order convergence in the $L^2$ norm. Additionally, the error of the fully discrete finite element method, with a second-order projection method along the trajectories of the evolving mesh, is shown to be $O(\ln(1/τ+1)τ^{2}+\ln(1/h+1)h^{r+1})$ in the discrete $L^\infty(0,T; L^2)$ norm using newly developed energy techniques and backward parabolic duality arguments that are applicable to the Stokes equations with an evolving mesh. To maintain consistency between the notations of the numerical scheme in a moving domain and those in a fixed domain, we introduce the equivalence class of finite element spaces across time levels. Numerical examples are provided to support the theoretical analysis and to illustrate the performance of the method in simulating Navier--Stokes flow in a domain with a rotating propeller.