[Paper Review] Eulerian time-stepping schemes for the non-stationary Stokes equations on time-dependent domains

Erik Burman, Stefan Frei+1 more|arXiv (Cornell University)|Oct 7, 2019

Advanced Numerical Methods in Computational Mathematics4 citations

TL;DR

This paper presents a novel Eulerian time-stepping scheme for the non-stationary Stokes equations on time-dependent domains using a geometrically unfitted finite element method with Nitsche's method for boundary conditions and ghost penalty stabilisation to implicitly extend solution values from previous time steps. The key contribution is a complete a priori error analysis proving optimal $L^2(L^2)$-norm error bounds for velocity errors, validated numerically.

ABSTRACT

This article is concerned with the discretisation of the Stokes equations on time-dependent domains in an Eulerian coordinate framework. Our work can be seen as an extension of a recent paper by Lehrenfeld & Olshanskii [ESAIM: M2AN, 53(2):585-614, 2019], where BDF-type time-stepping schemes are studied for a parabolic equation on moving domains. For space discretisation, a geometrically unfitted finite element discretisation is applied in combination with Nitsche's method to impose boundary conditions. Physically undefined values of the solution at previous time-steps are extended implicitly by means of so-called ghost penalty stabilisations. We derive a complete a priori error analysis of the discretisation error in space and time, including optimal $L^2(L^2)$-norm error bounds for the velocities. Finally, the theoretical results are substantiated with numerical examples.

Motivation & Objective

• To develop a stable and accurate time-stepping scheme for the non-stationary Stokes equations on time-dependent domains where the computational domain changes over time.
• To overcome the challenge of transferring solution data from one time-dependent domain to the next, where the solution at $ t_{n-1} $ is defined on $ \Omega(t_{n-1}) $ but needed on $ \Omega(t_n) $.
• To provide a complete a priori error analysis for the space-time discretisation, including optimal convergence rates in the $ L^2(L^2) $-norm for velocity errors.
• To extend the applicability of unfitted finite element methods to time-dependent problems by incorporating ghost penalty stabilisation for implicit solution extension across time steps.

Proposed method

• A geometrically unfitted finite element method is employed, where the background mesh is independent of the domain boundary, enabling arbitrary domain motion without remeshing.
• Nitsche's method is used to weakly enforce boundary conditions on the moving boundary, ensuring stability and consistency without requiring body-fitted meshes.
• Ghost penalty stabilisation is applied to implicitly extend the solution from the previous time step $ \bm{u}_h(t_{n-1}) $ to the current domain $ \Omega(t_n) $, avoiding explicit projection or interpolation.
• Backward differentiation formula (BDF)-type time discretisation is used, with the time derivative approximated in a variational formulation on the current domain.
• The analysis is conducted in a reference configuration via a transformation $ \bm{T}(t) $, allowing the use of standard finite element tools on a fixed domain $ \hat{\Omega} $, followed by pullback to the physical domain.
• A priori error estimates are derived using energy methods, including Gronwall's inequality, to bound the $ L^2(L^2) $-norm of the velocity error.

Experimental results

Research questions

• RQ1 Can a stable and convergent time-stepping scheme be developed for the non-stationary Stokes equations on time-dependent domains without requiring domain remeshing or ALE mapping?
• RQ2 How can physically undefined solution values from previous time steps be extended to the current domain in a stable and consistent manner?
• RQ3 What is the optimal convergence rate of the velocity error in the $ L^2(L^2) $-norm for a space-time discretisation on moving domains?
• RQ4 Can ghost penalty stabilisation effectively replace explicit projection or interpolation in time-stepping schemes for moving domain problems?

Key findings

• The proposed scheme achieves optimal $ L^2(L^2) $-norm error bounds for the velocity, confirming optimal convergence in both space and time.
• The a priori error analysis establishes stability and convergence of the scheme under standard assumptions on the domain motion and finite element regularity.
• Ghost penalty stabilisation enables implicit extension of past solution values to the current domain, avoiding explicit projection and preserving accuracy.
• The method is robust under large domain deformations and topology changes, as it does not rely on continuous mesh mappings.
• Numerical examples confirm the theoretical error bounds and demonstrate the method’s robustness and efficiency on moving domains.