Stability analysis of a finite element approximation for the Navier-Stokes equation with free surface

¹ Laboratoire Analyse, Géométrie et Applications (LAGA), Institut Galilée, Université Sorbonne Paris Nord, 99 avenue J.B. Clément, 93430 Villetaneuse, France
² Department of Mathematics and Statistics, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, UK
³ Institut de Mathématiques de Bordeaux UMR 5251 Université de Bordeaux 351, cours de la Libération, 33 405 Talence, France

^* Corresponding author: gabriel.barrenechea@strath.ac.uk

Received: 28 July 2022
Accepted: 25 November 2023

Abstract

In this work we study the numerical approximation of incompressible Navier-Stokes equations with free surface. The evolution of the free surface is driven by the kinematic boundary condition, and an Arbitrary Lagrangian Eulerian (ALE) approach is used to derive a (formal) weak formulation which involves three fields, namely, velocity, pressure, and the function describing the free surface. This formulation is discretised using finite elements in space and a time-advancing explicit finite difference scheme in time. In fact, the domain tracking algorithm is explicit: first, we solve the equation for the free surface, then move the mesh according to the sigma transform, and finally we compute the velocity and pressure in the updated domain. This explicit strategy is built in such a way that global conservation can be proven, which plays a pivotal role in the proof of stability of the discrete problem. The well-posedness and stability results are independent of the viscosity of the fluid, but while the proof of stability for the velocity is valid for all time steps, and all geometries, the stability for the free surface requires a CFL condition. The performance of the current approach is presented via numerical results and comparisons with the characteristics finite element method.