FREE BOUNDARY PROBLEMS IN FLUID MECHANICS

We consider a class of free boundary problems governed by the incom-pressible Navier-Stokes equations. Our objective is to develop robust nu-merical algorithms that capture the shape of the free boundary. In our approach, we combine a recent algorithm of Liu, Liu, and Pego for solving the Navier-Stokes equations with a prediction-correction technique for locat-ing the free boundary. The Liu, Liu, and Pego algorithm calls for the rather uncommon C1-compatible finite elements. A major part of our work is to develop software that implements their algorithm, then adapt it to handle free boundaries. 1. Numerical method for Navier-Stokes equations Fluid flow is governed by the Navier-Stokes equations. For incompressible viscous fluid flow in a domain Ω ⊂ RN (N ≥ 2), they take the following form: ut + (∇u)u+∇p = ν∆u+ f in Ω × (0, T],(1) divu = 0 in Ω × (0, T],(2) where u is the fluid velocity, p is the pressure, ν is the kinematic viscosity, and f is an external force per unit mass. The system (1)–(2) is completed with an initial condition for velocity and appropriate boundary conditions. Some of the most popular numerical methods for the incompressible Navier-Stokes equations are splitting schemes that decouple the computation of the velocity and the pressure. In [4], Liu, Liu, and Pego have proposed such a scheme, based on a time-discretization explicit in pressure and convection terms. At each time step it is only required to solve a Poisson equation for the pressure and an elliptic boundary-value problem for the velocity. The corresponding fully discrete finite element method with C1 elements for ve-locity and C0 elements for pressure is unconditionally stable and does not require any compatibility conditions between the finite element spaces for velocity and pressure