Semi-Lagrangian finite element exterior calculus for incompressible flows

Abstract

We develop a semi-Lagrangian discretization of the time-dependent incompressible Navier-Stokes equations with free boundary conditions on arbitrary simplicial meshes. We recast the equations as a nonlinear transport problem for a momentum 1-form and discretize in space using methods from finite element exterior calculus. Numerical experiments show that the linearly implicit fully discrete version of the scheme enjoys excellent stability properties in the vanishing viscosity limit and is applicable to inviscid incompressible Euler flows. We obtain second-order convergence and conservation of energy is achieved through a Lagrange multiplier.