A Higher Order Mixed Discontinuous FiniteElement Method For Incompressible Flows

A higher order mixed discontinous Galerkin finite element method is employed for the un-

steady Navier Stokes equations. In order to iteratively solve the Navier Stokes equations

several problems are tackled. The coupling of the physical quantities velocity and pressure is

resolved with help of the standard pressure correction scheme. The temporal discretisation

relies on a finite difference scheme (implicit midpoint rule) and the nonlinearity due to the

advection term is resolved with a Picard iteration. On top of that, for spatial discretisation

finite elements are used, which are fully discontinous for the pressure and partially disconti-

nous (in the tangential component) for the velocity approximation. The partial discontinuity

introduces local fluxes in the weak formulation of the finite element methods, here defined

as upwind fluxes. Furthermore, the jumps in the tangential component are penalised in

the diffusion term according to the symmetric interior penalty, which also introduces fur-

ther terms for consistency and symmetry reasons. Generally, in mixed discontinous Galerkin

methods the continuity equation, which ensures incompressibility of the flow, is solved exact

by retaining the continuity in the normal component and incorporation of the continuity

equation in the residual formulation of the momentum equation, where the according trial

function acts as a Lagrange multiplier. On top of that, the finite elements are chosen to be

quadrilateral, since their tensor product structure opens more possibilities of assembly op-

timisation algorithms.