Implementation of a divergence preserving finite element discretisation
The Navier-Stokes equations describing the flow of incompressible fluids consist of two parts: the time-dependent momentum equations including diffusion and convection terms and the continuity equation that guarantees the conservation of mass at any point in time. Standard discretisations lead to discrete approximations of the fluid velocities that fulfill the continuous continuity equation only approximatly. This leads to serious problems in the finite element approximation theory, a violation of energy conservation, and instabilities for high Reynolds number flow simulations. Numerous more or less complicated methods to tackle this problem are described in literature.
We have developed a very simple finite element discretisation for which the fulfillment of the discrete continuity equation automatically implies the fulfillment of the continuous equation (see basis functions above) in 2D. This approach shall be generalised to the 3D case.
Summary of project steps
- derivation of the three-dimensional divergence preserving basis
- computation of the resulting diffusion and convection operators (Maple or similar)
- analysis of possibilities to reduce the number of required operations by a rotation of the coordinate system (using the respective 2D analysis as a guideline),
- implementation of the new discretisation in Peano
- good C++ programming skills
- basic knowledge on finite element methods