JASS 2005: The Interplay of Mathematical Modelling and Numerical Simulation A Conservative FE-discretisation of the Navier-Stokes-Equation

In this work we present conservative FE-elements which were introduced by Cornelia Blanke in her diploma thesis 2004 [Bla04]. The usual way to solve a partial differential equation (PDE) with the finite element method (FEM) is to derive the weak formulation and discretize it with the functions in a suitable choosen finite-dimensional function space. For reasons of simplicity one mostley uses linear or bilinear elements for the function space. This choice of the function space isn’t always suitable, because the laws of conservation aren’t fulfilled. This may lead to instabilities and unphysical results. In this paper we present this new type of FE-elements which were invented by Cornelia Blanke. With a more physical approach to solve e.g. the Navier-Stokes-Equation with FEM we get the so-called conservative FE-elements. Particularly the laws of conservation for mass and energy in incompressible fluids are fulfilled by construction, which leads to an inherent stability for this method. 1