Implementation of conservative schemes in an edge-based finite element code

Abstract

In this work a novel edge-based finite element implementation applied to specific equations is presented. It contains a full description on how we obtained it for the diffusion equation, stabilized convection-diffusion equation and stabilized Navier-Stokes equations. Additionally, classical benchmark problems are solved to show the capabilities of the new implementation. As the differential equations we are interested in represent conservation statements, it would be desirable that the finite element approximation was exactly conservative (at least globally) independently of the mesh used. The present work revolves around that main objective. The initial available edge-based approximation is not totally conservative. Of course it becomes more and more conservative as the mesh is refined. It has good h-convergence features and produces ’good solutions’ (in the sense that the method does not introduces spurious oscillations and is numerically stable). On the other hand, the edge-based approximation proposed is exactly globally conservative. Additionally it has good h-convergence features and produces ’good solutions’.