Formulation and Analysis of a Parameter-Free Stabilized Finite Element Method

We present a stabilized finite element method for the scalar advection-diffusion equation, which does not require tunable mesh-dependent parameters. Stabilization is achieved by using diffusive fluxes extracted from an edge element lifting of Scharfetter-Gummel edge fluxes into the elements. Although the method is formally first-order accurate, qualitative numerical studies suggest that it occupies a middle ground between an artificial diffusion and a streamline-upwind Petrov-Galerkin formulations. The method is substantially less dissipative than the former, while having much smaller overshoots and undershoots than the latter.