Control Volume Finite Element Method with Multidimensional Edge Element Scharfetter-Gummel upwinding. Part 1. Formulation

We develop a new formulation of the Control Volume Finite Element Method (CVFEM) with a multidimensional Scharfetter-Gummel (SG) upwinding for the drift-diffusion equations. The formulation uses standard nodal elements for the concentrations and expands the flux in terms of the lowest-order Nedelec H(curl,Ω)-compatible finite element basis. The SG formula is applied to the edges of the elements to express the Nedelec element degree of freedom on this edge in terms of the nodal degrees of freedom associated with the endpoints of the edge. The resulting upwind flux incorporates the upwind effects from all edges and is defined at the interior of the element. This allows for accurate evaluation of integrals on the boundaries of the control volumes for arbitrary quadrilateral elements. The new formulation admits efficient implementation through a standard loop over the elements in the mesh followed by loops over the element nodes (associated with control volume fractions in the element) and element edges (associated with flux degrees of freedom). The quantities required for the SG formula can be precomputed and stored for each edge in the mesh for additional efficiency gains. For clarity the details are presented for two-dimensional quadrilateral grids. Extension to other element shapes and three dimensions is straightforward.