A new control volume finite element method for the stable and accurate solution of the drift-diffusion equations on general unstructured grids

We present a new Control Volume Finite Element Method (CVFEM) for the drift-diffusion equations. The method combines a conservative formulation of the current continuity equations with a novel definition of an exponentially fitted elemental current density. An edge element representation of the nodal CVFEM current density in the diffusive limit motivates this definition. We prove that in the absence of carrier drift the nodal current is sum of edge currents, which solve one-dimensional diffusion problems, times H(curl)conforming edge basis functions. Replacement of the edge diffusion problems by one-dimensional driftdiffusion equations extends this representation to the general case. The resulting H(curl,Ω)-conforming, exponentially fitted current (EFC) density field combines the upwind effect from all edges and enables accurate computation of current density integrals on arbitrary surfaces inside the elements. This obviates the need for the control volumes to be topologically dual to the finite elements and results in a method that is stable and accurate on general unstructured finite element grids. This sets apart our approach from other schemes, such as the Scharfetter-Gummel Box Integration Method, which require topologically dual grids. Numerical studies of the CVFEM-EFC for a suite of standard advection-diffusion test problems on nonuniform grids confirms the accuracy and the robustness of the new formulation. Simulation of an nchannel MOSFET device tests the method in a more realistic setting. Copyright c © 2012 John Wiley & Sons, Ltd.