Convergence of a second-order accurate Petrov–Galerkin scheme for convection–diffusion problems in semiconductors

Abstract

Sacco, R., Convergence of a second-order accurate Petrov–Galerkin scheme for convection–diffusion problems in semiconductors, Applied Numerical Mathematics 11 (1993) 517–528. In this work we give a proof of a convergence theorem for a Petrov–Galerkin finite element scheme for the solution in the one-dimensional case of the convection–diffusion equation in semiconductors − J ′−(μ( u′− uψ′))′ƒ. The method is based on the well-known Scharfetter–Gummel [5] approximation of the current density J , which is assumed to be divergence-free over each element. A suitable trial space I h for the unknown u is directly derived from the Scharfetter–Gummel argumentation. Regarding the choice of the test space V h , we define at every mesh node x j ( j1,… N) a discrete Green's function G j ( x,x j ) for the convection–diffusion operator and we characterize V h by taking each test function v j equal to the Green's function G j normalized to 1. This gives rise to a Petrov–Galerkin scheme which is proved to be second-order accurate with respect to the mesh size h; we also point out that the method is exactly equivalent to an optimal upwinding finite element scheme, thus making the convergence theorem proved in this work valid also for the upwinding method. A highly convection-dominated semiconductor device test problem is then examined; plots of both numerical solution and O( h 2) convergence rate are included to show the performances of the Scharfetter–Gummel Petrov–Galerkin method.