A Petrov‐Galerkin mixed finite element method with exponential fitting

Abstract

We discuss a Petrov-Galerkin mixed finite element formulation of the semiconductor continuity equations on a rectangular domain. We give error estimates for equations that are in principle degenerate in the singularly perturbed case. We give arguments that indicate that the method is also effective in the singularly perturbed case. We develop a discretization that gives a higher-order accurate solution for use in an a posteriori error estimator. 0 1995 John Wiley & Sons, Inc.

On Some Mixed Finite Element Methods with Numerical Integration

“…V,sect. 34] and [23]) and turns out to be a sound indication for good behavior of a numerical scheme to deal with advection-dominated flow problems, as previously remarked in [44,38].…”

“…In this latter application the mixed finite volume scheme (5.7) with the quadrature (8.3) can be regarded as a two-dimensional generalization of the classical exponentially fitted Scharfetter-Gummel method [41]. The resulting scheme can be proved to recover the exact solution (u, J) at the nodes of the dual mesh when µ = e ax+by , a, b ∈ R, x, y ∈ Ω, σ = 0, f = 0, and suitable Dirichlet-Neumann boundary conditions are assumed in problem (2.1) (see for the proof [40] and [44], respectively, in the case of triangles and rectangles). This nice property is a special instance of the "patch-test" (see [36,Chap.…”

An a priori error estimate for a monotone mixed finite-element discretization of a convection–diffusion problem

“…The space V h,ψ is in some sense the nonconforming analogue of the conforming exponential test space that was introduced by Van Nooyen in [27] specifically for rectangular meshes. A rectangular grid allows to separate the components of the vector valued test functions.…”

“…Several other generalizations of this idea can be found, especially if monotone schemes are desired, cf. e.g., [31,27] and more recently [30]. In contrast to the edge-averaged finite element method presented in [31] our method preserves additionally to the maximum principle also the continuity of the normal component of the flux density.…”

“…On the other hand we need for proving the maximum principle that the triangulation is of weakly acute type whereas in edge-averaging case a Delauney triangulation is sufficient. We will comment on some similarities of our method to a method of Van Nooyen in [27] in Sect. 2.…”