Volume 25, Issue 3

This paper provides an overview of the new features of the finite element library deal.II version 8.5.

In this paper, we study the large-time behavior of a numerical scheme discretizing drift–diffusion systems for semiconductors. The numerical method is finite volume in space, implicit in time, and the numerical fluxes are a generalization of the classical Scharfetter–Gummel scheme which allows to consider both linear or nonlinear pressure laws. We study the convergence of approximate solutions towards an approximation of the thermal equilibrium state as time tends to infinity, and obtain a decay rate by controlling the discrete relative entropy with the entropy production. This result is proved under assumptions of existence and uniform in time L ∞ -estimates for numerical solutions, which are then discussed. We conclude by presenting some numerical illustrations of the stated results.

We introduce and analyze a family of finite element methods for elliptic partial differential equations in non-variational form with non-differentiable coefficients. The finite element method studied is a variant of the one recently proposed in [Lakkis & Pryer, SIAM J. Sci. Comput ., 2011], where a finite element Hessian is introduced as an auxiliary unknown. We modify the definition of the finite element Hessian rendering the auxiliary variable completely local, thus resulting in a more efficient scheme. We show that the method is stable under general conditions on the coefficient matrix and derive error estimates in a discrete H 2 -norm provided the discretization parameter is sufficiently small. Numerical experiments are presented which verify the theoretical results.

We consider the discretization in time of inhomogeneous parabolic equations, using the technique of Laplace inversion along a contour located in the complex left half-plane which, after transformation to a finite interval, is then evaluated to high accuracy by a contour quadrature rule. This reduces the problem to a finite set of elliptic equations with complex coefficients, which may be solved in parallel. A serious problem is how to treat the source term f ( t ), because at each quadrature node along the contour we need its Laplace forward transform, which unfortunately is often unavailable. In this paper, we propose a new contour quadrature which does not require direct use of the Laplace forward transform of f ( t ). Compared to the existing contour quadratures, error analysis shows that the new quadrature possesses competitive asymptotic order of accuracy and numerical results show that when regularity of the initial term and/or differentiability of f ( t ) is not satisfied, the new quadrature is more accurate.