## Convergent Finite Element Discretizations of the Density Gradient Equation for Quantum Semiconductors

• We study nonlinear finite element discretizations for the density gradient equation in the quantum drift diffusion model. Especially, we give a finite element description of the so--called nonlinear scheme introduced by {it Ancona}. We prove the existence of discrete solutions and provide a consistency and convergence analysis, which yields the optimal order of convergence for both discretizations. The performance of both schemes is compared numerically, especially with respect to the influence of approximate vacuum boundary conditions.

Author: Rene Pinnau, Jorge Mauricio Ruiz urn:nbn:de:hbz:386-kluedo-14943 Berichte der Arbeitsgruppe Technomathematik (AGTM Report) (270) Preprint English 2007 2007 Technische Universität Kaiserslautern 2007/05/11 consistency ; convergence ; density gradient equation ; nonlinear finite element method ; numerics Fachbereich Mathematik 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik 35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Jxx Elliptic equations and systems [See also 58J10, 58J20] / 35J60 Nonlinear elliptic equations 35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Jxx Elliptic equations and systems [See also 58J10, 58J20] / 35J70 Degenerate elliptic equations 65-XX NUMERICAL ANALYSIS / 65Nxx Partial differential equations, boundary value problems / 65N12 Stability and convergence of numerical methods 65-XX NUMERICAL ANALYSIS / 65Nxx Partial differential equations, boundary value problems / 65N15 Error bounds 65-XX NUMERICAL ANALYSIS / 65Nxx Partial differential equations, boundary value problems / 65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 76-XX FLUID MECHANICS (For general continuum mechanics, see 74Axx, or other parts of 74-XX) / 76Yxx Quantum hydrodynamics and relativistic hydrodynamics [See also 82D50, 83C55, 85A30] / 76Y05 Quantum hydrodynamics and relativistic hydrodynamics [See also 82D50, 83C55, 85A30] Standard gemäß KLUEDO-Leitlinien vor dem 27.05.2011

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