A hybrid mixed discontinuous Galerkin finite-element method for convection-diffusion problems

Abstract

We propose and analyse a new finite element method for convection diffusion problems based on the combination of a mixed method for the elliptic and a discontinuous Galerkin method for the hyperbolic part of the problem. The two methods are made compatible via hybridization and the combination of both is appropriate for the solution of intermediate convection-diffusion problems. By construction, the discrete solutions obtained for the limiting subproblems coincide with the ones obtained by the mixed method for the elliptic and the discontinuous Galerkin method for the limiting hyperbolic problem, respectively. We present a new type of analysis that explicitly takes into account the Lagrange-multipliers introduced by hybridization. The use of adequate energy norms allows to treat the purely diffusive, the convection dominated, and the hyperbolic regime in a unified manner. In numerical tests, we illustrated the efficiency of our approach and compare to results obtained with other methods for convection diffusion problems.

Rectification properties of conically shaped nanopores: consequences of miniaturization

“…This nondimensionalization allows stable and unit independent simulations. (2) is discretized using a hybrid discontinuous Galerkin method with upwind stabilization 39 . This stabilization ensures stability of the numerical scheme for large applied voltages.…”

“…The pores are embedded in membranes with thicknesses from 90 nm to 12-µm. MsSimPore is based on a efficient finite element, hybrid discontinuous Galerkin scheme 39 , which is a novel approach for the simulation of ion transport in nanopores. Electrolyte reservoirs are considered explicitly by the use of Dirichlet boundary conditions, so that short pores can be modeled as well.…”

Boltzmann and Fokker-Planck Equations Modelling Opinion Formation in the Presence of Strong Leaders

“…To illustrate the long-time behaviour of the proposed model we discretize the nonlinear Fokker-Planck system (3.1) using a hybrid discontinuous Galerkin (DG) method introduced by Egger and Schöberl (2008). This hybrid DG method was initially developed for convection diffusion equations and yields stable discretizations for convection dominated problems as well as hyperbolic ones.…”

“…< t m = T , and define ∆t j = t j+1 − t j . We consider the following linearization of the Fokker-Planck equations (3.1), which fits into the framework of Egger & Schöberl (2008),…”

Uma Formulação Hibridizada de Elementos Finitos para Problemas Parabólicos

“…Além disso, as formulações hibridizadas, como a FHCD, apresentam estabilidade independentes da escolha dos espaços de aproximações (V h e M h ), tanto para o tipo do polinômio interpolante (polinômios de Lagrange, Legendre e outros) como para a ordem (grau) de interpolação dos mesmos, podendo também as malhas serem tomadas estruturadas ou não-estruturadas. Desse modo, notam-se importantes vantagens sobre alguns métodos usuais de elementos finitos tais como os métodos mistos clássicos [4,5,10], ondeé exigido um comprometimento entre as escolhas dos espaços de aproximação para que sejam asseguradas unicidade e existência de solução.…”