Finite element methods for time-dependent convection – diffusion – reaction equations with small diffusion

Article history: Received 14 May 2008 Received in revised form 25 August 2008 Accepted 28 August 2008 Available online 12 September 2008

[1]  Gustavo Benitez Alvarez,et al.  A new stabilized finite element formulation for scalar convection–diffusion problems: the streamline and approximate upwind/Petrov–Galerkin method , 2003 .

[2]  J. Guermond.  Stabilization of Galerkin approximations of transport equations by subgrid modelling , 1999 .

[3]  A. Galeão,et al.  Feedback Petrov-Galerkin methods for convection-dominated problems , 1991 .

[4]  T. Hughes.  Multiscale phenomena: Green's functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods , 1995 .

[5]  Stefan Turek,et al.  High-resolution FEM?FCT schemes for multidimensional conservation laws , 2004 .

[6]  Sashikumaar Ganesan,et al.  Stabilization by Local Projection for Convection–Diffusion and Incompressible Flow Problems , 2010, J. Sci. Comput..

[7]  Volker John,et al.  A Finite Element Variational Multiscale Method for the Navier-Stokes Equations , 2005, SIAM J. Sci. Comput..

[8]  R. LeVeque.  High-resolution conservative algorithms for advection in incompressible flow , 1996 .

[9]  Gert Lube,et al.  Residual-based stabilized higher-order FEM for advection-dominated problems , 2006 .

[10]  R. Codina.  Comparison of some finite element methods for solving the diffusion-convection-reaction equation , 1998 .

[11]  Dmitri Kuzmin,et al.  Algebraic Flux Correction I. Scalar Conservation Laws , 2005 .

[12]  Lutz Tobiska,et al.  Numerical Methods for Singularly Perturbed Differential Equations , 1996 .

[13]  Volker John,et al.  On Finite Element Methods for 3D Time-Dependent Convection-Diffusion-Reaction Equations with Small Diffusion , 2008 .

[14]  Tobias Knopp,et al.  Stabilized finite element methods with shock capturing for advection–diffusion problems , 2002 .

[15]  L. Tobiska,et al.  STABILISATION OF LOCAL PROJECTION TYPE APPLIED TO CONVECTION-DIFFUSION PROBLEMS WITH MIXED BOUNDARY CONDITIONS , 2007 .

[16]  R. Codina.  On stabilized finite element methods for linear systems of convection-diffusion-reaction equations , 2000 .

[17]  Roland Becker,et al.  A finite element pressure gradient stabilization¶for the Stokes equations based on local projections , 2001 .

[18]  Alexandre Ern,et al.  The discrete maximum principle for stabilized finite element methods , 2003 .

[19]  Guillermo Hauke,et al.  A simple subgrid scale stabilized method for the advection–diffusion-reaction equation , 2002 .

[20]  Volker John,et al.  On the performance of SOLD methods for convection-diffusion problems with interior layers , 2007, Int. J. Comput. Sci. Math..

[21]  T. Hughes,et al.  Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations , 1990 .

[22]  Erik Burman,et al.  Local Projection Stabilization for the Oseen Problem and its Interpretation as a Variational Multiscale Method , 2006, SIAM J. Numer. Anal..

[23]  Stefan Turek,et al.  Flux correction tools for finite elements , 2002 .

[24]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[25]  T. Hughes,et al.  MULTI-DIMENSIONAL UPWIND SCHEME WITH NO CROSSWIND DIFFUSION. , 1979 .

[26]  J. Maubach,et al.  Nonconforming streamline-diffusion-finite-element-methods for convection-diffusion problems , 1997 .

[27]  Dmitri Kuzmin,et al.  Explicit and implicit FEM-FCT algorithms with flux linearization , 2009, J. Comput. Phys..

[28]  A. H. Schatz,et al.  Crosswind Smear and Pointwise Errors in Streamline Diffusion Finite Element Methods , 1987 .

[29]  Volker John,et al.  On large Eddy simulation and variational multiscale methods in the numerical simulation of turbulent incompressible flows , 2006 .

[30]  Pavel B. Bochev,et al.  Stability of the SUPG finite element method for transient advection-diffusion problems , 2004 .

[31]  P. Hansbo,et al.  Edge stabilization for Galerkin approximations of convection?diffusion?reaction problems , 2004 .

[32]  Mohamed Hamdy Doweidar,et al.  Fourier analysis of semi-discrete and space–time stabilized methods for the advective–diffusive–reactive equation: I. SUPG , 2005 .

[33]  S. Zalesak.  Fully multidimensional flux-corrected transport algorithms for fluids , 1979 .

[34]  K. Sundmacher,et al.  Simulations of Population Balance Systems with One Internal Coordinate using Finite Element Methods , 2009 .

[35]  Alexandre Ern,et al.  Stabilized Galerkin approximation of convection-diffusion-reaction equations: discrete maximum principle and convergence , 2005, Math. Comput..

[36]  T. Hughes,et al.  Large Eddy Simulation and the variational multiscale method , 2000 .

[37]  L. Franca,et al.  On an Improved Unusual Stabilized Finite Element Method for theAdvective-Reactive-Diffusive Equation , 1999 .

[38]  Timothy A. Davis,et al.  Algorithm 832: UMFPACK V4.3---an unsymmetric-pattern multifrontal method , 2004, TOMS.

[39]  Gunar Matthies,et al.  MooNMD – a program package based on mapped finite element methods , 2004 .

[40]  Victor M. Calo,et al.  YZβ discontinuity capturing for advection‐dominated processes with application to arterial drug delivery , 2007 .

[41]  Regina C. Almeida,et al.  A stable Petrov-Galerkin method for convection-dominated problems , 1997 .

[42]  Erik Burman,et al.  Stabilized finite element methods for the generalized Oseen problem , 2007 .

[43]  Tayfun E. Tezduyar,et al.  Discontinuity-capturing finite element formulations for nonlinear convection-diffusion-reaction equations , 1986 .

[44]  Abimael F. D. Loula,et al.  Finite element analysis of convection dominated reaction-diffusion problems , 2004 .

[45]  Tayfun E. Tezduyar,et al.  Finite Element Methods for Fluid Dynamics with Moving Boundaries and Interfaces , 2004 .

[46]  W. Layton,et al.  A two-level variational multiscale method for convection-dominated convection-diffusion equations , 2006 .

[47]  Lutz Tobiska,et al.  Necessary L2-uniform convergence conditions for difference schemes for two-dimensional convection-diffusion problems , 1995 .

[48]  W. Wall,et al.  A space-time formulation and improved spatial reconstruction for the "divide-and-conquer" multiscale method , 2008 .

[49]  A. Galeão,et al.  A consistent approximate upwind Petrov—Galerkin method for convection-dominated problems , 1988 .

[50]  Ramon Codina,et al.  A discontinuity-capturing crosswind-dissipation for the finite element solution of the convection-diffusion equation , 1993 .

[51]  Alexandre Ern,et al.  Nonlinear diffusion and discrete maximum principle for stabilized Galerkin approximations of the convection-diffusion-reaction equation , 2002 .

[52]  J. Peraire,et al.  Finite Element Flux-Corrected Transport (FEM-FCT) for the Euler and Navier-Stokes equations , 1987 .

[53]  Gunar Matthies,et al.  A UNIFIED CONVERGENCE ANALYSIS FOR LOCAL PROJECTION STABILISATIONS APPLIED TO THE OSEEN PROBLEM , 2007 .

[54]  Volker John,et al.  A comparison of spurious oscillations at layers diminishing (SOLD) methods for convection-diffusion equations. - Part I , 2005 .