Steady-state convection-diffusion problems

In convection-diffusion problems, transport processes dominate while diffusion effects are confined to a relatively small part of the domain. This state of affairs means that one cannot rely on the formal ellipticity of the differential operator to ensure the convergence of standard numerical algorithms. Thus new ideas and approaches are required. The survey begins by examining the asymptotic nature of solutions to stationary convection-diffusion problems. This provides a suitable framework for the understanding of these solutions and the difficulties that numerical techniques will face. Various numerical methods expressly designed for convection-diffusion problems are then presented and extensively discussed. These include finite difference and finite element methods and the use of special meshes.

[1]  D. N. De G. Allen,et al.  RELAXATION METHODS APPLIED TO DETERMINE THE MOTION, IN TWO DIMENSIONS, OF A VISCOUS FLUID PAST A FIXED CYLINDER , 1955 .

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

[3]  Volker John,et al.  A numerical study of a posteriori error estimators for convection–diffusion equations , 2000 .

[4]  Willy Dörfler,et al.  Uniform A Priori Estimates for Singularly Perturbed Elliptic Equations in Multidimensions , 1999 .

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

[6]  J. Cole,et al.  Multiple Scale and Singular Perturbation Methods , 1996 .

[7]  C. Schwab.  P- and hp- finite element methods : theory and applications in solid and fluid mechanics , 1998 .

[8]  Giancarlo Sangalli.  Quasi Optimality of the SUPG Method for the One-Dimensional Advection-Diffusion Problem , 2003, SIAM J. Numer. Anal..

[9]  Endre Süli,et al.  Stabilised hp-Finite Element Approximation of Partial Differential Equations with Nonnegative Characteristic Form , 2001, Computing.

[10]  HANS-GäRG Roos,et al.  A note on the conditioning of upwind schemes on Shishkin meshes , 1996 .

[11]  A. Wathen,et al.  On parameter choice and iterative convergence for stabilised discretisations of advection-diffusion problems , 1999 .

[12]  Martin Stynes,et al.  Finite Element Methods for Convection-Diffusion Problems using Exponential Splines on Triangles , 1998 .

[13]  Hong Wang,et al.  A summary of numerical methods for time-dependent advection-dominated partial differential equations , 2001 .

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

[15]  O. A. Ladyzhenskai︠a︡,et al.  Linear and quasilinear elliptic equations , 1968 .

[16]  Thomas Apel,et al.  Anisotropic interpolation with applications to the finite element method , 1991, Computing.

[17]  R. Bruce Kellogg,et al.  Corner singularities and boundary layers in a simple convection–diffusion problem☆ , 2005 .

[18]  Hans-Görg Roos,et al.  The Discontinuous Galerkin Finite Element Method for Singularly Perturbed Problems , 2003 .

[19]  I. Babuska,et al.  A DiscontinuoushpFinite Element Method for Diffusion Problems , 1998 .

[20]  Lutz Tobiska,et al.  A finite difference analysis of a streamline diffusion method on a Shishkin mesh , 2004, Numerical Algorithms.

[21]  Martin Stynes,et al.  EFFICIENT GENERATION OF ORIENTED MESHES FOR SOLVING CONVECTION-DIFFUSION PROBLEMS , 1997 .

[22]  Torsten Linß,et al.  The necessity of Shishkin decompositions , 2001, Appl. Math. Lett..

[23]  Torsten Linß,et al.  Defect correction on Shishkin-type meshes , 2001, Numerical Algorithms.

[24]  R. Kellogg,et al.  Analysis of some difference approximations for a singular perturbation problem without turning points , 1978 .

[25]  JohnM . Miller,et al.  Robust Computational Techniques for Boundary Layers , 2000 .

[26]  Jens Markus Melenk,et al.  hp-Finite Element Methods for Singular Perturbations , 2002 .

[27]  Hans-Görg Roos,et al.  Ten ways to generate the Il'in and related schemes , 1994 .

[28]  G. I. Shishkin.  Approximation of the solutions of singularly perturbed boundary-value problems with a parabolic boundary layer , 1991 .

[29]  Natalia Kopteva,et al.  A Robust Adaptive Method for a Quasi-Linear One-Dimensional Convection-Diffusion Problem , 2001, SIAM J. Numer. Anal..

[30]  Paul Houston,et al.  Discontinuous hp-Finite Element Methods for Advection-Diffusion-Reaction Problems , 2001, SIAM J. Numer. Anal..

[31]  Hans-Görg Roos,et al.  Interior penalty discontinuous approximations of convection–diffusion problems with parabolic layers , 2005, Numerische Mathematik.

[32]  Guido Kanschat,et al.  A multilevel discontinuous Galerkin method , 2003, Numerische Mathematik.

[33]  Martin Stynes,et al.  A singularly perturbed convection–diffusion problem in a half-plane , 2006 .

[34]  Natalia Kopteva,et al.  Error Expansion for an Upwind Scheme Applied to a Two-Dimensional Convection-Diffusion Problem , 2003, SIAM J. Numer. Anal..

[35]  H. Elman,et al.  Iterative methods for stabilized discrete convection-diffusion problems , 2000 .

[36]  Giancarlo Sangalli,et al.  A robust a posteriori estimator for the Residual-free Bubbles method applied to advection-diffusion problems , 2001, Numerische Mathematik.

[37]  Claes Johnson,et al.  Computational Differential Equations , 1996 .

[38]  Torsten Linß,et al.  Layer-adapted meshes for convection-diffusion problems , 2003 .

[39]  Martin Stynes,et al.  A hybrid difference scheme on a Shishkin mesh for linear convection-diffusion problems , 1999 .

[40]  Endre Süli,et al.  Residual-free bubbles for advection-diffusion problems: the general error analysis , 2000, Numerische Mathematik.

[41]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[42]  Harold A. Buetow,et al.  Adaptive finite element methods for differential equations , 2003, Lectures in mathematics.

[43]  Hans-Görg Roos,et al.  A Priori Estimates for the Solution of Convection-Diffusion Problems and Interpolation on Shishkin Meshes , 1997 .

[44]  Mary F. Wheeler,et al.  A Priori Error Estimates for Finite Element Methods Based on Discontinuous Approximation Spaces for Elliptic Problems , 2001, SIAM J. Numer. Anal..

[45]  Tayfun E. Tezduyar,et al.  Calculation of the advective limit of the SUPG stabilization parameter for linear and higher-order elements , 2004 .

[46]  Torsten Linß,et al.  Numerical methods on Shishkin meshes for linear convection-diffusion problems , 2001 .

[47]  Alessandro Russo,et al.  CHOOSING BUBBLES FOR ADVECTION-DIFFUSION PROBLEMS , 1994 .

[48]  Guohui Zhou,et al.  How accurate is the streamline diffusion finite element method? , 1997, Math. Comput..

[49]  T. Apel.  Anisotropic Finite Elements: Local Estimates and Applications , 1999 .

[50]  J. Tinsley Oden,et al.  A Posteriori Error Estimation , 2002 .

[51]  A. M. Ilʹin,et al.  Matching of Asymptotic Expansions of Solutions of Boundary Value Problems , 1992 .

[52]  Klaus Böhmer.  Defect correction methods - theory and applications , 1984, Computing : Supplementum.

[53]  Niall Madden,et al.  Linear enhancements of the streamline diffusion method for convection-diffusion problems , 1996 .

[54]  Riccardo Sacco,et al.  A nonconforming exponentially fitted finite element method for two‐dimensional drift‐diffusion models in semiconductors , 1999 .

[55]  Achi Brandt,et al.  Inadequacy of first-order upwind difference schemes for some recirculating flows , 1991 .

[56]  J..  A Posteriori Error Estimation , 2006 .

[57]  J. J. Miller,et al.  Fitted Numerical Methods for Singular Perturbation Problems , 1996 .

[58]  M. Giles,et al.  Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality , 2002, Acta Numerica.

[59]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[60]  Lutz Tobiska,et al.  The SDFEM for a Convection-Diffusion Problem with a Boundary Layer: Optimal Error Analysis and Enhancement of Accuracy , 2003, SIAM J. Numer. Anal..

[61]  Torsten Linß.  On a convection-diffusion problem with a weak layer , 2005, Appl. Math. Comput..

[62]  R. B. Kellogg,et al.  Differentiability properties of solutions of the equation -ε 2 δ u + ru = f ( x,y ) in a square , 1990 .

[63]  Douglas N. Arnold,et al.  Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems , 2001, SIAM J. Numer. Anal..

[64]  J. Oden,et al.  A Posteriori Error Estimation in Finite Element Analysis , 2000 .

[65]  Torsten Linß,et al.  Asymptotic Analysis and Shishkin-Type Decomposition for an Elliptic Convection–Diffusion Problem , 2001 .

[66]  Natalia Kopteva.  Maximum Norm A Posteriori Error Estimates for a One-Dimensional Convection-Diffusion Problem , 2001, SIAM J. Numer. Anal..

[67]  Torsten Liní.  On a convection-diffusion problem with a weak layer , 2005 .

[68]  Natalia Kopteva,et al.  How accurate is the streamline-diffusion FEM inside characteristic (boundary and interior) layers? , 2004 .

[69]  Rüdiger Verfürth,et al.  Robust A Posteriori Error Estimates for Stationary Convection-Diffusion Equations , 2005, SIAM J. Numer. Anal..

[70]  Klaus Böhmer,et al.  Defect Correction Methods , 1984, Computing Supplementum.

[71]  P. Grisvard.  Elliptic Problems in Nonsmooth Domains , 1985 .