Table of Contents
ISRN Applied Mathematics
Volume 2012, Article ID 379547, 30 pages
http://dx.doi.org/10.5402/2012/379547
Review Article

Robust Numerical Methods for Singularly Perturbed Differential Equations: A Survey Covering 2008–2012

TU Dresden, 01062 Dresden, Germany

Received 22 October 2012; Accepted 7 November 2012

Academic Editors: S. W. Gong, X. Liu, and S. Sture

Copyright © 2012 Hans-Görg Roos. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. H.-G. Roos, M. Stynes, and L. Tobiska, Robust Numerical Methods for Singularly Perturbed Differential Equations, vol. 24 of Springer Series in Computational Mathematics, Springer, Berlin, Germany, 2nd edition, 2008, Convection-diffusion-reaction and flow problems.
  2. T. Linß, Layer-Adapted Meshes for Reaction-Convection-Diffusion Problems, vol. 1985 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2010. View at Publisher · View at Google Scholar
  3. M. K. Kadalbajoo and V. Gupta, “A brief survey on numerical methods for solving singularly perturbed problems,” Applied Mathematics and Computation, vol. 217, no. 8, pp. 3641–3716, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. L. P. Franca, G. Hauke, and A. Masud, “Revisiting stabilized finite element methods for the advective-diffusive equation,” Computer Methods in Applied Mechanics and Engineering, vol. 195, no. 13–16, pp. 1560–1572, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. G. Lube, “Stabilized fem for incompressible ow. critical review and new trends,” in European Conference on Computational Fluid Dynamics (ECCOMAS CFD '06), P. Wesseling, E. Onate, and J. Periaux, Eds., pp. 1–20, TU Delft, Dordrecht, The Netherlands, 2006.
  6. F. Brezzi and A. Russo, “Choosing bubbles for advection-diffusion problems,” Mathematical Models & Methods in Applied Sciences, vol. 4, no. 4, pp. 571–587, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. J.-L. Guermond, “Stabilization of Galerkin approximations of transport equations by subgrid modeling,” Mathematical Modelling and Numerical Analysis, vol. 33, no. 6, pp. 1293–1316, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. T. J. R. Hughes, “Multiscale phenomena: green's functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods,” Computer Methods in Applied Mechanics and Engineering, vol. 127, no. 1–4, pp. 387–401, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. V. John, S. Kaya, and W. Layton, “A two-level variational multiscale method for convection-dominated convection-diffusion equations,” Computer Methods in Applied Mechanics and Engineering, vol. 195, no. 33–36, pp. 4594–4603, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. H. C. Elman, D. J. Silvester, and A. J. Wathen, Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, NY, USA, 2005.
  11. A. Serghini Mounim, “A stabilized finite element method for convection-diffusion problems,” Numerical Methods for Partial Differential Equations, vol. 28, no. 6, pp. 1916–1943, 2012. View at Google Scholar
  12. P. Knobloch, “On the definition of the SUPG parameter,” Electronic Transactions on Numerical Analysis, vol. 32, pp. 76–89, 2008. View at Google Scholar · View at Zentralblatt MATH
  13. P. Knobloch, “On the choice of the SUPG parameter at outflow boundary layers,” Advances in Computational Mathematics, vol. 31, no. 4, pp. 369–389, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. V. John, P. Knobloch, and S. B. Savescu, “A posteriori optimization of parameters in stabilized methods for convection-diffusion problems—part I,” Computer Methods in Applied Mechanics and Engineering, vol. 200, no. 41–44, pp. 2916–2929, 2011. View at Publisher · View at Google Scholar
  15. V. John and P. Knobloch, “On spurious oscillations at layers diminishing (SOLD) methods for convection-diffusion equations. I. A review,” Computer Methods in Applied Mechanics and Engineering, vol. 196, no. 17–20, pp. 2197–2215, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. V. John and P. Knobloch, “On the performance of SOLD methods for convection-diffusion problems with interior layers,” International Journal of Computing Science and Mathematics, vol. 1, no. 2–4, pp. 245–258, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. V. John and P. Knobloch, “On spurious oscillations at layers diminishing (SOLD) methods for convection-diffusion equations. II. Analysis for P1 and Q1 finite elements,” Computer Methods in Applied Mechanics and Engineering, vol. 197, no. 21–24, pp. 1997–2014, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. M. Augustin, A. Caiazzo, A. Fiebach et al., “An assessment of discretizations for convection-dominated convection-diffusion equations,” Computer Methods in Applied Mechanics and Engineering, vol. 200, no. 47-48, pp. 3395–3409, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. E. Burman and P. Hansbo, “Edge stabilization for Galerkin approximations of convection-diffusion-reaction problems,” Computer Methods in Applied Mechanics and Engineering, vol. 193, no. 15-16, pp. 1437–1453, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. F. Schieweck, “On the role of boundary conditions for CIP stabilization of higher order finite elements,” Electronic Transactions on Numerical Analysis, vol. 32, pp. 1–16, 2008. View at Google Scholar · View at Zentralblatt MATH
  21. E. Burman and A. Ern, “Stabilized Galerkin approximation of convection-diffusion-reaction equations: discrete maximum principle and convergence,” Mathematics of Computation, vol. 74, no. 252, pp. 1637–1652, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. P. Knobloch, “A generalization of the local projection stabilization for convection-diffusion-reaction equations,” SIAM Journal on Numerical Analysis, vol. 48, no. 2, pp. 659–680, 2010. View at Publisher · View at Google Scholar
  23. R. Codina, “Stabilization of incompressibility and convection through orthogonal sub-scales in finite element methods,” Computer Methods in Applied Mechanics and Engineering, vol. 190, no. 13-14, pp. 1579–1599, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. R. Becker and M. Braack, “A finite element pressure gradient stabilization for the Stokes equations based on local projections,” Calcolo, vol. 38, no. 4, pp. 173–199, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. G. Matthies, P. Skrzypacz, and L. Tobiska, “A unified convergence analysis for local projection stabilisations applied to the Oseen problem,” Mathematical Modelling and Numerical Analysis, vol. 41, no. 4, pp. 713–742, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  26. R. Codina, “Analysis of a stabilized finite element approximation of the Oseen equations using orthogonal subscales,” Applied Numerical Mathematics, vol. 58, no. 3, pp. 264–283, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  27. P. Knobloch and L. Tobiska, “On the stability of finite-element discretizations of convection-diffusion-reaction equations,” IMA Journal of Numerical Analysis, vol. 31, no. 1, pp. 147–164, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  28. P. Knobloch and G. Lube, “Local projection stabilization for advection-diffusion-reaction problems: one-level vs. two-level approach,” Applied Numerical Mathematics, vol. 59, no. 12, pp. 2891–2907, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  29. L. He and L. Tobiska, “The two-level local projection stabilization as an enriched one-level approach,” Advances in Computational Mathematics, vol. 36, no. 4, pp. 503–523, 2012. View at Publisher · View at Google Scholar
  30. D. Juhnke and L. Tobiska, “A local projection type stabilization with exponential enrichments applied to one-dimensional advection-diffusion equations,” Computer Methods in Applied Mechanics and Engineering, vol. 201–204, pp. 179–190, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  31. G. R. Barrenechea and F. Valentin, “A residual local projection method for the Oseen equation,” Computer Methods in Applied Mechanics and Engineering, vol. 199, no. 29–32, pp. 1906–1921, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  32. G. R. Barrenechea, V. John, and P. Knobloch, A Local Projection Stabilisation Finite Element Method Wiith Nonlinear Crosswind Diffusion for Convection-Diffusion-Reaction Equations, WIAS, Berlin, Germany, 2012.
  33. B. Rivière, Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation, vol. 35 of Frontiers in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, USA, 2008. View at Publisher · View at Google Scholar
  34. J. S. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications, vol. 54 of Texts in Applied Mathematics, Springer, New York, NY, USA, 2008. View at Publisher · View at Google Scholar
  35. G. Kanschat, Discontinuous Galerkin Methods for Viscous Incompressible Flow, Advances in Numerical Mathematics. Vieweg, 2008.
  36. D. A. Di Pietro and A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods, vol. 69 of Mathématiques & Applications, Springer, Berlin, 2012.
  37. Y. Bazilevs and T. J. R. Hughes, “Weak imposition of Dirichlet boundary conditions in fluid mechanics,” Computers & Fluids, vol. 36, no. 1, pp. 12–26, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  38. Y. Bazilevs, C. Michler, V. M. Calo, and T. J. R. Hughes, “Weak Dirichlet boundary conditions for wall-bounded turbulent flows,” Computer Methods in Applied Mechanics and Engineering, vol. 196, no. 49–52, pp. 4853–4862, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  39. E. Burman, “A penalty-free nonsymmetric nitsche-type method for the weak imposition of boundary conditions,” SIAM Journal on Numerical Analysis, vol. 50, no. 4, pp. 1959–1981, 2012. View at Google Scholar
  40. R. E. Ewing, J. Wang, and Y. Yang, “A stabilized discontinuous finite element method for elliptic problems,” Numerical Linear Algebra with Applications, vol. 10, no. 1-2, pp. 83–104, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  41. B. Cockburn, J. Gopalakrishnan, and R. Lazarov, “Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems,” SIAM Journal on Numerical Analysis, vol. 47, no. 2, pp. 1319–1365, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  42. P. Causin and R. Sacco, “A discontinuous Petrov-Galerkin method with Lagrangian multipliers for second order elliptic problems,” SIAM Journal on Numerical Analysis, vol. 43, no. 1, pp. 280–302, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  43. B. Cockburn, B. Dong, J. Guzmán, M. Restelli, and R. Sacco, “A hybridizable discontinuous Galerkin method for steady-state convection-diffusion-reaction problems,” SIAM Journal on Scientific Computing, vol. 31, no. 5, pp. 3827–3846, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  44. H. Egger and J. Schöberl, “A hybrid mixed discontinuous Galerkin finite-element method for convection-diffusion problems,” IMA Journal of Numerical Analysis, vol. 30, no. 4, pp. 1206–1234, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  45. C. L. Bottasso, S. Micheletti, and R. Sacco, “A multiscale formulation of the discontinuous Petrov-Galerkin method for advective-diffusive problems,” Computer Methods in Applied Mechanics and Engineering, vol. 194, no. 25-26, pp. 2819–2838, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  46. P. Causin, R. Sacco, and C. L. Bottasso, “Flux-upwind stabilization of the discontinuous Petrov-Galerkin formulation with Lagrange multipliers for advection-diffusion problems,” Mathematical Modelling and Numerical Analysis, vol. 39, no. 6, pp. 1087–1114, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  47. L. Demkowicz and J. Gopalakrishnan, “A class of discontinuous Petrov-Galerkin methods. II. Optimal test functions,” Numerical Methods for Partial Differential Equations, vol. 27, no. 1, pp. 70–105, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  48. N. Heuer and L. Demkowicz, Robust Dpg Method for Convection-Dominated Diffusion Problems, University of Texas at Austin, ICES.
  49. J. Chan, N. Heuer, T. Bui-Thanh, and D. Demkowicz, “Robust dpg method for convection-dominated diffusion problems ii: a natural in flow condition,” submitted.
  50. R. J. Labeur and G. N. Wells, “A Galerkin interface stabilisation method for the advection-diffusion and incompressible Navier-Stokes equations,” Computer Methods in Applied Mechanics and Engineering, vol. 196, no. 49–52, pp. 4985–5000, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  51. G. N. Wells, “Analysis of an interface stabilized finite element method: the advection-diffusion-reaction equation,” SIAM Journal on Numerical Analysis, vol. 49, no. 1, pp. 87–109, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  52. D. Kuzmin and S. Turek, “Flux correction tools for finite elements,” Journal of Computational Physics, vol. 175, no. 2, pp. 525–558, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  53. D. Kuzmin, Free Cfd Book: A Guide to Numerical Methods for Transport Equations, University Erlangen-Nürnberg, 2010.
  54. E. Burman and M. A. Fernández, “Finite element methods with symmetric stabilization for the transient convection-diffusion-reaction equation,” Computer Methods in Applied Mechanics and Engineering, vol. 198, no. 33–36, pp. 2508–2519, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  55. N. Ahmed, G. Matthies, L. Tobiska, and H. Xie, “Discontinuous Galerkin time stepping with local projection stabilization for transient convection-diffusion-reaction problems,” Computer Methods in Applied Mechanics and Engineering, vol. 200, no. 21-22, pp. 1747–1756, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  56. M. Feistauer, J. Hájek, and K. Svadlenka, “Space-time discontinuous Galerkin method for solving nonstationary convection-diffusion-reaction problems,” Applications of Mathematics, vol. 52, no. 3, pp. 197–233, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  57. E. Burman and A. Ern, “Implicit-explicit Runge-Kutta schemes and finite elements with symmetric stabilization for advection-diffusion equations,” Mathematical Modelling and Numerical Analysis, vol. 46, no. 4, pp. 681–707, 2012. View at Publisher · View at Google Scholar
  58. V. Dolejší, M. Feistauer, V. Kučera, and V. Sobotíková, “An optimal L(L2)-error estimate for the discontinuous Galerkin approximation of a nonlinear non-stationary convection-diffusion problem,” IMA Journal of Numerical Analysis, vol. 28, no. 3, pp. 496–521, 2008. View at Publisher · View at Google Scholar
  59. V. Dolejší and M. Vlasák, “Analysis of a BDF-DGFE scheme for nonlinear convection-diffusion problems,” Numerische Mathematik, vol. 110, no. 4, pp. 405–447, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  60. V. John and J. Novo, “Error analysis of the SUPG finite element discretization of evolutionary convection-diffusion-reaction equations,” SIAM Journal on Numerical Analysis, vol. 49, no. 3, pp. 1149–1176, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  61. E. Burman and G. Smith, “Analysis of the space semi-discretized SUPG method for transient convection-diffusion equations,” Mathematical Models and Methods in Applied Sciences, vol. 21, no. 10, pp. 2049–2068, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  62. E. Burman, “Consistent SUPG-method for transient transport problems: stability and convergence,” Computer Methods in Applied Mechanics and Engineering, vol. 199, no. 17–20, pp. 1114–1123, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  63. H.-G. Roos and M. Schopf, “Error estimation in energy norms: is it necessary to fit the mesh to the boundary layer?” submitted.
  64. R. Lin, “Discontinuous discretization for least-squares formulation of singularly perturbed reaction-diffusion problems in one and two dimensions,” SIAM Journal on Numerical Analysis, vol. 47, no. 1, pp. 89–108, 2008. View at Publisher · View at Google Scholar
  65. T. K. Nilssen, X.-C. Tai, and R. Winther, “A robust nonconforming H2-element,” Mathematics of Computation, vol. 70, no. 234, pp. 489–505, 2001. View at Publisher · View at Google Scholar
  66. J. Guzmán, D. Leykekhman, and M. Neilan, “A family of non-conforming elements and the analysis of Nitsche's method for a singularly perturbed fourth order problem,” Calcolo, vol. 49, no. 2, pp. 95–125, 2012. View at Publisher · View at Google Scholar
  67. O. Axelsson, E. Glushkov, and N. Glushkova, “The local Green's function method in singularly perturbed convection-diffusion problems,” Mathematics of Computation, vol. 78, no. 265, pp. 153–170, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  68. L. Demkowicz and J. Gopalakrishnan, “A class of discontinuous Petrov-Galerkin methods. Part I: the transport equation,” Computer Methods in Applied Mechanics and Engineering, vol. 199, no. 23-24, pp. 1558–1572, 2010. View at Publisher · View at Google Scholar
  69. L. Demkowicz, J. Gopalakrishnan, and A. H. Niemi, “A class of discontinuous Petrov-Galerkin methods. Part III: adaptivity,” Applied Numerical Mathematics, vol. 62, no. 4, pp. 396–427, 2012. View at Publisher · View at Google Scholar
  70. H. Han and Z. Huang, “Tailored finite point method based on exponential bases for convection-diffusion-reaction equation,” Mathematics of Computation, vol. 82, no. 281, pp. 213–226, 2013. View at Publisher · View at Google Scholar
  71. X. Li, X. Yu, and G. Chen, “Error estimates of the finite element method with weighted basis functions for a singularly perturbed convection-diffusion equation,” Journal of Computational Mathematics, vol. 29, no. 2, pp. 227–242, 2011. View at Google Scholar
  72. H. Riou and P. Ladevéze, “A new numerical strategy for the resolution of high- péclet advection-diffusion problems,” Computer Methods in Applied Mechanics and Engineering, vol. 241–244, pp. 302–310, 2012. View at Google Scholar
  73. A. L. Lombardi and P. Pietra, “Exponentially fitted discontinuous galerkin schemes for singularly perturbed problems,” Numerical Methods for Partial Differential Equations, vol. 28, no. 6, pp. 1747–1777, 2012. View at Google Scholar
  74. F. Brezzi, L. D. Marini, and P. Pietra, “Two-dimensional exponential fitting and applications to drift-diffusion models,” SIAM Journal on Numerical Analysis, vol. 26, no. 6, pp. 1342–1355, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  75. S. Holst, A. Jüngel, and P. Pietra, “A mixed finite-element discretization of the energy-transport model for semiconductors,” SIAM Journal on Scientific Computing, vol. 24, no. 6, pp. 2058–2075, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  76. R. B. Kellogg and M. Stynes, “Corner singularities and boundary layers in a simple convection-diffusion problem,” Journal of Differential Equations, vol. 213, no. 1, pp. 81–120, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  77. R. B. Kellogg and M. Stynes, “Sharpened bounds for corner singularities and boundary layers in a simple convection-diffusion problem,” Applied Mathematics Letters, vol. 20, no. 5, pp. 539–544, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  78. T. Linß and M. Stynes, “Asymptotic analysis and Shishkin-type decomposition for an elliptic convection-diffusion problem,” Journal of Mathematical Analysis and Applications, vol. 261, no. 2, pp. 604–632, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  79. E. O'Riordan and G. I. Shishkin, “A technique to prove parameter-uniform convergence for a singularly perturbed convection-diffusion equation,” Journal of Computational and Applied Mathematics, vol. 206, no. 1, pp. 136–145, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  80. N. S. Bakhvalov, “On the optimization of the methods for solving boundary value problems in the presence of a boundary layer,” Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki, vol. 9, pp. 841–859, 1969. View at Google Scholar
  81. T. Linß, “Layer-adapted meshes for convection-diffusion problems,” Computer Methods in Applied Mechanics and Engineering, vol. 192, no. 9-10, pp. 1061–1105, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  82. T. Linß, Layer Adapted Meshes for Convection-Diffusion Problems, Habilitation TU Dresden, 2006.
  83. H.-G. Roos, “Error estimates for linear finite elements on Bakhvalov-type meshes,” Applications of Mathematics, vol. 51, no. 1, pp. 63–72, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  84. T. Apel and G. Lube, “Anisotropic mesh refinement in stabilized Galerkin methods,” Numerische Mathematik, vol. 74, no. 3, pp. 261–282, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  85. Z. Xie and Z. Zhang, “Uniform superconvergence analysis of the discontinuous Galerkin method for a singularly perturbed problem in 1-D,” Mathematics of Computation, vol. 79, no. 269, pp. 35–45, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  86. J. M. Melenk, hp-Finite Element Methods for Singular Perturbations, vol. 1796 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2002. View at Publisher · View at Google Scholar
  87. T. Linß, H.-G. Roos, and M. Schopf, “Nitsche-mortaring for singularly perturbed convection-diffusion problems,” Advances in Computational Mathematics, vol. 36, no. 4, pp. 581–603, 2012. View at Publisher · View at Google Scholar
  88. H.-G. Roos and T. Skalický, “A comparison of the finite element method on Shishkin and Gartland-type meshes for convection-diffusion problems,” CWI Quarterly, vol. 10, no. 3-4, pp. 277–300, 1997, International Workshop on the Numerical Solution of Thin-Layer Phenomena (Amsterdam). View at Google Scholar
  89. S.-T. Liu and Y. Xu, “Graded Galerkin methods for the high-order convection-diffusion problem,” Numerical Methods for Partial Differential Equations, vol. 25, no. 6, pp. 1261–1282, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  90. M. Stynes and E. O'Riordan, “A uniformly convergent Galerkin method on a Shishkin mesh for a convection-diffusion problem,” Journal of Mathematical Analysis and Applications, vol. 214, no. 1, pp. 36–54, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  91. M. Dobrowolski and H.-G. Roos, “A priori estimates for the solution of convection-diffusion problems and interpolation on Shishkin meshes,” Zeitschrift für Analysis und ihre Anwendungen, vol. 16, no. 4, pp. 1001–1012, 1997. View at Google Scholar · View at Zentralblatt MATH
  92. H.-G. Roos and T. Linß, “Sufficient conditions for uniform convergence on layer-adapted grids,” Computing, vol. 63, no. 1, pp. 27–45, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  93. H.-G. Roos and M. Schopf, “Analysis of finite element methods on Bakhvalov-type meshes for linear convection-diffusion problems in 2D,” Applications of Mathematics, vol. 57, no. 2, pp. 97–108, 2012. View at Publisher · View at Google Scholar
  94. R. G. Durán and A. L. Lombardi, “Finite element approximation of convection diffusion problems using graded meshes,” Applied Numerical Mathematics, vol. 56, no. 10-11, pp. 1314–1325, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  95. H.-G. Roos and M. Schopf, “Finite elements on locally uniform meshes for convection-diffusion problems with boundary layers,” Computing, vol. 92, no. 4, pp. 285–296, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  96. S. Franz and T. Linß, “Superconvergence analysis of the Galerkin FEM for a singularly perturbed convection-diffusion problem with characteristic layers,” Numerical Methods for Partial Differential Equations, vol. 24, no. 1, pp. 144–164, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  97. P. Sun, L. Chen, and J. Xu, “Numerical studies of adaptive finite element methods for two dimensional convection-dominated problems,” Journal of Scientific Computing, vol. 43, no. 1, pp. 24–43, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  98. T. Linß and M. Stynes, “Numerical methods on shishkin meshes for linear convection- diffusion problems,” Computer Methods in Applied Mechanics and Engineering, no. 190, pp. 3527–3542, 2001. View at Google Scholar
  99. C. Grossmann and H.-G. Roos, Numerical Treatment of Partial Differential Equations, Universitext, Springer, Berlin, Germany, 2007. View at Publisher · View at Google Scholar
  100. M. Stynes and L. Tobiska, “The SDFEM for a convection-diffusion problem with a boundary layer: optimal error analysis and enhancement of accuracy,” SIAM Journal on Numerical Analysis, vol. 41, no. 5, pp. 1620–1642, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  101. J. Zhang and L. Mei, “Pointwise error estimates of the bilinear sdfem on shishkin meshes,” Numerical Methods for Partial Differential Equations. In press.
  102. N. Kopteva, “How accurate is the streamline-diffusion FEM inside characteristic (boundary and interior) layers?” Computer Methods in Applied Mechanics and Engineering, vol. 193, no. 45–47, pp. 4875–4889, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  103. S. Franz, T. Linß, and H.-G. Roos, “Superconvergence analysis of the SDFEM for elliptic problems with characteristic layers,” Applied Numerical Mathematics, vol. 58, no. 12, pp. 1818–1829, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  104. Q. Lin and J. Lin, Finite Element Methods: Accuracy and Improvement, Mathematics Monograph Series, Science Press, 2006.
  105. R. G. Durán, A. L. Lombardi, and M. I. Prieto, “Superconvergence for finite element approximation of a convection-diffusion equation using graded meshes,” IMA Journal of Numerical Analysis, vol. 32, no. 2, pp. 511–533, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  106. R. G. Durán, A. L. Lombardi, and M. I. Prieto, “Supercloseness on graded meshes for Q1 finite element approximation of a reaction-diffusion equation,” Journal of Computational and Applied Mathematics, vol. 242, pp. 232–242, 2013. View at Google Scholar
  107. G. Zhu and S. Chen, “Convergence and superconvergence analysis of an anisotropic nonconforming finite element methods for singularly perturbed reaction-diffusion problems,” Journal of Computational and Applied Mathematics, vol. 234, no. 10, pp. 3048–3063, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  108. S. Franz, “Convergence phenomena for Qp-elements for convection-diffusion problems,” Numerical Methods for Partial Differential Equations, vol. 29, pp. 280–296, 2013. View at Google Scholar
  109. M. Stynes and L. Tobiska, “Using rectangular Qp elements in the SDFEM for a convection-diffusion problem with a boundary layer,” Applied Numerical Mathematics, vol. 58, no. 12, pp. 1789–1802, 2008. View at Publisher · View at Google Scholar
  110. S. Franz, “Superconvergence for pointwise interpolation in convection-diffusion problems”.
  111. S. Franz, T. Linß, H.-G. Roos, and S. Schiller, “Uniform superconvergence of a finite element method with edge stabilization for convection-diffusion problems,” Journal of Computational Mathematics, vol. 28, no. 1, pp. 32–44, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  112. S. Franz, “Continuous interior penalty method on a Shishkin mesh for convection-diffusion problems with characteristic boundary layers,” Computer Methods in Applied Mechanics and Engineering, vol. 197, no. 45–48, pp. 3679–3686, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  113. G. Matthies, “Local projection stabilisation for higher order discretisations of convection-diffusion problems on Shishkin meshes,” Advances in Computational Mathematics, vol. 30, no. 4, pp. 315–337, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  114. G. Matthies, “Local projection methods on layer-adapted meshes for higher order discretisations of convection-diffusion problems,” Applied Numerical Mathematics, vol. 59, no. 10, pp. 2515–2533, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  115. S. Franz and G. Matthies, “Local projection stabilisation on S-type meshes for convection-diffusion problems with characteristic layers,” Computing, vol. 87, no. 3-4, pp. 135–167, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  116. H.-G. Roos and H. Zarin, “A supercloseness result for the discontinuous Galerkin stabilization of convection-diffusion problems on Shishkin meshes,” Numerical Methods for Partial Differential Equations, vol. 23, no. 6, pp. 1560–1576, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  117. Z. Xie and Z. Zhang, “Superconvergence of DG method for one-dimensional singularly perturbed problems,” Journal of Computational Mathematics, vol. 25, no. 2, pp. 185–200, 2007. View at Google Scholar · View at Zentralblatt MATH
  118. H. Zhu and Z. Zhang, “Convergence analysis of the LDG method applied to singularly perturbed problems,” Numerical Methods for Partial Differential Equations. In press.
  119. H. Zhu and Z. Zhang, “Uniform convergence of the LDG method for a singularly perturbed problem with the exponential boundary layer”.
  120. S. Franz, L. Tobiska, and H. Zarin, “A new approach to recovery of discontinuous Galerkin,” Journal of Computational Mathematics, vol. 27, no. 6, pp. 697–712, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  121. S. Franz, F. Liu, H.-G. Roos, M. Stynes, and A. Zhou, “The combination technique for a two-dimensional convection-diffusion problem with exponential layers,” Applications of Mathematics, vol. 54, no. 3, pp. 203–223, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  122. F. Liu, N. Madden, M. Stynes, and A. Zhou, “A two-scale sparse grid method for a singularly perturbed reaction-diffusion problem in two dimensions,” IMA Journal of Numerical Analysis, vol. 29, no. 4, pp. 986–1007, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  123. H.-G. Roos and M. Schopf, “Convergence and stability in balanced norms of finite element methods on shishkin meshes for reaction-diffusion problems”.
  124. R. Lin and M. Stynes, “A balanced finite element method for singularly perturbed reaction-diffusion problems,” SIAM Journal on Numerical Analysis, vol. 50, pp. 2729–2743, 2012. View at Google Scholar
  125. S. Franz and H. G. Roos, “Error estimation in a balanced norm for a convection- diffusion problem with two different boundary layers,” submitted.
  126. B. Garcia-Archilla, “Shishkin mesh simulation: a new stabilization technique for convection-diffusion problems”.
  127. V. B. Andreev and N. Kopteva, “Pointwise approximation of corner singularities for a singularly perturbed reaction-diffusion equation in an L-shaped domain,” Mathematics of Computation, vol. 77, no. 264, pp. 2125–2139, 2008. View at Publisher · View at Google Scholar
  128. V. B. Andreev, “Uniform grid approximation of nonsmooth solutions of a mixed boundary value problem for a singularly perturbed reaction-diffusion equation in a rectangle,” Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki, vol. 48, no. 1, pp. 90–114, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  129. S. Franz, R. B. Kellogg, and M. Stynes, “Galerkin and streamline diffusion finite element methods on a Shishkin mesh for a convection-diffusion problem with corner singularities,” Mathematics of Computation, vol. 81, no. 278, pp. 661–685, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  130. S. Franz, R. B. Kellogg, and M. Stynes, “On the choice of mesh for a singularly perturbed problem with a corner singularity,” in Proceedings of the Boundary and Interior Layers, Computational and Asymptotic Methods (BAIL '10), vol. 81 of Lecture Notes in Computational Science and Engineering, pp. 119–126, Springer, Heidelberg, Germany, 2011.
  131. H. G. Roos and L. Ludwig, “Finite element superconvergence on shishkin meshes for convection-diffusion problems with corner singularities,” submitted.
  132. H.-G. Roos and L. Ludwig, “Superconvergence for convection-diffusion problems with low regularity,” Proceedings in Applied Mathematics and Mechanics, pp. 173–187, 2012. View at Google Scholar
  133. V. B. Andreev, “Uniform grid approximation of nonsmooth solutions of a singularly perturbed convection-diffusion equation in a rectangle,” Differentsial'nye Uravneniya, vol. 45, no. 7, pp. 954–964, 2009. View at Publisher · View at Google Scholar
  134. V. B. Andreev, “Pointwise approximation of corner singularities for singularly perturbed elliptic problems with characteristic layers,” International Journal of Numerical Analysis and Modeling, vol. 7, no. 3, pp. 416–427, 2010. View at Google Scholar · View at Zentralblatt MATH
  135. C. Clavero, J. L. Gracia, and M. Stynes, “A simpler analysis of a hybrid numerical method for time-dependent convection-diffusion problems,” Journal of Computational and Applied Mathematics, vol. 235, no. 17, pp. 5240–5248, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  136. L. Kaland and H.-G. Roos, “Parabolic singularly perturbed problems with exponential layers: robust discretizations using finite elements in space on Shishkin meshes,” International Journal of Numerical Analysis and Modeling, vol. 7, no. 3, pp. 593–606, 2010. View at Google Scholar · View at Zentralblatt MATH
  137. V. Dolejší and H.-G. Roos, “BDF-FEM for parabolic singularly perturbed problems with exponential layers on layers-adapted meshes in space,” Neural, Parallel & Scientific Computations, vol. 18, no. 2, pp. 221–235, 2010. View at Google Scholar
  138. H.-G. Roos and M. Vlasak, “An optimal uniform a priori error estimate for an un- steady singularly perturbed problem,” International Journal of Numerical Analysis and Modeling Computing and Information, vol. 9, no. 1, pp. 56–72, 2012. View at Google Scholar
  139. R. Verfürth, “Robust a posteriori error estimates for stationary convection-diffusion equations,” SIAM Journal on Numerical Analysis, vol. 43, no. 4, pp. 1766–1782, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  140. L. El Alaoui, A. Ern, and E. Burman, “A priori and a posteriori analysis of non-conforming finite elements with face penalty for advection-diffusion equations,” IMA Journal of Numerical Analysis, vol. 27, no. 1, pp. 151–171, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  141. B. Achchab, M. El Fatini, A. Ern, and A. Souissi, “A posteriori error estimates for subgrid viscosity stabilized approximations of convection-diffusion equations,” Applied Mathematics Letters, vol. 22, no. 9, pp. 1418–1424, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  142. B. Achchab, M. El Fatini, and A. Souissi, “A posteriori error analysis for multiscale stabilization of convection-diffusion-reaction problems: unsteady state,” International Journal of Mathematics and Statistics, vol. 4, no. S09, pp. 3–11, 2009. View at Google Scholar
  143. T. Apel, S. Nicaise, and D. Sirch, “A posteriori error estimation of residual type for anisotropic diffusion-convection-reaction problems,” Journal of Computational and Applied Mathematics, vol. 235, no. 8, pp. 2805–2820, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  144. B. Achchab, A. Agouzal, M. El Fatini, and A. Souissi, “Robust hierarchical a posteriori error estimators for stabilized convection-diffusion problems,” Numerical Methods for Partial Differential Equations, vol. 28, no. 5, pp. 1717–1728, 2012. View at Google Scholar
  145. B. Achchab, A. Benjouad, M. El Fatini, A. Souissi, and G. Warnecke, “Robust a posteriori error estimates for subgrid stabilization of non-stationary convection dominated diffusive transport,” Applied Mathematics and Computation, vol. 218, no. 9, pp. 5276–5291, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  146. I. Cheddadi, R. Fučík, M. I. Prieto, and M. Vohralík, “Guaranteed and robust a posteriori error estimates for singularly perturbed reaction-diffusion problems,” Mathematical Modelling and Numerical Analysis, vol. 43, no. 5, pp. 867–888, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  147. M. Vohralík, A Posteriori Error Estimates for Efficiency and Error Control in Numerical Simulations, vol. 6 of Lecture Notes, Université Pierre et Marie Curie, Paris, France, 2012.
  148. M. Ainsworth, A. Allendes, G. R. Barrenechea, and R. Rankin, Fully Computable a Posteriori Error Bounds for Stabilized Fem Approximations of Convection-Diffusion-Reaction Problems in Three Dimensions, University of Strathclyde.
  149. L. Zhu and D. Schötzau, “A robust a posteriori error estimate for hp-adaptive DG methods for convection-diffusion equations,” IMA Journal of Numerical Analysis, vol. 31, no. 3, pp. 971–1005, 2011. View at Publisher · View at Google Scholar
  150. A. Ern, A. F. Stephansen, and M. Vohralík, “Guaranteed and robust discontinuous Galerkin a posteriori error estimates for convection-diffusion-reaction problems,” Journal of Computational and Applied Mathematics, vol. 234, no. 1, pp. 114–130, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  151. D. Kuzmin and S. Korotov, “Goal-oriented a posteriori error estimates for transport problems,” Mathematics and Computers in Simulation, vol. 80, no. 8, pp. 1674–1683, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  152. V. Dolejší, A. Ern, and M. Vohralík, “A framework for robust a posteriori error control in unsteady nonlinear advection-diffusion problems”.
  153. T. Linß, “A posteriori error estimation for higher order fem applied to one-dimensional reaction-diffusion problems”.
  154. N. Kopteva, “Maximum norm a posteriori error estimate for a 2D singularly perturbed semilinear reaction-diffusion problem,” SIAM Journal on Numerical Analysis, vol. 46, no. 3, pp. 1602–1618, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  155. S. Franz and N. Kopteva, “Green's function estimates for a singularly perturbed convection-diffusion problem,” Journal of Differential Equations, vol. 252, no. 2, pp. 1521–1545, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  156. S. Franz and N. Kopteva, “Green's function estimates for a singularly perturbed convection-diffusion problem in three dimensions,” International Journal of Numerical Analysis and Modeling B, vol. 2, no. 2-3, pp. 124–141, 2011. View at Google Scholar
  157. N. Kopteva and T. Linß, “Maximum norm a posteriori error estimation for a time-dependent reaction-diffusion problem,” Computational Methods in Applied Mathematics, vol. 12, pp. 189–205, 2012. View at Google Scholar
  158. T. Richter, “A posteriori error estimation and anisotropy detection with the dual-weighted residual method,” International Journal for Numerical Methods in Fluids, vol. 62, no. 1, pp. 90–118, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  159. W. Huang, L. Kamenski, and J. Lang, “A new anisotropic mesh adaptation method based upon hierarchical a posteriori error estimates,” Journal of Computational Physics, vol. 229, no. 6, pp. 2179–2198, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  160. G. Kunert, “Toward anisotropic mesh construction and error estimation in the finite element method,” Numerical Methods for Partial Differential Equations, vol. 18, no. 5, pp. 625–648, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  161. R. Schneider, “A review of anisotropic refinement methods for triangular meshes in FEM,” in Advanced Finite Element Methods and Applications, vol. 66 of Lecture Notes in Applied and Computational Mechanics, pp. 133–152, Springer, Heidelberg, Germany, 2013. View at Publisher · View at Google Scholar
  162. T. Linss and M. Stynes, “Numerical solution of systems of singularly perturbed differential equations,” Computational Methods in Applied Mathematics, vol. 9, no. 2, pp. 165–191, 2009. View at Google Scholar · View at Zentralblatt MATH
  163. T. Linß and N. Madden, “Layer-adapted meshes for a linear system of coupled singularly perturbed reaction-diffusion problems,” IMA Journal of Numerical Analysis, vol. 29, no. 1, pp. 109–125, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  164. J. M. Melenk, L. Oberbroeckling, and C. Xenophontos, “Analytic regularity for a singularly perturbed system of reaction-diffusion equations with multiple scales: a road map,” submitted.
  165. T. Linß, “Analysis of an upwind finite-difference scheme for a system of coupled singularly perturbed convection-diffusion equations,” Computing, vol. 79, no. 1, pp. 23–32, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  166. H.-G. Roos and C. Reibiger, “Numerical analysis of a system of singularly perturbed convection-diffusion equations related to optimal control,” Numerical Mathematics: Theory, Methods and Applications, vol. 4, no. 4, pp. 562–575, 2011. View at Publisher · View at Google Scholar
  167. E. O'Riordan and M. Stynes, “Numerical analysis of a strongly coupled system of two singularly perturbed convection-diffusion problems,” Advances in Computational Mathematics, vol. 30, no. 2, pp. 101–121, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  168. H.-G. Roos and C. Reibiger, “Analysis of a strongly coupled system of two convection-diffusion equations with full layer interaction,” Zeitschrift für Angewandte Mathematik und Mechanik, vol. 91, no. 7, pp. 537–543, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  169. T. Linss, “Analysis of a system of singularly perturbed convection-diffusion equations with strong coupling,” SIAM Journal on Numerical Analysis, vol. 47, no. 3, pp. 1847–1862, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  170. H.-G. Roos, “Special features of strongly coupled systems of convection-diffusion equations with two small parameters,” Applied Mathematics Letters, vol. 25, no. 8, pp. 1127–1130, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH