Publications since 1991

Books

H.-G.Roos, M.Stynes & L.Tobiska, Numerical Methods for Singularly Perturbed Differential Equations -- Convection-Diffusion and Flow Problems, Springer Series in Computational Mathematics Volume 24, ISBN 3-540-60718-8, Springer-Verlag, Berlin, 1996. Postscript file List of Contents

 

C.Grossmann, H.-G.Roos & M.Stynes, Numerical Treatment of Partial Differential Equations, Universitext Series, Springer-Verlag, Berlin, 2007. cover table of contents

 

H.-G.Roos, M.Stynes & L.Tobiska, Robust Numerical Methods for Singularly Perturbed Differential Equations, 2nd edition, Springer Series in Computational Mathematics Volume 24, ISBN 978-3-540-34466-7, Springer-Verlag, Berlin, 2008.

Book Chapter

M.Stynes, Steady-state convection-diffusion problems, in Acta Numerica 2005 (A.Iserles, ed.), Cambridge University Press, Cambridge, 2005, pages 445-508.

Papers

  1. E.O'Riordan & M.Stynes, An analysis of some exponentially fitted finite element methods for singularly perturbed elliptic problems, Chapter 7 of Computational Methods for Boundary and Interior Layers in Several Dimensions (J.J.H.Miller, ed.), Boole Press, Dublin, 1991, 138-153.
  2. M.Stynes & E.O'Riordan, An analysis of a singularly perturbed two-point boundary value problem using only finite element techniques, Math. Comp. 56 (1991), 663-675.
  3. E.O'Riordan, A.Hegarty & M.Stynes, Construction and analysis of some exponentially upwinded finite element methods for singularly perturbed two-dimensional elliptic problems, in Numerical Methods in Singularly Perturbed Problems - Proc. of Intl. Seminar on Applied Math '91, Dresden, Germany (H.-G.Roos, A.Felgenhauer, L.Angermann, eds.), Technische Universität Dresden, 1991, 91-96.
  4. M.Stynes & W.Guo, Finite element analysis of an advection-diffusion equation, in Numerical Methods in Singularly Perturbed Problems - Proc. of Intl. Seminar on Applied Math '91, Dresden, Germany (H.-G.Roos, A.Felgenhauer, L.Angermann, eds.), Technische Universität Dresden, 1991, 123-128.
  5. E.O'Riordan & M.Stynes, A globally uniformly convergent finite element method for a singularly perturbed elliptic problem in two dimensions, Math. Comp. 57 (1991), 47-62.
  6. H.-G.Roos & M.Stynes, A uniformly convergent discretization method for a fourth-order singular perturbation problem, Bonn. Math. Scrift. 228 (1991), 30-40.
  7. A.F.Hegarty, E.O'Riordan & M.Stynes, A comparison of uniformly convergent difference schemes for two-dimensional convection-diffusion problems, J. Comp. Phys. 105 (1993), 24-32.
  8. M.Stynes, Numerical solution of convection-diffusion problems, Bull. Irish Math. Soc. 30 (1993), 41-55.
  9. W.Guo & M.Stynes, Finite element analysis of an exponentially fitted lumped scheme for time-dependent convection-diffusion problems, Numer. Math. 66 (1993), 347-371.
  10. W.Guo & M.Stynes, Pointwise error estimates for a streamline diffusion scheme on a Shishkin mesh for a convection-diffusion problem, in Proc. CFD94 (J.J.Gottlieb & C.R.Ethier, eds.), University of Toronto, 1994, 479-486.
  11. G.Sun & M.Stynes, Finite element methods on piecewise equidistant meshes for interior turning point problems, Numer. Algorithms 8 (1994), 111-129.
  12. K.W.Morton & M.Stynes, An analysis of the cell vertex method, Modél. Math. Anal. Numér. 28 (1994), 699-724.
  13. W.Guo & M.Stynes, Finite element analysis of an exponentially fitted non-lumped scheme for advection-diffusion equations, Appl. Numer. Math. 15 (1994), 375-393.
  14. M.Stynes, A streamline diffusion finite element method on a Shishkin mesh for a convection-diffusion problem, Rendiconti Mat. Poli. Milano 64 (1994), 129-140.
  15. G.Sun & M.Stynes, Finite element methods for singularly perturbed higher order elliptic two-point boundary value problems I: reaction-diffusion type, IMA J. Numer. Anal. 15 (1995), 117-139.
  16. G.Sun & M.Stynes, Finite element methods for singularly perturbed higher order elliptic two-point boundary value problems II: convection-diffusion type, IMA J. Numer. Anal. 15 (1995), 197-219.
  17. D.Adam, A.Felgenhauer, H.-G.Roos & M.Stynes, A nonconforming finite element method for a singularly perturbed boundary value problem, Computing 54 (1995), 1-25.
  18. M.Stynes & L.Tobiska, Necessary L2-uniform convergence conditions for difference schemes for two-dimensional convection-diffusion problems, Comput. Math. Applic. 29 (1995), 45-53.
  19. G.Sun & M.Stynes, An almost fourth order uniformly convergent difference scheme for a semilinear singularly perturbed reaction-diffusion problem, Numer. Math. 70 (1995), 487-500.
  20. M.Stynes, Finite volume methods for convection-diffusion problems, invited plenary lecture at Modelling 94 Conf., Prague, Czech Republic, J. Comput. Appl. Math. 63 (1995), 83-90.
  21. G.Sun & M.Stynes, A uniformly convergent method for a singularly perturbed semilinear reaction-diffusion problem with multiple solutions, Math. Comp. 65 (1996), 1085-1109.
  22. H.-G.Roos & M.Stynes, Necessary conditions for uniform convergence of finite difference schemes for convection-diffusion problems with exponential and parabolic layers, Appl. Math. 41 (1996), 269-280. Postscript file
  23. N.Madden & M.Stynes, Linear enhancements of the streamline diffusion method for convection-diffusion problems, Computers Math. Applic. 32 (1996), 29-42. Abstract
  24. W.Guo & M.Stynes, Pointwise error estimates for a streamline diffusion scheme on a Shishkin mesh for a time-dependent convection-diffusion problem, IMA J. Numer. Anal. 17(1997), 29-59. Postscript file
  25. N.Madden & M.Stynes, Efficient generation of oriented meshes for solving convection-diffusion problems, Int. J. Numer. Methods Engrg. 40 (1997), 565-576. Gzipped postscript file
  26. W.Guo & M.Stynes, An analysis of a cell vertex finite volume method for a parabolic convection-diffusion problem, Math. Comp. 66 (1997), 105-124. postscript file
  27. M.Stynes & H.-G.Roos, The midpoint upwind scheme, Appl. Numer. Math. 23 (1997), 361-374. Abstract
  28. R.B.Kellogg & M.Stynes, Optimal approximability of solutions of singularly perturbed two-point boundary value problems, SIAM J. Numer. Anal. 34 (1997), 1808-1816. Postscript file
  29. K.W.Morton, M.Stynes & E.Süli, Analysis of a cell-vertex finite volume method for convection-diffusion problems, Math. Comp. 66 (1997), 1389-1406. Postscript file
  30. M.Stynes & E.O'Riordan, A uniformly convergent Galerkin method on a Shishkin mesh for a convection-diffusion problem, J. Math. Anal. Appl. 214 (1997), 36-54. AbstractPostscript file
  31. M.Stynes & L.Tobiska, Error estimates and numerical experiments for streamline-diffusion-type methods on arbitrary and Shishkin meshes, CWI Quarterly 10 (1997), 337-352. Abstract
  32. R.Sacco & M.Stynes, Finite element methods for convection-diffusion problems using exponential splines on triangles, Computers Math. Applic. 35 (1998), 35-45. Gzipped postscript file
  33. L.P.Franca, A.Nesliturk & M.Stynes, On the stability of residual-free bubbles for convection-diffusion problems and their approximation by a two-level finite element method, Comput. Methods Appl. Mech. Engrg. 166 (1998), 35-49. Postscript file
  34. M.Stynes & L.Tobiska, A finite difference analysis of a streamline diffusion method on a Shishkin mesh, Numer. Algorithms 18 (1998), 337-360.
  35. N.Madden, M.Stynes & G.P.Thomas, On the development of complete flow models for wave-current interactions, in Proc. Coastal Dynamics 97 (E.B.Thornton, ed.), ASCE, Reston, Virginia, 1998, 405-414. Abstract
  36. T.Linss & M.Stynes, A hybrid difference scheme on a Shishkin mesh for linear convection-diffusion problems, Appl. Numer. Math. 31 (1999), 255-270.
  37. R.B.Kellogg & M.Stynes, n-widths and singularly perturbed boundary value problems, SIAM J. Numer. Anal. 36 (1999), 1604-1620. Abstract
  38. L.Tobiska, G.Matthies & M.Stynes, Convergence properties of the streamline-diffusion finite element method on a Shishkin mesh for singularly perturbed elliptic equations with exponential layers, in Analytical and Numerical Methods for Convection-Dominated and Singularly Perturbed Problems, Nova Science Publishers, New York, 2000. Abstract
  39. M.Stynes & L.Tobiska, The streamline-diffusion method for nonconforming $Q_1^{rot}$ elements on rectangular tensor-product meshes, IMA J. Numer. Anal. 21 (2001), 123-142 . Abstract
  40. T.Linss & M.Stynes, The SDFEM on Shishkin meshes for linear convection-diffusion problems, Numer. Math. 87 (2001), 457-484. Abstract
  41. T.Linss & M.Stynes, Numerical methods on Shishkin meshes for linear convection-diffusion problems, Comput. Methods Appl. Mech. Engrg. 190 (2001), 3527-3542. Abstract
  42. M.Stynes & L.Tobiska, Analysis of the streamline-diffusion finite element method on a piecewise uniform mesh for a convection-diffusion problem with exponential layers, East-West J. Numer. Math. 9 (2001), 59-76. Abstract
  43. R.B.Kellogg & M.Stynes, n-widths and singularly perturbed boundary value problems II, SIAM J. Numer. Anal. 39 (2001), 690-707. Abstract
  44. T.Linss & M.Stynes, Asymptotic analysis and Shishkin-type decomposition for an elliptic convection-diffusion problem, J. Math. Anal. Appl. 261 (2001), 604-632. Abstract
  45. M.Stynes & R.B.Kellogg, Quasioptimality of the streamline diffusion finite element parameter in 3D, in Finite element methods. Three-dimensional problems (P.Neittaanmäki & M.Kri˛ek, eds.), GAKUTO Internat. Ser. Math. Sci. Appl., 15, 2001, 302--309.
  46. N.Kopteva & M.Stynes, Approximation of derivatives in a convection-diffusion two-point boundary value problem, Appl. Numer. Math. 39 (2001), 47-60.
  47. M.C.Natividad & M.Stynes, An extrapolation technique for a singularly perturbed problem on a Shishkin mesh, in Proceedings of the Third Asian Mathematical Conference, 2000 (Diliman) (T.Sunada, P.W. Sy and Y.Lo, eds.), World Sci. Publishing, River Edge, NJ, 2002, 383-388.
  48. N.Kopteva & M.Stynes, A robust adaptive method for a quasilinear one-dimentional convection-diffusion problem, SIAM J. Numer. Anal. 39 (2001), 1446-1467. Abstract  Zipped postscript file
  49. B.Kellogg & M.Stynes, n-widths for singularly perturbed problems, Math. Bohem. 127 (2002), 343-352.
  50. M.C.Natividad & M.Stynes, Richardson extrapolation for a convection-diffusion problem using a Shishkin mesh, Appl. Numer. Math. 45 (2003), 315-329. Abstract Zipped postscript file
  51. N.Madden & M.Stynes, A uniformly convergent numerical method for a coupled system of two singularly perturbed linear reaction-diffusion problems, IMA J. Numer. Anal. 23 (2003), 627-644. Abstract Zipped postscript file
  52. M.Stynes, A jejune heuristic mesh theorem, Comput. Methods Appl. Math. 3 (2003), 488-492. Abstract Postscript file
  53. M.Stynes & L.Tobiska, The SDFEM for a convection-diffusion problem with a boundary layer: optimal error analysis and enhancement of accuracy, SIAM J. Numer. Anal. 41 (2003), 1620-1642. Abstract  Zipped postscript file
  54. M.Stynes, Numerical methods for convection-diffusion problems or The 30 years war, in Proceedings of the 20th Biennial Conference on Numerical Analysis, 2003 (Dundee)(D.F.Griffiths & G.A.Watson, eds.), Numerical Analysis Report NA/217, University of Dundee, 95-103. Abstract PDF file
  55. P.Baucour, K.Cronin & M.Stynes, Process optimization strategies to diminish variability in the quality of discrete packaged foods during thermal processing, J. Food Engrg. 60 (2003), 147—155.
  56. N.Kopteva & M.Stynes, Numerical analysis of a singularly perturbed nonlinear reaction-diffusion  problem with multiple solutions, Appl. Numer. Math. 51 (2004), 273-288. Abstract Postscript file
  57. R.B.Kellogg & M.Stynes, Corner singularities and boundary layers in a simple convection-diffusion problem, J. Differential Equations 213 (2005), 81-120. Abstract
  58. N.Kopteva, N.Madden & M.Stynes, Grid equidistribution for reaction-diffusion problems in one dimension, Numer. Algorithms 40(2005), 305-322.
  59. M.Stynes & R.B.Kellogg, Pointwise bounds on derivatives of solutions of elliptic convection-diffusion problems, in Proceedings of the 10th International Conference on Mathematical Modelling (MMA2005) and 2nd International Conference Computational Methods in Applied Mathematics (CMAM2), Trakai, Lithuania, 1-5 June 2005. R. {\v C}iegis (Ed.). Technika, Vilnius,  2005, pages 265-270.
  60. M.Stynes & N.Kopteva, Numerical analysis of singularly perturbed nonlinear reaction-diffusion problems with multiple solutions, Comput. Math. Appl. 51 (2006), 857-864.
  61. R.B.Kellogg & M.Stynes, A singularly perturbed convection-diffusion problem in a half-plane, Applicable Anal. 85 (2006), 1471--1485. Abstract Postscript file
  62. R.B.Kellogg & M.Stynes, Sharpened bounds for corner singularities and boundary layers in a simple convection-diffusion problem, Appl. Math. Lett. 20 (2007), 539-544.
  63. M.Stynes, Convection-diffusion problems, SDFEM/SUPG and a priori meshes, Int. J. Computing Sci. Math. 1 (2007), 412-431.
  64. R.B.Kellogg, T.Linß & M.Stynes,  A finite difference method on layer-adapted meshes for an elliptic reaction-diffusion system in two dimensions, Math. Comp. 77 (2008), 2085-2096.
  65. R.B.Kellogg, N.Madden & M.Stynes, A parameter-robust numerical method for a system of reaction-diffusion equations in two dimensions, Numer. Methods Partial Differential Equations, 24 (2008), 312-334.
  66. E.O'Riordan, J.Stynes & M.Stynes, A parameter-uniform finite difference method for a coupled system of convection-diffusion two-point boundary value problems, Numer. Math. Theor. Meth. Appl. 1 (2008), 176-197.
  67. R.B.Kellogg & M.Stynes, Layers and corner singularities in singularly perturbed elliptic problems, BIT 48 (2008), 309-314.
  68. M.Stynes & L.Tobiska, Using rectangular $Q_p$ elements in the SDFEM for a convection-diffusion problem with a boundary layer, Appl. Numer. Math. 58 (2008), 1709-1802.
  69. N.Cordero, G.Shishkin, L.Shishkina, O.Slattery & M. Stynes, Finite difference scheme for the accurate modelling of boundary layers in microchannels, Proc. EuroSimE 2008, Freiburg, Germany, 21--23 April 2008, 585-592.
  70. N.Cordero, K.Cronin, G.Shishkin, L.Shishkina and M. Stynes, Finite difference scheme for a singularly perturbed parabolic equations in the presence of initial and boundary layers, Math. Model. Anal. 13 (2008), 483-492.
  71. T.Linss & M.Stynes, Numerical solution of systems of singularly perturbed differential equations, Comput. Methods Appl. Math. 9 (2009), no. 2, 165-191.
  72. E.O'Riordan and M.Stynes, Numerical analysis of a strongly coupled system of two singularly perturbed convection-diffusion problems, Adv. Comput. Math. 30 (2009), 101-121.
  73. S.Franz, F.Liu, H.-G.Roos, M.Stynes & A.Zhou, The combination technique for a two-dimensional convection-diffusion problem with exponential layers, Appl. Math. 54 (2009), no.3, 203-223.
  74. A.Naughton & M.Stynes, Singularly perturbed convection-diffusion problems in one dimension: bounds on derivatives, Comput. Methods Appl. Math. 9 (2009), no.3, 281-291.
  75. E.O'Riordan, J.Stynes & M.Stynes, An iterative numerical algorithm for a strongly coupled system of singularly perturbed convection-diffusion problems, in Numerical Analysis and Its Applications (Proc. of 4th International Conference NAA 2008, Lozenetz, Bulgaria, 16-20 June 2008), Lecture Notes in Computer Science Vol. 5434, S.Margenov, L.G.Vulkov, J.Wasniewski (Eds.), 2009, 104-115.
  76. A.Naughton, R.B.Kellogg & M.Stynes, Regularity and derivative bounds for a convection-diffusion problem with a Neumann outflow condition, J. Differential Equations 247 (2009), 2495-2516.
  77. G.Shishkin, L.Shishkina, K.Cronin, M.Stynes & M.Viscor, Heat-mass-transfer for granular drying; mathematical modeling, in Mathematical and Informational Technologies in Management, Engineering and Education. Issue 5. Applied aspects of informational and analytical modelling and information processing. (Proc. of 3rd International Scientific Conference, Ekaterinburg) UGTU/UPI, 2009, p. 16-23 (in Russian).
  78. F.Liu, N.Madden, M.Stynes & A.Zhou, A two-scale sparse grid method for a singularly perturbed reaction-diffusion problem in two dimensions, IMA J. Numer. Anal. 29 (2009), 986-1007.
  79. A.Naughton & M.Stynes, Regularity and derivative bounds for a convection-diffusion problem with Neumann boundary conditions on characteristic boundaries, Z. Anal. Anwend. 29 (2010), 163-181.
  80. M.Viscor & M.Stynes, A robust finite difference method for a singularly perturbed degenerate parabolic problem, Part I, Int. J. Numer. Anal. Model. 7 (2010), no.3, 549-566.
  81. G.Shishkin, L.P.Shishkina, K.Cronin, M.Stynes & M. Viscor, A numerical method for a Stefan-type problem, Math. Model. Anal. 16 (2011), no.1, 119--142.
  82. N.Kopteva & M.Stynes, Approximation of interior-layer solutions in a singularly perturbed semilinear reaction-diffusion problem Numer. Math. (to appear).  
  83. F.Liu, M.Stynes & A.Zhou, Improved accuracy of two-scale finite element discretizations by postprocessing, Part I, SIAM J. Numer. Anal. (to appear).
  84. C.Clavero, J.L.Gracia & M.Stynes, A simpler analysis of a hybrid numerical method for time-dependent convection-diffusion problems, J. Comput. Appl. Math. 235 (2011), no.17, 5240-5248.
  85. S.Franz, R.B.Kellogg & M.Stynes, Galerkin and streamline diffusion finite element methods on a Shishkin mesh for a convection-diffusion problem with corner singularities, Math. Comp. (to appear).
  86. M.Viscor & M.Stynes, A robust finite difference method for a singularly perturbed degenerate parabolic problem, Part II (submitted for publication).
  87. R.Lin & M.Stynes, A balanced finite element method for singularly perturbed reaction-diffusion problems (submitted for publication).

                                                                 

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