Homepage of Prof. Dr. rer. nat. habil. H.-G. Roos

                Address: Prof. Dr. rer. nat. habil. H.-G. Roos
Institute of Numerical Mathematics
Department of Mathematics
Technical University of Dresden

Phone: (+49)(0351) 463 34154, Fax: (+49) (0351) 463 34268

E-Mail: Mail jetzt senden

Position: Seniorprofessorship Numerical methods for Partial Differential Equations

Main field of interests:
  • Numerical analysis of discretisation methods for pde's
  • Singularly perturbed differential equations
  • Applications of splines and wavelets for differential equations
  • Reliable error estimators
  • Flow problems

  • Publications:
    • Papers
    • Books and Proceedings
      • Goering, H.; Roos, H.-G.; Tobiska, L.:
        Die Finite-Elemente-Methode für Anfänger.

        Wiley, Berlin, 2010.    Link to the publisher
      • Roos, H.-G.; Stynes, M.; Tobiska, L.:
        Robust Numerical Methods for Singularly Perturbed Differential Equations -- Convection-Diffusion-Reaction and Flow Problems.

        Springer Series in Computational Mathematics , Vol. 24, 2nd ed., 2008, 616 pages, ISBN: 978-3-540-34466-7
        Table of Contents    Link to the publisher
      • Grossmann, C.; Roos, H.-G.; Stynes, M.:
        Numerical treatment of partial differential equations.

        Springer, Heidelberg-Berlin, 2007.    Link to the publisher


Papers:2015 2014 2013 2012 2011 2010 2009  2008  2007   2006   2005   2004   2003   2002   2001   1999   1998   1997


  • (with H. Zarin)
    On the discontinuous Galerkin method for reaction-diffusion problems: error estimates in energy and balanced norms
    submitted, arXiv:1705.04126v1
  • (with D. Savvvidou and C. Xenophontos)
    On the finite element approximation of fourth order singularly perturbed eigenvalue problems
  • Error estimates in balanced norms of finite element methods on layer-adapted meshes for second order reaction-diffusion problems
  • (with S. Franz)
    On robust error estimation for singularly perturbed fourth order problems


  • Error estimates in balanced norms of finite element methods on Shishkin meshes for reaction-diffusion problems
    arXiv.org/abs/1604.05120v1, see also Modeling and Analysis of Information Systems, 23, no 3(2016), 357-363
  • (with S. Franz)
    Robust error estimation in energy and balanced norms for singularly perturbed fourth order problems
    Comput. and Math. with Applications, 72(2016), 233-247
  • (with H. Zarin, L. Teofanova)
    A continuous interior penalty finite element method for a third-order singularly perturbed boundary value problem
    to appear in J. Comput. Appl. Math.
  • (with L. Ludwig)
    Convergence and supercloseness of a finite element method for a singularly perturbed convection-diffusion problem on the L-shaped domain.
    IMA J. Numer. Anal., 16(3), 2016, 1261-1280
  • 2015:

  • (with M. Stynes)
    Some open problems in the numerical analysis of singularly perturbed differential equations
    CMAM, 15(4),2015,531-550
  • (with M. Schopf)
    An optimal a priori error estimate in the maximum norm for the Il'in scheme in 2D.
    BIT, 55(2015), 4, 1169-1186
  • (with L. Teofanov, Z. Ucelaz)
    Graded meshes for higher order FEM.
    JCM, 33, No1, 2015, 1-16
  • Some remarks on strongly coupled systems of convection-diffusion equations in 2D.
  • (with M. Schopf)
    Convergence and stability in balanced norms of finite element methods on Shishkin meshes for reaction-diffusion problems.
    ZAMM 95, No 6, 551-565 (2015)
  • (with S. Becher)
    Richardson extrapolation for a singularly perturbed turning point problem with exponential boundary layers
    J. Comput. Appl. Math., 290(2015), 334-351, doi 10.1016/j.cam.2015.05.022
  • (with M. Schopf)
    Layer structure and the Galerkin finite element method for a system of weakly coupled singularly perturbed convection-diffusion equations with multiple scales
    ESAIM, 49(5),2015,1525-1547
  • (with L. Teofanov, Z. Ucelaz)
    Uniformly convergent difference schemes for a singularly perturbed third order boundary value problem
    Appl. Num. Math.,96(2015), 108-117: doi 10.1016/j.apnum.205.06.002
    • 2014:

    • (with L. Ludwig)
      Finite element superconvergence on Shishkin meshes for convection-diffusion problems with corner singularities.
      IMA J. Numer. Anal., 34(2014), 782-799
    • (with M. Vlasak)
      An optimal uniform a priori error estimate for an unstaedy singularly perturbed problem.
      IJNAM, 11, 1(2014), 24-33
    • (with S. Franz, A. Wachtel)
      A C^0 interior penalty method for a fourth-order elliptic problem on a layer-adapted mesh.
      Num. Meth. partial diff. equ., 30(2014), 838-861
    • (with L. Teofanov, Z. Ucelaz)
      A modified Bakhvalov mesh.
      Appl. Math. Letters, 31(2014), 7-11
    • (with S. Franz)
      Error estimation in a balanced norm for a convection-diffusion problem with two different boundary layers.
      accepted, Calcolo, 51(2014), 423-440
    • (with S. Franz)
      Superconvergence for higher-order elements in convection-diffusion problems.
      NMTMA 7(2014), 356-373


    • (with Z. Ucelaz)
      Qualocation for a singularly perturbed boundary value problem.
      JCAM, 237(2013), 556-564
    • (with M. Schopf)
      Error estimation in energy norms: Is it necessary to fit the mesh to boundary layers ?
      Dimov, Farago, Vulkov (Eds.): NAA 2012, LNCS 8236, pp. 95-109, 2013


    • Robust numerical methods for singularly perturbed differential equations: a survey covering 2008-2012.
      ISRN Applied Mathematics, vol. 2012, ID 379547, doi:10.5042/2012/379547
    • (with M. Vlasak)
      Optimal error estimates for nonstationary singularly perturbed problems for low order discretization orders.
      ACC Journal, TU Liberec, XVIII 4/2012, 146-152
    • Special features of strongly coupled systems of convection-diffusion equations with two small parameters.
      Appl. Math. Letters, 25(2012), 1127-1130
    • (with S. Franz, R. Gaertner, A.Voigt)
      A note on the convergence analysis of a diffuse-domain approach.
      CMAM, 12(2012), 153-167
    • (with L. Ludwig)
      Superconvergence for convection-diffusion problems with low regularity.
      Proc. of Applications of Mathematics 2012, Prague 2-5, 2012, 173-187
    • (with M. Schopf)
      Analysis of finite element methods on Bakhvalov-type meshes for linear convection-diffusion problems in 2D.
      Appl. of Math., 57(2012), 97-108
    • (with T. Linss, M. Schopf)
      Nitsche-mortaring for singularly perturbed convection-diffusion problems.
      Advances Comput. Math., 4(2012), 581-603


    • (with S. Franz)
      The capriciousness of numerical methods for singular perturbations.
      SIAM review, 53, 2011, N0 1, 157-173
    • (with C. Großmann, L. Ludwig)
      Layer-adapted methods for a singularly perturbed singular problem.
      CMAM, 11 (2011), 192-205
    • (with C. Reibiger)
      Numerical analysis of a strongly coupled system of two convection-diffusion equations with full layer interaction.
      ZAMM 91(2011), 537-543
    • (with Ch. Reibiger)
      Numerical analysis of a system of singularly perturbed convection-diffusion equations related to optimal control.
      NMTMA, 4(2011), 562-575
    • (with M. Krizek)
      Two-sided bounds of the discretization error for finite elements.
      ESAIM, 45(2011), 915-924
    • (with M. Schopf)
      Finite elements on locally uniform meshes for convection-diffusion problems with boundary layers.
      Computing, 92(2011), 285-296
    • (with Ch. Grossmann, R. K. Mohanty)
      A direct higher order discretization in singular perturbations via domain split--a computational approach.
      Appl. Math. Comp., 217(2011), 9302-9312


    • (with V. Dolejsi)
      BDF-FEM for parabolic singularly perturbed problems with exponential layers on layer-adapted meshes in space.
      NeuralParallelSci. Comput., 18(2010), no.2, 221-235
    • (with S. Franz, T. Linss, S. Schiller)
      Uniform convergence of finite element methods with edge stabilization on Shishkin meshes.
      J. Comput. Math.,28(2010),32-44
    • (with l. Kaland)
      Parabolic singularly perturbed problems with exponential layers: robust discretizations using finite elements in space on Shishkin meshes.
      IJNAM, 7(2010), nr 3, 593-606
    • Two remarks on numerical methods for singularly perturbed problems.
      Preprint MATH-NM-06-2010, TU Dresden


    • Stabilized FEM for convection-diffusion problems on layer-adapted meshes.
      J. Comput. Math., 27(2009), 266-279
    • (with S. Franz, F. Liu, M. Stynes, A. Zhou)
      The combination technique for two-dimensional convection-diffusion problems with exponential layers.
      Applications of Math., 54(2009), 203-223


    • (with L. Teofanov)
      A singularly perturbed problem with two small parameters II:
      The Galerkin finite element method on a Shishkin mesh.
      J. Comput. Appl. Math., 212(2008), 374-389
    • (with T. Apel)
      Remarks on the analysis of finite element methods on a Shishkin mesh: are Scott-Zhang interpolants applicable?
      Preprint MATH-NM-06-2008, TU Dresden
    • (with S. Franz and T. Linss) Superconvergence analysis of the SDFEM for elliptic problems with characteristic layers.
      Appl. Numer. Math., 58(2008),1818-1829


    • (with R. Vanselow) A comparison of four- and five-point difference approximations for stabilizing the one-dimensional stationary convection-diffusion equation.
      ETNA, 32(2008), 63-75
    • A link between local projection stabilizations and the continuous interior penalty method for convection-diffusion problems.
      Preprint MATH-NM-05-07       pdf-file
    • (with H. Zarin) A supercloseness result for the discontinuous Galerkin stabilization of convection-diffusion problems.
      Numer. Meth. f. Partial Diff. Equ., 23(2007), 1560-1576
    • (with L. Teofanov) A singularly perturbed problem with two parameters I:
      Solution decomposition
      J. Comput. Appl. Math., 206(2007), 1082-1097


    • Error estimates for linear finite elements on Bakhvalov-type meshes.
      Applications of Mathematics, 51(2006),63-72
    • Superconvergence on a hybrid R-T-mesh for singularly perturbed problems with exponential layers.
      ZAMM, 86(2006), 649-655


    • (with H. Zarin) Interior penalty discontinuous approximations of a convection-diffusion
      problem  with parabolic layers.
      Numerische Mathematik, 100(2005),735-759


    • A uniformly convergent scheme for a singularly perturbed eigenvalue problem.
      Proceedings of BAIL 2004
    • (with T. Linss) Analysis of a finite difference scheme for a singularly perturbed problem with two small parameters.
      JMAA, 289(2004), 355-366
    • (with H. Zarin) The discontinuous Galerkin method for singularly perturbed problems.
      in: Numerical Mathematics and advanced Applications, eds.: M. Feistauer et all., Springer 2004, 736-745


    • On the streamline-diffusion stabilization for convection-diffusion problems
      Report MATH-NM-05(2003), TUD
    • (with H. Zarin) The streamline-diffusion method for a convection-diffusion problem with a point source.
      JCAM , 150(2003), 109-128
    • (with Z. Uzelac) The SDFEM for a convection-diffusion problem with two small parameters.
      CMAM, 3(2003), No.3, 443-458
    • (with H.Zarin) The discontinous Galerkin finite element method for singularly perturbed problems.
      in: Lecture Notes in Comput. Science and Engin., vol. 35 (2003), 246-267
    • (with H. Zarin) Some properties of the discontinuous Galerkin-method for reaction-diffusion and
      convection-diffusion problems in 1D.
      Novi Sad J. Math., 33(2003), no 2, 33-42


    • Optimal uniform convergence of basic schemes for elliptic problems with strong parabolic boundary layers.
      JMAA, 267(2002), 194-208
    • (with H. Zarin) A second order scheme for singularly perturbed differential equations with discontinuous source term.
      Journal Numerical Math., 10(2002), 275-289
    • (with D. Wollstein, T. Linss) A uniformly accurate FVM discretization for a convection-diffusion problem.
      ETNA 13 (2002), 1-11


    • (with A. Froehner, T. Linss) Defect correction on Shishkin.type meshes.
      Numerical Algorithms, 26(2001), 281-299
    • (with T. Linss) Gradient recovery for singularly perturbed boundary value problems (II).
      M3AS, 11, No.7(2001), 1169-1179
    • (with T. Linss, R. Vulanovic) Uniform pointwise convergence on Shishkin-type meshes for convection-diffusion problems.
      SINUM , 38, 897-912 (2001)
    • (with T.Linss, D. Schneider) Uniform convergence of an upwind finite element method on layer-adapted grids.
      Computer Meth. Appl. Mech. Eng., 190 (2001), 4519-4530
    • (with T.Linss) Gradient recovery for singularly perturbed problems (I)
      Computing, 66, 163-178 (2001)
    • On a stabilization effect of thin submeshes for convection-diffusion-problems.
      ZAMM , 81 (2001), 637-639
    • (with A. Froehner ) The uniform convergence of a defect correction method on a Shishkin mesh.
      Appl. Num. Math. , 37, 79-94 (2001)


    • (with T.Skalicky) Anisotropic mesh refinement for problems with internal and boundary layers.
      International J. f. Numer. Methods in Engineering, 46 (1999), 1933-1953
    • (with T. Linss) Sufficient conditions for uniform convergence on layer adapted grids.
      Computing, 63, 27-45 (1999)
    • (with B. Bagaev) The finite element method on an adapted mesh for a two-dimensional convection-diffusion problem.
      Sibirian Numerical J., no 4, 1999, 309-320


    • (with T. Skalicky) Galerkin/Least-squares finite element method for convection-dominated problems on Gartland-type meshes.
      Report MATH-NM-12, 1998, TU Dresden
    • Layer-adapted grids for singularly perturbed boundary value problems,
      Z. Angew. Math. Mech., 78(5): 291-309,1998


    • (with M.Dobrowolski):A priori estimates for the solution of convection-diffusion problems and interpolation on Shishkin meshes,
      Zeitschrift für Analysis u. Anw., 16(4), 1001-1012, 1997
    • (with T.Skalicky):A comparison of the finite element method on Shishkin and Gartland-type meshes for convection-diffusion problems.
      CWI Quarterly, 10(3&4), 277-300, 1997

    Books and Proceedings:

    Last modified: August, 2008