Prof. Dr. Hans-Görg Roos

Senior Professorship Numerical methods for Partial Differential Equations

Institute of Numerical Mathematics
Department of Mathematics
TUD Technical University of Dresden

contact
by e-mail: hans-goerg.roos@tu-dresden.de
or via the secretariat of the Institute

Table of contents

      1. Main field of interests
      2. Conferences
      3. Books and Proceedings
      4. Papers

Main field of interests

  • Numerical analysis of discretization methods for pde's
  • Singularly perturbed differential equations
  • Applications of splines and wavelets for differential equations
  • Reliable error estimators
  • Flow problems

Conferences

  • BAIL 2020, Buenos Aires, July 13th - 17th, 2020
  • 16th Annual Workshop Numerical methods for problems with layer phenomena, Sevilla, February 14-15, 2019
  • 12th Annual Workshop Numerical methods for problems with layer phenomena, Dresden, April 9 - 10, 2015
  • Chess and Mathematics Dresden, November 21 - 23, 2008

Books and Proceedings

Papers

2019/2020

  • (with M. Schopf)
    A convection-diffusion problem with a small variable diffusion coefficient
    arXiv:2001.04115
  • Layer-adapted meshes: Milestones in 50 years of history
    arXiv:1909.08273
  • (with S. Franz)
    Error estimates in balanced norms of finite element methods for higher order reaction-diffusion problems
    IJNAM, 17:4 (2020), 532-542
  • (with M. Brdar and S. Franz)
    Numerical treatment for singularly perturbed fourth-order two-parameter problems
    ETNA 51, 2019, 50-62
  • Monotone discretization of elliptic problems with mixed derivatives on anisotropic meshes: a counterexample
    arXiv:1901.05321

2018

  • Robust Numerical Methods for Singularly Perturbed Differential Equations-- Supplements
    November 2018
  • (with H. Zarin, L. Teofanova)
    A continuous interior penalty finite element method for a third-order singularly perturbed boundary value problem
    J. Comput. Appl. Math, 37(2018), 175-190
  • Remarks on balanced norm error estimates for systems of reaction-diffusion equations
    Appl. of Math, 63(2018), 273-279
  • (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
    submitted

2017

  • Error estimates in balanced norms of finite element methods on layer-adapted meshes for second order reaction-diffusion problems
    Boundary and Interior Layers, BAIL 2016, eds. Huang, Z., Stynes, M. and Z. Zhang, Springer 2017, 1-18
  • (with S. Franz)
    On robust error estimation for singularly perturbed fourth order problems
    Boundary and Interior Layers, BAIL 2016, eds. Huang, Z., Stynes, M. and Z. Zhang, Springer 2017, 77-85

2016

  • 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 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.
    arXiv.org/abs/1502.04473
  • (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

2013

  • (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

2012

  • 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

2011

  • (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.
    CMAM 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

2010

  • (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), no 3, 593-606
  • Two remarks on numerical methods for singularly perturbed problems.
    Preprint MATH-NM-06-2010, TU Dresden

2009

  • 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

2008

  • (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

2007

  • (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

2006

  • 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

2005

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

2004

  • 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

2003

  • 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 Engineering, 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

2002

  • 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

2001

  • (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)

1999

  • (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.
    Siberian Numerical J., no 4, 1999, 309-320

1998

  • (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

1997

  • (with M.Dobrowolski):A priori estimates for the solution of convection-diffusion problems and interpolation on Shishkin meshes,
    Journal of Analysis and Application, 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

About this page

Markus Herrich
Last modified: Mar 17, 2026