# Prof. Dr. rer. nat. habil. H.-G. Roos

Seniorprofessorship Numerical methods for Partial Differential Equations

Institute of Numerical Mathematics

Department of Mathematics

Technical University of Dresden

Kontakt

per E-Mail: hans-goerg.roos@tu-dresden.de

oder über Sekretariat des Institutes

## Table of contents

### 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

### Conferences

- BAIL 2020, Buenos Aires, 13th - 17th July 2020
- 16. Annual Workshop Numerical methods for problems with layer phenomena, Sevilla, 14./15. February 2019
- 12. Annual Workshop Numerical methods for problems with layer phenomena, Dresden , 9.4. - 10.4.2015
- Chess and Mathematics Dresden, 21. - 23.11.2008

### Books and Proceedings

- (with M. Stynes and L. Tobiska)

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 - (with Ch. Grossmann, M. Stynes)

Numerical treatment of partial differential equations

Springer-Verlag, Heidelberg-Berlin, 2007, 596 pages, ISBN 978-3-540-71582-5 - (with Ch. Grossmann)

Numerische Behandlung partieller Differentialgleichungen

Teubner-Verlag, Stuttgart, November 2005, third extended edition, 570 pages, ISBN 3-519-22089-X - (with H. Schwetlick)

Numerische Mathematik: Das Grundwissen für jedermann (Teubner, September 99) - (with M.Stynes, L.Tobiska)

Numerical methods for singularly perturbed differential equations,

350 pages, Springer-Verlag, 1996 - (with Ch. Grossmann)

Numerik partieller Differentialgleichungen,

478 pages, Teubner-Verlag, Stuttgart, 1992, second edition 1994 - (with A.Felgenhauer, L.Angermann, eds.)

Numerical Methods in Singularly Perturbed Problems,

University of Technology Dresden, 1991 - (with E.Adams, R.Ansorge, Ch.Grossmann, eds.)

Discretization in Differential Equations and Enclosures, Akademie-Verlag, Berlin 1987 - (with H.Goering, L.Tobiska)

Finite-Element-Methode,

Akademie-Verlag, Berlin 1985 (third extended edition 1993) - (with H.Goering, A.Felgenhauer, G.Lube, L.Tobiska)

Singularly perturbed differential equations,

Akademie-Verlag, Berlin 1983

### 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.

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

**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), nr 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.

Numerische Mathematik, 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 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

**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.

Sibirian 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,

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