Publications

Habilitation thesis

PhD thesis

Diploma thesis

Papers in edited journals

  1. P. Constantinou, S. Franz, L. Ludwig, Chr. Xenophontos:
    Finite element approximation of reaction-diffusion problems using an exponentially graded mesh
    Computer Math. Appl. (2018), 76 (2018), 2523-2534
  2. P. Constantinou, S. Franz, L. Ludwig, Chr. Xenophontos:
    A mixed hp FEM for the approximation of 4th order singularly perturbed problems on smooth domains
    Numer. Methods Partial Differential Equations, (2018)
  3. S. Franz, M. Waurick:
    Resolvent estimates and numerical implementation for the homogenisation of one-dimensional periodic mixed type problems
    ZAMM, 98 (2018) 7, 1284-1294
  4. S. Franz, G. Matthies:
    A Unified Framework for Time-Dependent Singularly Perturbed Problems with Discontinuous Galerkin in Time
    Math. Comp., 87 (2018) 313, 2113-2132
  5. S. Franz, K. Höhne, G. Matthies:
    Grad-Div stabilisation applied to inf-sup stable discretisations with S-type meshes for the Oseen problem
    IMA J. Numer. Anal., 38 (1) (2018), 299-329
  6. S. Franz, S. Trostorff, M. Waurick:
    Numerical methods for changing type systems
    IMA J. Numer. Anal., (2017)
  7. S. Franz, K. Höhne, M. Waurick:
    A solution decomposition for a singularly perturbed 4th order problem
    Analysis, 37 (2017) 2, 77-100
  8. S. Franz, Chr. Xenophontos:
    A Short Note on the connection between layer-adapted exponentially graded and S-type meshes
    CMAM, 18 (2017) 2, 199-202
  9. S. Franz:
    Convergence of LPS-FEM for convection-diffusion problems on layer-adapted meshes
    BIT Numerical Mathematics, 57(3) (2017), 771-786
  10. S. Franz:
    Analysis for a family of continuous-discontinuous Galerkin FEM for convection-diffusion problems.
    Appl. Numer. Math., 110 (2016), 93-109
  11. S. Franz, L. Ludwig, Chr. Xenophontos:
    Finite element approximation of convection-diffusion problems using an exponentially graded mesh
    Computer Math. Appl., 72 (2016) 6, 1532-1540
  12. S. Franz, H.-G. Roos:
    Robust error estimation in energy and balanced norms for singularly perturbed forth order problems
    Computer Math. Appl., 72 (2016) 1, 233-247
  13. S. Franz, H.-G. Roos:
    Error estimation in a balanced norm for a convection-diffusion problem with two different boundary layers.
    Calcolo, 51 (2014) 3, 423-440 , Addendum
  14. S. Franz, H.-G. Roos:
    Superconvergence of a Galerkin FEM for Higher-Order Elements in Convection-Diffusion Problems.
    Numer. Math. Theor. Meth. Appl., 7 (2014) 3, 356-373 , arXiv:1307.7543
  15. S. Franz, H.-G. Roos, A. Wachtel:
    A C0 interior penalty method for a singularly-perturbed fourth-order elliptic problem on a layer-adapted mesh.
    Numer. Methods Partial Differential Equations, 30 (2014) 3, 838-861
  16. S. Franz:
    Superconvergence Using Pointwise Interpolation in Convection-Diffusion Problems.
    Appl. Numer. Math., 76 (2014), 132-144 , arXiv:1304.7443
  17. S. Franz:
    Convergence Phenomena of Q_p-Elements for Convection-Diffusion Problems.
    Numer. Methods Partial Differential Equations, 29 (2013) 1, 280-296
  18. S. Franz, R. Gärtner, H.-G. Roos and A. Voigt:
    A Note on the Convergence Analysis of a Diffuse-Domain Approach.
    CMAM, 12 (2012) 2, 153-167
  19. 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., 81 (2012) 278, 661-685, preprint als pdf
  20. S. Franz, N. Kopteva:
    Green's function estimates for a singularly perturbed convection-diffusion problem.
    Journal of Differential Equations, 252 (2012), 1521-1545, preprint als pdf
  21. S. Franz, N. Kopteva:
    Green's function estimates for a singularly perturbed convection-diffusion problem in three dimensions.
    Int. J. Numer. Anal. Model. Ser. B, 2 (2011) 2-3, 124-141 , preprint als pdf
  22. S. Franz, G. Matthies:
    Convergence on Layer-adapted Meshes and Anisotropic Interpolation Error Estimates of Non-Standard Higher Order Finite Elements.
    Appl. Numer. Math., 61 (2011), 723-737 , corrected preprint
  23. S. Franz:
    SDFEM with non-standard higher-order finite elements for a convection-diffusion problem with characteristic boundary layers.
    BIT Numerical Mathematics, 51 (2011) 3, 631-651 , preprint
  24. S. Franz, H.-G. Roos:
    The Capriciousness of Numerical Methods for Singular Perturbations.
    SIAM Review, 53 (2011) 1, 157-173 , preprint
  25. S. Franz, G. Matthies:
    Local projection stabilisation on S-type meshes for convection-diffusion problems with characteristic layers.
    Computing, 87 (2010) 3-4, 135-167 , preprint
  26. S. Franz, T. Linß, H.-G. Roos, S. Schiller:
    Uniform superconvergence of a finite element method with edge stabilization for convection-diffusion problems.
    J. Comp. Math., 28 (2010) 1, 32-44 , preprint
  27. S. Franz, L. Tobiska, H. Zarin:
    A New Approach to Recovery of Discontinuous Galerkin.
    J. Comp. Math., 27 (2009) 6, 697-712 , preprint
  28. 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) 3, 203-223 , preprint
  29. S. Franz, T. Linß, H.-G. Roos:
    Superconvergence analysis of the SDFEM for elliptic problems with characteristic layers,
    Appl. Numer. Math., 58 (2008) 12, 1818-1829 , preprint
  30. S. Franz:
    Continuous interior penalty method on a Shishkin mesh for convection-diffusion problems with characteristic boundary layers,
    Comput. Meth. Appl. Mech. Engng., 197 (2008) 45-48, 3679-3686 , preprint
  31. S. Franz, T. Linß:
    Superconvergence analysis of the Galerkin FEM for a singularly perturbed convection-diffusion problem with characteristic layers,
    Numer. Methods Partial Differential Equations, 24 (2008) 1, 144-164 , preprint
  32. N.A. Adams, S. Hickel, S. Franz:
    Implicit subgrid-scale modeling by adaptive deconvolution,
    J. Comput. Phys. 200 (2004) 2, 412-431

Papers in Proceedings

  1. S. Franz, H.-G. Roos:
    On robust error estimation for singularly perturbed fourth-order problems
    In Boundary and Interior Layers, Computational and Asymptotic Methods BAIL 2016
    Lecture Notes in Computational Science and Engineering 120 (2017), 77-85, Springer, Berlin Heidelberg,
  2. S. Franz, H.-G. Roos:
    Robust error estimation in energy and balanced norms for singularly perturbed forth order problems
    Model. Anal. Inform. Sist., 23:3 (2016), 364-369
  3. S. Franz, J. Rubisch:
    A discontinuous Galerkin method for fractional differential equations
    PAMM, 16 (2016), 761-762
  4. S. Franz, N. Kopteva:
    On the Sharpness of Green's function estimates for a convection-diffusion problem.
    In Proceedings of the Fifth Conference on Finite Difference Methods: Theory and Applications (FDM: T&A 2010), (2011), 44-57, Rousse University Press, arXiv:1102.4520v2
  5. S. Franz, R. B. Kellogg, M. Stynes:
    On the choice of mesh for a singularly perturbed problem with a corner singularity.
    In BAIL 2010 - Boundary and Interior Layers, Computational and Asymptotic Methods
    Lecture Notes in Computational Science and Engineering 81 (2010), 119-126, Springer, Berlin Heidelberg, preprint as pdf
  6. S. Franz, G. Matthies:
    Local projection stabilisation on layer-adapted meshes for convection-diffusion problems with characteristic layers (Part I and II)
    In BAIL 2010 - Boundary and Interior Layers, Computational and Asymptotic Methods
    Lecture Notes in Computational Science and Engineering 81 (2010), 127-138, Springer, Berlin Heidelberg, preprint as pdf
  7. S. Franz, T. Linß:
    Superconvergence analysis of Galerkin FEM and SDFEM for Elliptic Problems with characteristic layers,
    Proceedings in Applied Mathematics and Mechanics 6 (2006) 1, 737-738.
  8. N.A. Adams, S. Franz, S. Hickel, P. Koumoutsakos:
    Optimization of an implicit subgrid-scale model for LES,
    In Proceedings of the 21st International Congress of Theoretical and Applied Mechanics, Warsaw, Poland, 2004

Preprints and submitted papers

  1. L. Teofanov, M. Brdar, S. Franz, H. Zarin:
    SDFEM for a singularly perturbed problem with two parameters.
    submitted
  2. M. Brdar, S. Franz, H.-G. Roos:
    Numerical treatment for singularly perturbed fourth-order two-parameter problems.
    submitted
  3. S. Franz, N. Kopteva:
    Full analysis of the Green's function for a singularly perturbed convection-diffusion problem in three dimensions.
    arXiv:1103.2948v2
  4. S. Franz:
    A recovery operator for the combination method.
    Preprint MATH-NM-09-2008, 10/2008, preprint as pdf

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Author: Sebastian Franz
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