Books

  1. Thomas Apel: Anisotropic finite elements: Local estimates and applications. Series "Advances in Numerical Mathematics", Teubner, Stuttgart, 1999. ISBN 3-519-02744-5. Vergriffen! (Habilitationsschrift. Preprint SFB393/99-03, TU Chemnitz, 1999.) [ full text: ps.gz pdf ]
  2. Thomas Apel: Finite-Elemente-Methoden über lokal verfeinerten Netzen für elliptische Probleme in Gebieten mit Kanten. PhD thesis, TU Chemnitz, 1991

Editorial work

  1. Thomas Apel, Matthias Gerdts (eds.): Special Issue Dedicated to Joachim Gwinner. Journal of Applied and Numerical Optimization 3(2021)2, pp. 239-434. [ preface ]
  2. Thomas Apel, Ulrich Langer, Arnd Meyer, Olaf Steinbach (eds.): Advanced Finite Element Methods with Applications - Selected Papers from the 30th Chemnitz Finite Element Symposium 2017. Lecture Notes in Computational Science and Engineering 128. Springer, Berlin, 2019.
  3. Thomas Apel, Olaf Steinbach (eds.): Special issue on the occasion of the 65th birthdays of Ulrich Langer and Arnd Meyer.  Computing and Visualization in Science 19(2018)5--6, pp. 1--89. [ preface ]
  4. Thomas Apel, Olaf Steinbach (eds.): Advanced Finite Element Methods and Applications. Lecture Notes in Applied and Computational Mechanics 66. Springer, Berlin, 2012. [ alternative link ]
  5. Thomas Apel, Albrecht Böttcher, Gundolf Haase, Bernd Heinrich, Michael Jung, Ulrich Langer, Arnd Meyer, Arnd Rösch, Olaf Steinbach (eds.): Selected papers from the 20th Chemnitz Finite Element Symposium. ETNA 32(2008), pp. 1-209
  6. Thomas Apel, Arnd Meyer (eds.): Selected papers from the 16th Chemnitz Finite Element Symposium 2003. Appl. Numer. Math. 54(2005)3-4, pp. 311-579

Invited contributions

  1. Thomas Apel, Markus Melenk: Interpolation and quasi-interpolation in h- and hp-version finite element spaces. In E. Stein, R. de Borst and T. J. R. Hughes (eds.): Encyclopedia of Computational Mechanics, Second Edition. Wiley, Chichester, 2016.
  2. Thomas Apel, Johannes Pfefferer, and Arnd Rösch: Locally refined meshes in optimal control for elliptic partial differential equations - an overview. In: G. Leugering, P. Benner, S. Engell, A. Griewank, H. Harbrecht, M. Hinze, R. Rannacher, and S. Ulbrich (eds.): Trends in PDE Constrained Optimization. International Series of Numerical Mathematics 165. Birkhäuser, Basel, 2014, pp. 285-302.
  3. Thomas Apel, Dieter Sirch: A priori mesh grading for distributed optimal control problems. In: G. Leugering, S. Engell, A. Griewank, M. Hinze, R. Rannacher, V. Schulz, M. Ulbrich, and S. Ulbrich (eds.): Constrained Optimization and Optimal Control for Partial Differential Equations. International Series of Numerical Mathematics 160. Springer, Basel, 2012, pp. 377-389.
  4. Thomas Apel: Interpolation in h-version finite element spaces. In E. Stein, R. de Borst and T. J. R. Hughes (eds.): Encyclopedia of Computational Mechanics. Wiley, Chichester, 2004, pp. 55-72.

Journal papers

  1. Thomas Apel, Philipp Zilk: Isogeometric analysis of the Laplace eigenvalue problem on circular sectors: Regularity properties, and graded meshes. Computers & Mathematics with Applications 175(2024), 236-254. [ Preprint | Paper ]
  2. Thomas Apel, Mariano Mateos, Arnd Rösch: Non-coercive Neumann boundary control problems. Results Math 79(2024), 227. [ Preprint | Paper ]
  3. Thomas Apel, Volker Kempf: Pressure-robust error estimate of optimal order for the Stokes equations: domains with re-entrant edges and anisotropic mesh grading.  Calcolo 58(2021) 15. [ Preprint | Paper ]
  4. Thomas Apel, Volker Kempf, Alexander Linke, Christian Merdon: A non-conforming pressure-robust finite element method for the Stokes equations on anisotropic meshes. IMA J. Numer. Anal. 42(2022)1, 392–416 [ Preprint | Paper ]
  5. Thomas Apel, Serge Nicaise: Regularity of the solution of the scalar Signorini problem in polygonal domains.  Results in Mathematics (RIMA) 75(2020). [ Preprint | Paper ]
  6. Thomas Apel, Volker Kempf: Brezzi-Douglas-Marini interpolation of any order on anisotropic triangles and tetrahedra. SIAM J. Numer. Anal. (SINUM) 58(2020), 1696–1718. [Preprint | Paper]
  7. Thomas Apel, Johannes Pfefferer, Sergejs Rogovs, Max Winkler: L∞-error estimates for Neumann boundary value problems on graded meshes. IMA J. Numer. Anal. 40(2018), 474–497. [Preprint | Paper]
  8. Thomas Apel, Mariano Mateos, Johannes Pfefferer, Arnd Rösch: Error estimates for Dirichlet control problems in polygonal domains.  Math. Control Relat. Fields 8(2018), 217–245. [ Preprint | paper ]
  9. Thomas Apel, Johannes Pfefferer, Max Winkler: Error estimates for the postprocessing approach applied to Neumann boundary control problems in polyhedral domains. Accepted by IMA J. Numer. Anal. 2017 [  Preprint | Paper ]
  10. Thomas Apel, Serge Nicaise, Johannes Pfefferer: Adapted numerical methods for the numerical solution of the Poisson equation with L2 boundary data in non-convex domains. SIAM J. Numer. Anal. 55(2017), 1937-1957 [Preprint | Paper]
  11. Thomas Apel, Olaf Steinbach, Max Winkler: Error estimates for Neumann boundary control problems with energy regularization. J. Numer. Math. 24(2016), 207-233. [  preprint | paper ]
  12. Thomas Apel, Serge Nicaise, Johannes Pfefferer: Discretization of the Poisson equation with non-smooth boundary data and emphasis on non-convex domains. Numer. Methods PDE, 32(2016), 1433–1454. [ Preprint | Paper ]
  13. Thomas Apel, Mariano Mateos, Johannes Pfefferer, Arnd Rösch: On the regularity of the solutions of Dirichlet optimal control problems in polygonal domains. SIAM J. Control Optim. 53(2015), 3620–3641. [ Preprint | Paper]
  14. Thomas Apel, Johannes Pfefferer, Max Winkler: Local mesh refinement for the discretisation of Neumann boundary control problems on polyhedra. Math. Meth. Appl. Sci. 39(2016), 1206–1232 [  Preprint | Paper ]
  15. Thomas Apel, Ariel Lombardi, and Max Winkler: Anisotropic mesh refinement in polyhedral domains: error estimates with data in L2(Omega). M2AN 48(2014), 1117-1145. [ Preprint | paper ]
  16. Thomas Apel, Johannes Pfefferer, and Arnd Rösch: Finite element error estimates on the boundary with application to optimal control. Math. Comp. 84(2015), 33-70. [  Preprint | paper ]
  17. Thomas Apel, Thomas Flaig ,and Serge Nicaise: A priori error estimates for finite element methods for H(2,1)-elliptic equations. Numer. Funct. Anal. Optimization 35(2014)2, 153-176 [Paper |  Preprint]
  18. Thomas Apel, Thomas Flaig: Crank-Nicolson schemes for optimal control problems with evolution equations. SIAM J. Numer. Anal. 50(2012), 1484-1512 [ Paper |  Preprint]
  19. Thomas Apel, Johannes Pfefferer, and Arnd Rösch: Finite element error estimates for Neumann boundary control problems on graded meshes. Comput. Optim. Appl. 52(2012), 3-28 [ Paper |  Preprint ]
  20. Thomas Apel, Olaf Benedix, Dieter Sirch, and Boris Vexler: A priori mesh grading for an elliptic problem with Dirac right-hand side. SIAM J. Numer. Anal. 49(2011), 992-1005 [ Paper |  Preprint ]
  21. Thomas Apel, Dieter Sirch: L2-error estimates for Dirichlet and Neumann problems on anisotropic finite element meshes. Appl. Math. 56(2011)2, 177–206. [ paper |  Preprint ]
  22. Gabriel Acosta, Thomas Apel, Ricardo G. Durán and Ariel L. Lombardi: Error estimates for Raviart-Thomas interpolation of any order over anisotropic tetrahedra. Math. Comp. 80(2011), 141-163. [paper | Preprint]
  23. Thomas Apel, Serge Nicaise, Dieter Sirch: A posteriori error estimation of residual type for anisotropic diffusion-convection-reaction problems. J. Comp. Appl. Math. 235(2011)8, 2805-2820. [paper]
  24. Thomas Apel, Arnd Rösch and Dieter Sirch: L∞-error estimates on graded meshes with application to optimal control. SIAM J. Control Optim. 48(2009), 1771-1796. [paper |  Preprint]
  25. Thomas Apel, Gunter Winkler: Optimal control under reduced regularity Appl. Numer. Math., 59(2009), 2050--2064. [paper | Preprint]
  26. Thomas Apel, Dominique Leguillon, Cornelia Pester, Zohar Yosibash: Edge singularities and structure of the 3-D Williams expansion. Comptes Rendus M\'ecanique, 336(2008), 629-635. [ pdf-file]
  27. Gabriel Acosta, Thomas Apel, Ricardo G. Durán, Ariel L. Lombardi: Anisotropic error estimates for an interpolant defined via moments. Computing 82(2008), 1-9. [paper | preprint]
  28. Thomas Apel, Gunar Matthies: Non-conforming, anisotropic, rectangular finite elements of arbitrary order for the Stokes problem. SIAM J. Numer. Anal. 46(2008), 1867-1891. [paper | Preprint]
  29. Thomas Apel, Tobias Knopp, Gert Lube: Stabilized finite element methods with anisotropic mesh refinement for the Oseen problem. Appl. Numer. Math. 58(2008), 1830-1843. [ preprint | paper]
  30. Thomas Apel, Serge Nicaise: A posteriori error estimations of a SUPG method for anisotropic diffusion-convection-reaction problems. C. R. Acad. Sci. Paris, Ser. I, vol. 345, pp. 657-662, 2007. [pdf-file]
  31. Thomas Apel, Arnd Rösch, Gunter Winkler: Optimal control in nonconvex domains: a priori discretization error estimates. Calcolo 44(2007), 137-158. [paper |  preprint]
  32. Thomas Apel, Cornelia Pester: Clement-type interpolation on spherical domains - interpolation error estimates and application to a posteriori error estimation. IMA J. Numer. Anal. 25(2005), 310-336. [preprint | paper]
  33. Thomas Apel, Serge Nicaise: The inf-sup condition for low order elements on anisotropic meshes. Calcolo 41(2004), 89-113. [paper | Preprint]
  34. Thomas Apel, Sergei Grosman, Peter K. Jimack, Arnd Meyer: A new methodology for anisotropic mesh refinement based upon error gradients. Appl. Numer. Math. 50(2004), 329-341. [paper | preprint]
  35. Thomas Apel, Nico Düvelmeyer: Transformation of hexahedral finite element meshes into tetrahedral meshes according to quality criteria. Computing, 71(2003), 293--304. [paper | preprint]
  36. Thomas Apel, Anna-Margarete Sändig, Sergey I. Solov'ev: Computation of 3D vertex singularities for linear elasticity: Error estimates for a finite element method on graded meshes Math. Modeling Numer. Anal. (M2AN) 36(2002)6, 1043-1070. [paper | preprint], Corrigendum by Thomas Apel, Arnd Meyer and Cornelia Pester [pdf]
  37. Thomas Apel, Volker Mehrmann, David Watkins: Structured eigenvalue methods for the computation of corner singularities in 3D anisotropic elastic structures. Comput. Methods Appl. Mech. Engrg. 191(2002), 4459-4473, DOI: 10.1016/S0045-7825(02)00390-0 [paper | preprint]
  38. Thomas Apel, Joachim Schöberl: Multigrid methods for anisotropic edge refinement. SIAM J. Numer. Anal. 40(2002)5, 1993-2006. [paper | preprint]
  39. Thomas Apel, H. Maharavo Randrianarivony: Stability of discretizations of the Stokes problem on anisotropic meshes.Mathematics and Computers in Simulation 61(2003) 3-6, 437-447. [ paper | preprint]
  40. Thomas Apel, Serge Nicaise, Joachim Schöberl: A non-conforming finite element method with anisotropic mesh grading for the Stokes problem in domains with edges. IMA J. Num. Anal. 21(2001)4, 843-856. [paper | preprint]
  41. Thomas Apel, Serge Nicaise, Joachim Schöberl: Crouzeix-Raviart type finite elements on anisotropic meshes. Numer. Math. 89 (2001), 193-223 [preprint | paper]
  42. Thomas Apel, Serge Nicaise: The finite element method with anisotropic mesh grading for elliptic problems in domains with corners and edges. Math. Methods Appl. Sci. 21(1998), 519-549. [paper | preprint]
  43. Thomas Apel: Interpolation of non-smooth functions on anisotropic finite element meshes. Math. Modeling Numer. Anal. 33(1999)6, 1149-1185. [paper | preprint]
  44. Thomas Apel: Treatment of boundary layers with anisotropic finite elements. Z. Angew. Math. Mech. 78(1998)S3, S855-S856. [paper | preprint]
  45. Thomas Apel, Gert Lube: Anisotropic mesh refinement for a singularly perturbed reaction diffusion model problem. Appl. Numer. Math. 26(1998), 415-433 [paper | early preprint]
  46. Thomas Apel: Anisotropic interpolation error estimates for isoparametric quadrilateral finite elements. Computing 60(1998), 157-174 [paper | preprint]
  47. Thomas Apel, Gert Lube: Anisotropic mesh refinement in stabilized Galerkin methods. Numer. Math. 74(1996), 261-282 [paper | preprint.]
  48. Thomas Apel, Frank Milde: Comparison of various mesh refinement strategies near edges. Comm. Numer. Methods Engrg. 12(1996), 373-381 [paper | preprint]
  49. Thomas Apel, Anna-Margarete Sändig, John R. Whiteman: Graded mesh refinement and error estimates for finite element solutions of elliptic boundary value problems in non-smooth domains. Math. Methods Appl. Sci. 19(1996), 63-85 [paper | preprint |  BICOM report]
  50. Thomas Apel, Manfred Dobrowolski: Anisotropic interpolation with applications to the finite element method. Computing 47(1992), 277-293 [paper]
  51. Thomas Apel, Bernd Heinrich: Mesh refinement and windowing near edges for some elliptic problem. SIAM J. Numer. Anal. 31(1994), 695-708 [paper]

Submitted papers

  1. Thomas Apel, Philipp Zilk: Isogeometric analysis of the Laplace eigenvalue problem on circular sectors: Regularity properties, graded meshes & variational crimes. arXiv:2402.16589 [math.NA] [ preprint ]
  2. Reymart S. Lagunero, Klemens Fellner, Thomas Apel, Volker Kempf, Philipp Zilk: Lipolysis on Lipid Droplets: Mathematical Modelling and Numerical Discretisation. arXiv:2401.17935 [math.AP] [ preprint ]

Conference proceedings and other collections

  1. Thomas Apel, Leon Eckardt, Christof Haubner, Volker Kempf: The maximal angle condition on finite elements: useful or not? PAMM · Proc. Appl. Math. Mech. 20(2020)1,  e202000116.
  2. Thomas Apel, Mariano Mateos, Johannes Pfefferer, Arnd Rösch: Superconvergent graded meshes for an elliptic Dirichlet control problem. In: Thomas Apel, Ulrich Langer, Arnd Meyer, Olaf Steinbach (eds.): Advanced Finite Element Methods with Applications - Selected Papers from the 30th Chemnitz Finite Element Symposium 2017. Lecture Notes in Computational Science and Engineering 128. Springer, Berlin, 2019, pp. 1-16.
  3. Thomas Apel, Thomas G. Flaig: Simulation and mathematical optimization of the hydration of concrete for avoiding thermal cracks. In: K. Gürlebeck and C. Könke (eds.): Proceedings of the 18th International Conference on the Applications of Computer Science and Mathematics in Architecture and Civil Engineerin, Weimar, Germany, 2009 [ abstract |  paper ]
  4. Thomas Apel, Arnd Rösch, Gunter Winkler: Discretization error estimates for an optimal control problem in a nonconvex domain. In: A. Bermúdez de Castro and D. Gómez and P. Quintela and P. Salgado (eds.): Numerical Mathematics and Advanced Applications, Proceedings of ENUMATH 2005, the 6th European Conference on Numerical Mathematics and Advanced Applications, Santiago de Compostela, Spain, July 2005, Springer, Berlin, 2006, pp. 299-307 [paper | preprint]
  5. Gert Lube, Thomas Apel, Tobias Knopp: Stabilized finite element methods with anisotropic mesh refinement for the Oseen problem. In: Proceedings of the Int. Conference on Boundary and Interior Layers, BAIL 2006 [ pdf-file]
  6. Thomas Apel, Cornelia Pester: Quadratic eigenvalue problems in the analysis of cracks in brittle materials. In: Proceedings of the ECCOMAS Conference, Jyväskylä 2004 [ paper]
  7. Thomas Apel, Volker Mehrmann, David Watkins: Numerical solution of large scale structured polynomial or rational eigenvalue problems. In: F. Cucker and P. Olver (eds.): Foundations of Computational Mathematics, Minneapolis 2002. London Mathematical Society Lecture Note Series, vol. 312, Cambridge University Press, 2004. [ps-file]
  8. Gunter Winkler, Thomas Apel, Uwe Wystup: Valuation of options in Heston's stochastic volatility model using finite element methods. In J. Hakala and U. Wystup (eds.): Foreign Exchange Risk, Risk Books, London, 2001, 283-303.
  9. Thomas Apel, Serge Nicaise, Joachim Schöberl: Finite element methods with anisotropic meshes near edges. In M. Krizek and P. Neittaanmäki (eds.): Proc. Internat. Conf. Finite Element Methods: Three-dimensional Problems. GAKUTO Internat. Series, Math. Sci. Appl., vol. 15, Gakkotosho, Tokyo, 2001, 1-8. [paper]
  10. Thomas Apel, Martin Berzins, Peter Jimack, Gerd Kunert, Alexander Plaks, Igor Tsukerman, Mark Walkley: Mesh Shape and Anisotropic elements: Theory and Practice. In J. R. Whiteman (ed.): The Mathematics of Finite Elements and Applications X, Elsevier, Amsterdam, 2000, 367-376 [paper]
  11. Thomas Apel: Anisotropic mesh refinement for the treatment of boundary layers. In M. Bach, C. Constanda, G. C. Hsiao, A.-M. Sändig, P. Werner (eds.): Analysis, numerics and applications of differential and integral equations, Pitman Research Notes in Mathematics 379, Longman, Harlow, 1998, 12-16 [preprint]
  12. Thomas Apel, Serge Nicaise: Elliptic problems in domains with edges: anisotropic regularity and anisotropic finite element meshes. In J. Cea, D. Chenais, G. Geymonat, J. L. Lions (eds.): Partial Differential Equations and Functional Analysis (In Memory of Pierre Grisvard) Birkhäuser, Boston, 1996, 18-34
  13. Thomas Apel, Roland Mücke, John R. Whiteman: Incorporation of a-priori mesh grading into a-posteriori adaptive mesh refinement. In A. Casal, L. Gavete, C. Conde, J. Herranz (eds.): III Congreso Matematica Aplicada/XIII C.E.D.Y.A. Madrid, 1993 Madrid, 1995, 79-92 [preprint | paper]
  14. Thomas Apel, Arnd Meyer, Michael Theß: Poisson equation and Lame system over three-dimensional domains. In J. Knop, P. Schreiber (eds.): Power Explorer User Report. Parsytec Computer GmbH, Aachen, 1995, 176-178 [preprint]

Research Reports

      1. Thomas Apel, Markus Melenk: Interpolation and quasi-interpolation in h- and hp-version finite element spaces (extended version). ASC Report 39/2015, Institute for Analysis and Scientific Computing, Vienna University of Technology, Wien, 2015. [  paper ]
      2. Thomas Apel, Serge Nicaise, Johannes Pfefferer: A dual singular complement method for the numerical solution of the Poisson equation with L2 boundary data in non-convex domains. Preprint arXiv:1505.00414, 2015.
      3. Thomas Apel, Dieter Sirch, Gunter Winkler: Error estimates for control constrained optimal control problems: Discretization with anisotropic finite element meshes.  Preprint SPP1253-02-06, DFG Priority Program 1253, Erlangen, 2008.
      4. Thomas Apel, Hans-Görg Roos: Remarks on the analysis of finite element methods on a Shishkin mesh: are Scott-Zhang interpolants applicable? Preprint MATH-NM-06-2008, TU Dresden, 2008 [ paper]
      5. Thomas Apel, Uwe Reichel: Partitioning of finite element meshes for parallel computing: A case study. Preprint SFB393/96-18a, TU Chemnitz-Zwickau, 1997 [paper]
      6. Thomas Apel, Serge Nicaise: The finite element method with anisotropic mesh grading for the Poisson problem in domains with edges. [paper]
      7. Thomas Apel, Gundolf Haase, Arnd Meyer, Matthias Pester: Parallel solution of finite element equation systems: efficient inter-processor communication. Preprint SPC95_5, TU Chemnitz-Zwickau, 1995 [paper | note]
      8. Thomas Apel, Gert Lube: Local inequalities for anisotropic finite elements and their application to convection-diffusion problems. Preprint SPC94_26, TU Chemnitz-Zwickau, 1994 [paper]
      9. Thomas Apel, Roland Mücke, John R. Whiteman: An adaptive finite element technique with a-priori mesh grading. Report 9, BICOM Institute of Computational Mathematics, 1993 [paper |  BICOM report]
      10. Thomas Apel: Diskrete Modelle der Gasdynamik in gemischten Euler-Lagrange-Koordinaten. Diplomarbeit, TH Karl-Marx-Stadt (now TU Chemnitz), 1986
      11. Thomas Apel: Numerische Verfahren zur Berechnung instationärer Strömungen. Jahresarbeit, TH Karl-Marx-Stadt (now TU Chemnitz), 1985

Manuals

    1. Thomas Apel, Frank Milde, Uwe Reichel: SPC-PM Po 3D v4.0 - Programmer's Manual. Preprint SFB393/99-37, TU Chemnitz, 1999 [paper]
    2. Thomas Apel, Uwe Reichel: SPC-PM Po 3D v3.3 - User's Manual. Preprint SFB393/99-06, TU Chemnitz, 1999 [paper]
    3. Thomas Apel, Frank Milde, Michael Theß: Preprint SPC95_34, TU Chemnitz-Zwickau, 1995 SPC-PM Po 3D - Programmer's Manual. [ paper]
    4. Thomas Apel: SPC-PM Po 3D - User's Manual. Preprint SPC95_33, TU Chemnitz-Zwickau, 1995 [ paper]
    5. Thomas Apel: Programmbeschreibung zum 3D-Finite-Elemente-Programmsystem FEMPS3D. TU Chemnitz, 1990