Anisotropic Finite Elements

DFG-Funding in the framework of SFB 393 until December 2004, project AP 72/3, funding until December 2006
 
 Researchers and partners: 
  • Thomas Apel (Universität der Bundeswehr München)
  • Sergey Grosman (Universität der Bundeswehr München)
  • Gert Kunert (früher TU Chemnitz)
  • Gert Lube (Georg-August-Universität Göttingen)
  • Gunar Matthies (TU Dresden)
  • Serge Nicaise (Université de Valenciennes et du Hainaut Cambrésis, Valenciennes, Frankreich)
  • Joachim Schöberl (TU Wien)
 
 Content: 
  • Anisotropic local interpolation error estimates
  • Optimal mesh refinement strategies for boundary value problems with anisotropic solutions
  • Anisotropic error estimates and adaptive techniques for anisotropic finite element meshes
  • inf-sup stable finite element pairs
 

 Publications 

 Books and PhD Theses: 
  • Apel, T.: 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.) [Download]
  • Apel, T.: Finite-Elemente-Methoden über lokal verfeinerten Netzen für elliptische Probleme in Gebieten mit Kanten. PhD thesis, TU Chemnitz, 1991
  • Grosman, S.: Adaptivity in anisotropic finite element calculations. PhD thesis, TU Chemnitz, 2006. [Dissertation]

 

 Student Research Projects: 

  • Seidel, J.: Eine Auflösungsmethode für das Finite-Elemente-Gleichungssystem bei anisotroper Diskretisierung in der Umgebung einer Kante, Diplomarbeit, TU Chemnitz, 2002.
  • Grosman, S.: Robust local problem error estimation for a singularly perturbed reaction-diffusion problem on anisotropic finite element meshes, Master thesis, TU Chemnitz, 2001.
  • Randrianarivony, M.: Stability of mixed finite element methods with anisotropic meshes, Master thesis, TU Chemnitz, 2001.

 

 Articles in Scientific Journals: 

  • Apel, T., Lombardi, A. L., Winkler, M.: Anisotropic mesh refinement in polyhedral domains: error estimates with data in L2(Omega). ESAIM: Math. Model. Numer. Anal. 48(4): 1117-1145, 2014. [Preprint] / [Paper]
  • Apel, T., Sirch, D.: L2-error estimates for the Dirichlet and Neumann problem on anisotropic finite element meshes. Appl. Math. 56(2), 177–206, 2011. [Preprint]
  • Apel, T., Nicaise, S., Sirch, D.: A posteriori error estimation of residual type for anisotropic diffusion-convection-reaction problems. J. Comp. Appl. Math. 235(8): 2805-2820, 2011. [Paper]
  • Apel, T., Knopp, T., Lube, G.: Stabilized finite element methods with anisotropic mesh refinement for the Oseen problem, Appl. Numer. Math. 58(12): 1830-1843, 2008. [Paper]
  • Apel, T., Matthies, G.: Non-conforming, anisotropic, rectangular finite elements of arbitrary order for the Stokes problem, SIAM J. Numer. Anal. 46(4): 1867-1891, 2008. [Paper]
  • Acosta, G., Apel, T., Durán, R. G., Lombardi, A. L.: Error estimates for Raviart-Thomas interpolation of any order over anisotropic tetrahedra, Math. Comp. 80(273): 141-163, 2010. [Paper] / [Preprint]
  • Acosta, G., Apel, T., Durán, R. G., Lombardi, A. L.: Anisotropic error estimates for an interpolant defined via moments, Computing 82(1): 1-9, 2008. [Paper] / [Preprint]
  • Grosman, S.: An equilibrated residual method with a computable error approximation for a singularly perturbed reaction-diffusion problem on anisotropic finite element meshes, Math. Model. Numer. Anal. 40(2): 239-267, 2006. [Paper]
  • Apel, T., Schöberl, J.: Multigrid methods for anisotropic edge refinement, SIAM J. Numer. Anal. 40(5): 1993-2006, 2006. [Paper]
  • Apel, T., Nicaise, S.: The inf-sup condition for the Bernardi-Fortin-Raugel element on anisotropic meshes, Calcolo 41(2): 89-113, 2004. [Paper] / [Preprint]
  • Apel, T., Grosman, S., Jimack, P. K., Meyer, A.: A new methodology for anisotropic mesh refinement based upon error gradients, Appl. Numer. Math. 50(3-4): 329-341, 2004. [Paper] / [Preprint]
  • Apel, T., Randrianarivony, H. M.: Stability of discretizations of the Stokes problem on anisotropic meshes, Math. Comput. Simulation 61(3-6):437-447, 2003. [Paper] / [Preprint]
  • Apel, T., Nicaise, S., Schöberl, J.: A non-conforming finite element method with anisotropic mesh grading for the Stokes problem in domains with edges, IMA J. Numer. Anal. 21(4): 843-856, 2001. [Paper]
  • Apel, T., Nicaise, S., Schöberl, J.: Crouzeix-Raviart type finite elements on anisotropic meshes. Numer. Math. 89(2): 193-223, 2001. [Paper] / [Preprint]
  • Apel, T.: Interpolation of non-smooth functions on anisotropic finite element meshes, Math. Model. Numer. Anal 33(6): 1149-1185, 1999. [Preprint]
  • Apel, T., Nicaise, S.: The finite element method with anisotropic mesh grading for elliptic problems in domains with corners and edges. Math. Methods Appl. Sci. 21(6): 519-549, 1998. [Paper] / [Preprint]
  • Apel, T., Lube, G.: Anisotropic mesh refinement for a singularly perturbed reaction diffusion model problem, Appl. Numer. Math. 26(4): 415-433, 1998. [Paper] / [Preprint]
  • Apel, T.: Anisotropic interpolation error estimates for isoparametric quadrilateral finite elements, Computing 60(2): 157-174, 1998. [Paper] / [Preprint]
  • Apel, T., Lube, G.: Anisotropic mesh refinement in stabilized Galerkin methods, Numer. Math. 74(3): 261-282, 1996. [Paper] / [Preprint]
  • Apel, T., Milde, F.: Comparison of various mesh refinement strategies near edges, Comm. Numer. Methods Engrg. 12(7): 373-381, 1996. [Paper] / [Preprint]
  • Apel, T., Dobrowolski, M.: Anisotropic interpolation with applications to the finite element method. Computing 47(3-4): 277-293, 1992. [Paper]

 

 Proceedings: 

  • Apel, T., Nicaise, S., Schöberl, J.: 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]
  • Apel, T., Berzins, M., Jimack, P., Kunert, G., Plaks, A., Tsukerman, I., Walkley, M.: 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]
  • Apel, T.: Treatment of boundary layers with anisotropic finite elements. Z. Angew. Math. Mech. 78(1998)S3, S855-S856. [Preprint]
  • Apel, T.: 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]
  • Apel, T., Nicaise, S.: 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
  • Apel, T., Mücke, R., Whiteman, J. R.: 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

 

 Further Research Reports: 

  • Apel, T., Nicaise, S.: The finite element method with anisotropic mesh grading for the Poisson problem in domains with edges. [Paper]
  • Apel, T., Lube, G.: Local inequalities for anisotropic finite elements and their application to convection-diffusion problems. Preprint SPC94_26, TU Chemnitz-Zwickau, 1994. [Paper]