Forschungsprojekte Prof. Apel

Anisotrope finite Elemente

 Anisotropic Finite Elements

DFG-Förderung im Rahmen des SFB 393 bis Dezember 2004, Projekte AP 72/3, Förderungszeitraum bis Dezember 2006
 
 Bearbeiter und Kooperationspartner: 
  • Thomas Apel
  • Sergey Grosman
  • Gert Kunert
  • Gert Lube
  • Gunar Matthies
  • Serge Nicaise
  • Joachim Schöbert
 
 Inhalt: 
  • Anisotrope lokale Interpolationsfehlerabschätzungen
  • Optimale Netzverfeinerungsstrategien für Randwertprobleme mit anisotropen Lösungen
  • anisotrope Fehlerschätzer und adaptive Verfahren für anisotrope Finite-Elemente-Netze
  • inf-sup stabile Finite-Elemente-Paare
 

 Veröffentlichungen 

 Bücher und Dissertationen: 
  • 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]

 

 Studienarbeiten: 

  • 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.

 

 Artikel in referierten Zeitschriften: 

  • 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]

 

 Tagungsberichte: 

  • 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

 

 Weitere Forschungsberichte: 

  • 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]

Diskretisierung von Optimalsteuerproblemen

DFG-Projekt: Numerical analysis and discretization strategies for optimal control problems with singularities

im DFG-Schwerpunktprogramm 1253: Optimization with PDE constraints

 

 Leitung: 

  • Thomas Apel
  • Arnd Rösch
  • Boris Vexler

 

 Bearbeiter: 

  • Olaf Benedix
  • Thomas Flaig
  • Martin Naß
  • Johannes Pfefferer
  • Dieter Sirch
  • Gunter Winkler
  • Max Winkler

 

 Kooperationspartner: 

  • Roland Becker
  • Michael Hinze
  • Ronald W. Hoppe
  • Gert Lube
  • Rolf Rannacher
  • Fredi Tröltzsch

 

 Förderungszeiträume: 

  • 2. Förderungszeitraum: Oktober 2009 - Dezember 2013
  • 1. Förderungszeitraum: Oktober 2006 - September 2009

 

 Ziele: 

  • A priori Fehlerabschätzungen für Optimalsteuerprobleme mit Kontrollbeschränkungen
  • A priori Fehlerabschätzungen bei punktweisen Zustandsbeschränkungen
  • A posteriori Fehlerschätzer bei Kontrollbeschränkungen
  • A posteriori Fehlerschätzer bei Zustandsbeschränkungen
  • Verifizierung aller Untersuchungen durch entsprechende numerische Tests

 

 Abschlussarbeiten aus dem Projekt: 

  • Flaig, T. G.: Discretization strategies for optimal control problems with parabolic partial differential equations, PhD thesis, UniBw München, 2013. [im Buchhandel]
  • Sirch, D.: Finite Element Error Analysis for PDE-constrained Optimal Control Problems: The Control Constrained Case Under Reduced Regularity, PhD thesis, TU München, 2010. [Dissertation]
  • Winkler, G.: Control constrained optimal control problems in non-convex three dimensional polyhedral domains, PhD thesis, TU Chemnitz, 2008. [Dissertation]

 

 Veröffentlichungen aus dem Projekt: 

  • Apel, T., Pfefferer, J., Rösch, A.: 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.
  • Flaig, T. G., Meidner, D.,  Vexler, B.: Petrov-Galerkin Crank-Nicolson Scheme for Parabolic Optimal Control Problems on Nonsmooth Domains, In Günter Leugering, Peter Benner, Sebastian Engell, Andreas Griewank, Helmut Harbrecht, Michael Hinze, Rolf Rannacher, Stefan Ulbrich (ed.): Trends in PDE Constrained Optimization. International Series of Numerical Mathematics 165. Birkhäuser, Basel, 2014, pp. 421-435.
  • Apel, T., Flaig, T. G., Nicaise, S.: A priori error estimates for finite element methods for H^(2,1)-elliptic equations Numerical Functional Analysis and Optimization 35(2): 153-176, 2014. [Paper] [Preprint]
  • Flaig, T. G.: Implicit Runge-Kutta schemes for optimal control problems with evolution equations, submitted, 2013. [Preprint]
  • Apel, T., Flaig, T. G.: Crank-Nicolson Schemes for Optimal Control Problems with Evolution Equations, SIAM J. Numer. Anal. 50: 1484-1512, 2012. [Paper] / [Preprint]
  • Apel, T., Pfefferer, J., Rösch, A.: Finite element error estimates for Neumann boundary control problems on graded meshes. Computational Optimization and Applications 52(1): 3-28, 2012. [Paper] / [Preprint]
  • Apel, T., Sirch, D.: 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. [Preprint]
  • Apel, T., Benedix, O., Sirch, D., Vexler, B.: A priori mesh grading for an elliptic problem with Dirac right-hand side, SIAM Journal on Numerical Analysis 49(3): 992 - 1005, 2011. [Paper][Preprint]
  • 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., Flaig, T. G.: Simulation and Mathematical Optimization of the Hydration of Concrete for avoiding thermal Cracks., in K. Gürlebeck and C. Könke (eds.): 18th International Conference on the Application of Computer Science and Mathematics in Architecture and Civil Engineering, ISSN 1611-4086, Weimar 2009.
  • Nicaise, S., Sirch, D.: Optimal control of the Stokes equations: Conforming and non-conforming finite element methods under reduced regularity, Comput. Optim. Appl. 49(3): 567-600, 2009. [Paper] [Preprint]
  • Apel, T., Sirch, D., Winkler, G.: Error estimates for control constrained optimal control problems: Discretization with anisotropic finite element meshes, submitted, 2008. [Preprint]
  • Apel, T., Rösch, A., Sirch, D.: L-error estimates on graded meshes with application to optimal control, SIAM J. Control Optim. 48(2009), 1771-1796. [Paper][Preprint]
  • Apel, T., Rösch, A., Winkler, G.: Optimal control in nonconvex domains: a priori discretization error estimates, Calcolo 44(3): 137-158, 2007. [Paper][Preprint]
  • Apel, T., Winkler, G.: Optimal Control Under Reduced Regularity. Appl. Numer. Math. 59(9): 2050-2064, 2009. [Paper][Preprint]
  • Apel, T., Rösch, A., Winkler, G.: Discretization error estimates for an optimal control problem in a nonconvex domain, in: A. Bermúdez de Castro et al. (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, 299-307, Springer, Berlin, 2006. [Paper] / [Preprint]

Randsteuerungsprobleme auf polyedrischen Gebieten

 Boundary Control Problems in Polyhedral Domains 

ein Projekt im Graduiertenkolleg International Research Training Group (IGDK) 1754: Optimization and Numerical Analysis for Partial Differential Equations with Nonsmooth Structures

Das Graduiertenkolleg wird gemeinsam von der Deutschen Forschungsgemeinschaft (DFG) und dem österreichischen Fonds zur Förderung der wissenschaftlichen Forschung (FWF) finanziert.


 Leitung: 

  • Thomas Apel
  • Olaf Steinbach
  • Boris Vexler


 Bearbeiter: 

  • Max Winkler
  • Sergejs Rogovs


 Mentor: 

  • Johannes Pfefferer


 Förderungszeiträume: 

  • 1. Förderungszeitraum: März 2012 - August 2016
  • 2. Förderungszeitraum: September 2016 - Februar 2021


 Ziele: 

  • A priori Fehlerabschätzungen für optimale Randsteuerprobleme
  • Lokale Netzverfeinerung an Ecken und Kanten
  • Verifizierung der Untersuchungen durch entsprechende numerische Tests


 Abschlussarbeiten aus dem Projekt: 

  • Pfefferer, J.: Numerical analysis for elliptic Neumann boundary control problems on polygonal domains, PhD Thesis, UniBw München, 2014. [Dissertation]
  • Winkler, M.: Finite element error analysis for Neumann boundary control problems on polygonal and polyhedral domains, PhD Thesis, UniBw München, 2015.


 Veröffentlichungen aus dem Projekt: 

  • 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, accepted by SIAM J. Numer. Anal. 2017 [Preprint]
  • 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. [Paper] / [Preprint]
  • 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. [Paper] / [Preprint]
  • Neitzel, I., Pfefferer, J., Rösch, A.: Finite element discretization of state-constrained elliptic optimal control problems with semilinear state equation, SIAM J. Control. Optim, 53(2):974-904, 2015. [Paper] / [Preprint]
  • Apel, T., Pfefferer, J., Rösch, A.: Finite element error estimates on the boundary with application to optimal control, Math. Comp. 84(291): 33-70, 2015. [Paper] / [Preprint]
  • Apel, T., Lombardi, A. L., Winkler, M.: Anisotropic mesh refinement in polyhedral domains: error estimates with data in L^2(Ω), ESAIM. Math. Model. Numer. Anal. 48(4): 1117-1145, 2014. [Paper] / [Preprint]
  • Apel, T., Pfefferer, J., Winkler, M.: Local Mesh Refinement for the Discretization of Neumann Boundary Control Problems on Polyhedra, Math. Methods Appl. Sci., 39(5):1206-1232, 2016. [Paper] / [Preprint]
  • Apel, T., Steinbach, O., Winkler, M.: Error Estimates for Neumann Boundary Control Problems with Energy Regularization, J. Numer. Math 24(4):207-233, 2016. [Paper]
  • Klaus Krumbiegel, Johannes Pfefferer: Superconvergence for Neumann boundary control problems governed by semilinear elliptic equations, Comput. Optim. Appl., 61(2):373-408, 2015. [Paper] / [Preprint]
  • Grossmann, C., Winkler, M.: Mesh-Independent Convergence of Penalty Methods Applied to Optimal Control with Partial Differential Equations, Optimization 62(5): 629-647, 2013. [Paper]
  • Grossmann, C., Winkler, M.: A Mesh-Independence Principle for Quadratic Penalties Applied to Semilinear Elliptic Boundary Control, Schedae Informaticae 21: 9-26, 2012. [Paper]

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