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Numerical Study of Goal-Oriented Error Control for Stabilized Finite Element Methods

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Advanced Finite Element Methods with Applications (FEM 2017)

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Abstract

The efficient and reliable approximation of convection-dominated problems continues to remain a challenging task. To overcome the difficulties associated with the discretization of convection-dominated equations, stabilization techniques and a posteriori error control mechanisms with mesh adaptivity were developed and studied in the past. Here we combine the Dual Weighted Residual (DWR) method for goal-oriented error control with stabilized finite element methods. By a duality argument an error representation is derived on that an adaptive strategy is built. The key ingredient of this work is the application of a higher-order discretization of the dual problem in order to make a robust error control for user-chosen quantities of interest feasible. By numerical experiments in 2D and 3D we illustrate that this interpretation of the DWR methodology is capable to resolve layers and sharp fronts with high accuracy and to further reduce spurious oscillations.

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References

  1. Ahmed, N., John, V.: Adaptive time step control for higher order variational time discretizations applied to convection-diffusion equations. Comput. Methods Appl. Mech. Eng. 285, 83–101 (2015)

    Article  MathSciNet  Google Scholar 

  2. Angermann, L.: Balanced a posteriori error estimates for finite-volume type discretizations of convection-dominated elliptic problems. Computing 55(4), 305–323 (1995)

    Article  MathSciNet  Google Scholar 

  3. Arndt, D., Bangerth, W., Davydov, D., Heister, T., Heltai, L., Kronbichler, M., Maier, M., Pelteret, J.-P., Turcksin, B., Wells, D.: The deal.II library, Version 8.5. J. Numer. Math. 25(3), 137–146 (2017). https://doi.org/10.1515/jnma-2016-1045

  4. Bangerth, W., Rannacher, R.: Adaptive Finite Element Methods for Differential Equations. Birkhäuser, Basel (2003)

    Google Scholar 

  5. Bangerth, W., Hartmann, R., Kanschat, G.: Deal.II-A general purpose object oriented finite element library. ACM Trans. Math. Softw. 33(4), 24/1–24/27 (2007). https://doi.org/10.1145/1268776.1268779

  6. Barrenechea, G.R., John, V., Knobloch, P.: Some analytical results for an algebraic flux correction scheme for a steady convection-diffusion equation in one dimension. IMA J. Numer. Anal. 35(4), 1729–1756 (2015)

    Article  MathSciNet  Google Scholar 

  7. Barrenechea, G.R., John, V., Knobloch, P.: Analysis of algebraic flux correction schemes. SIAM J. Numer. Anal. 54(4), 2427–2451 (2016)

    Article  MathSciNet  Google Scholar 

  8. Bause, M., Schwegler, K.: Analysis of stabilized higher order finite element approximation of nonstationary and nonlinear convection-diffusion-reaction equations. Comput. Methods Appl. Mech. Eng. 209–212, 184–196 (2012)

    Article  MathSciNet  Google Scholar 

  9. Bause, M., Köcher, U., Radu, F.A., Schieweck, F.: Post-processed Galerkin approximation of improved order for wave equations. Math. Comput., submitted (2018). arXiv:1803.03005

    Google Scholar 

  10. Becker, R.: An optimal-control approach to a posteriori error estimation for finite element discretizations of the Navier–Stokes equations. East-West J. Numer. Math. 8, 257–274 (2000)

    MathSciNet  MATH  Google Scholar 

  11. Becker, R., Rannacher, R.: Weighted a posteriori error control in FE methods. In: Bock, H.G., et al. (eds.) ENUMATH 97. Proceedings of the 2nd European Conference on Numerical Mathematics and Advanced Applications, pp. 621–637. World Scientific, Singapore (1998)

    Google Scholar 

  12. Becker, R., Rannacher, R.: An optimal control approach to a posteriori error estimation in finite element methods. In: Iserles, A. (ed.) Acta Numerica, vol. 10, pp. 1–102. Cambridge University, Cambridge (2001)

    Google Scholar 

  13. Besier, M., Rannacher, R.: Goal-oriented space-time adaptivity in the finite element galerkin method for the computation of nonstationary incompressible flow. Int. J. Num. Methods Fluids 70(9), 1139–1166 (2012)

    Article  MathSciNet  Google Scholar 

  14. Bittl, M., Kuzmin, D.: An hp-adaptive flux-corrected transport algorithm for continuous finite elements. Computing 95, 27–48 (2013)

    Article  MathSciNet  Google Scholar 

  15. Braack, M., Ern, A.: A posteriori control of modeling errrors and discretization errors. Multiscale Model. Simul. 1(2), 221–238 (2003)

    Article  MathSciNet  Google Scholar 

  16. Brooks, A.N., Hughes, T.J.R.: Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 32(1–3), 199–259 (1982)

    Article  MathSciNet  Google Scholar 

  17. Carey, G.F., Oden, J.T.: Finite Elements, Computational Aspects, Vol. III (The Texas Finite Element Series). Prentice-Hall, Englewood Cliffs (1984)

    Google Scholar 

  18. Dörfler, W.: A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33, 1106–1124 (1996)

    Article  MathSciNet  Google Scholar 

  19. Endtmayer, B., Wick, T.: A partition-of-unity dual-weighted residual approach for multi-objective goal functional error estimation applied to elliptic problems. Comput. Methods Appl. Math. 17(4), (2017). https://doi.org/10.1515/cmam-2017-0001

  20. Ern, A., Schieweck, F.: Discontinuous Galerkin method in time combined with a stabilized finite element method in space for linear first-order pdes. Math. Comput. 85, 2099–2129 (2016)

    Article  MathSciNet  Google Scholar 

  21. Evans, L.C.: Partial Differential Equations. American Mathematical Society, Providence (2010)

    MATH  Google Scholar 

  22. Houston, P., Rannacher, R., Süli, E.: A posteriori error analysis for stabilised finite element approximations of transport problems. Comput. Methods Appl. Mech. Eng. 190, 1483–1508 (2000)

    Article  MathSciNet  Google Scholar 

  23. Hughes, T.J.R., Brooks, A.N.: A multidimensional upwind scheme with no crosswind diffusion. In: Hughes, T.J.R. (ed.) Finite Element Methods for Convection Dominated Flows, AMD, vol. 34, pp. 19–35. American Society of Mechanical Engineers (ASME), New York (1979)

    Google Scholar 

  24. Hughes, T.J.R., Mallet, M., Mizukami, A.: A new finite element formulation for computational fluid dynamics: II. Beyond SUPG. Comput. Methods Appl. Mech. Eng. 54, 341–355 (1986)

    Article  MathSciNet  Google Scholar 

  25. John, V., Knobloch, P.: Adaptive computation of parameters in stabilized methods for convection-diffusion problems. In: Cangiani, A., et al. (eds.) Numerical Mathematics and Advanced Applications 2011. Springer, Heidelberg (2013)

    Google Scholar 

  26. John, V., Knobloch, P., Savescu, S.B.: A posteriori optimization of parameters in stabilized methods for convection-diffusion problems-Part I. Comput. Methods Appl. Mech. Eng. 200, 2916–2929 (2011)

    Article  MathSciNet  Google Scholar 

  27. John, V., Novo, J.: A robust SUPG norm a posteriori error estimator for stationary convection-diffusion equations. Comput. Methods Appl. Mech. Eng. 255, 289–305 (2013)

    Article  MathSciNet  Google Scholar 

  28. John, V., Schmeyer, E.: Finite element methods for time-dependent convection-diffusion-reaction equations with small diffusion. Comput. Methods Appl. Mech. Eng. 198, 173–181 (2009)

    MathSciNet  MATH  Google Scholar 

  29. John, V., Schmeyer, E.: On finite element methods for 3D time-dependent convection-diffusion-reaction equations with small diffusion In: Hegarty A., et al. (eds.) BAIL 2008 - Boundary and Interior Layers. Lecture Notes in Computational Science and Engineering, vol. 69. Springer, Heidelberg (2009)

    Google Scholar 

  30. Köcher, U.: Variational Space-Time Methods for the Elastic Wave Equation and the Diffusion Equation, Dissertation. Helmut Schmidt University, Hamburg, urn:nbn:de:gbv:705-opus-31129 (2015)

    Google Scholar 

  31. Köcher, U., Bruchhäuser, M.P., Bause, M.: Efficient and scalable data structures and algorithms for goal-oriented adaptivity of space-time FEM codes. SoftwareX 100239 (2019). https://doi.org/10.1016/j.softx.2019.100239

  32. Kuzmin, D., Korotov, S.: Goal-oriented a posteriori error estimates for transport problems. Math. Comput. Simul. 80(8), 1674–1683 (2010)

    Article  MathSciNet  Google Scholar 

  33. Kuzmin, D., Turek, S.: Flux correction tools for finite elements. J. Comput. Phys. 175, 525–558 (2002)

    Article  MathSciNet  Google Scholar 

  34. Löhner, R., Morgan, K., Peraire, P., Vahdati, M.: Finite element flux-corrected transport (FEM-FCT) for the Euler and Navier-Stokes equations. Int. J. Numer. Methods Fluids 7(10), 1093–1109 (1987)

    Article  Google Scholar 

  35. Lube, G., Rapin, G.: Residual-based stabilized higher-order FEM for advection-dominated problems. Comput. Methods Appl. Mech. Eng. 195, 4124–4138 (2006)

    Article  MathSciNet  Google Scholar 

  36. Rannacher, R.: A posteriori error estimation in least-squares stabilized finite element schemes. Comput. Methods Appl. Mech. Eng. 166, 99–114 (1998)

    Article  MathSciNet  Google Scholar 

  37. Richter, T., Wick, T.: Variational localizations of the dual weighted residual estimator. J. Comput. Appl. Math. 279, 192–208 (2015)

    Article  MathSciNet  Google Scholar 

  38. Roos, H.-G., Stynes, M., Tobiska, L.: Robust Numerical Methods for Singularly Perturbed Differential Equations. Springer, Berlin (2008)

    MATH  Google Scholar 

  39. Schmich, M.: Adaptive Finite Element methods for computing nonstationary incompressible Flows, Dissertation. University of Heidelberg, Heidelberg urn:nbn:de:bsz:16-opus-102001 (2009)

    Google Scholar 

  40. Schwegler, K.: Adaptive goal-oriented error control for stabilized approximations of convection-dominated problems, Dissertation. Helmut Schmidt University Hamburg (2014). http://edoc.sub.uni-hamburg.de/hsu/volltexte/2014/3086/

    Google Scholar 

  41. Verfürth, R.: A Posteriori Error Estimation Techniques for Finite Element Methods. Oxford University, Oxford (2013)

    Book  Google Scholar 

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Acknowledgement

The authors wish to thank the anonymous reviewers for their help to improve the presentation of this paper.

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Authors and Affiliations

  1. Helmut Schmidt University, Hamburg, Germany

    Marius Paul Bruchhäuser, Kristina Schwegler & Markus Bause

Authors
  1. Marius Paul Bruchhäuser
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  2. Kristina Schwegler
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  3. Markus Bause
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Corresponding author

Correspondence to Marius Paul Bruchhäuser .

Editor information

Editors and Affiliations

  1. Institut für Mathematik & Computergestützte Simulation, Universität der Bundeswehr München, Neubiberg, Germany

    Thomas Apel

  2. Institute for Computational Mathematics, Johannes Kepler University Linz, Linz, Austria

    Ulrich Langer

  3. Fakultät für Mathematik, TU Chemnitz, Chemnitz, Germany

    Arnd Meyer

  4. Institut für Angewandte Mathematik, Technische Universität Graz, Graz, Austria

    Olaf Steinbach

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Bruchhäuser, M.P., Schwegler, K., Bause, M. (2019). Numerical Study of Goal-Oriented Error Control for Stabilized Finite Element Methods. In: Apel, T., Langer, U., Meyer, A., Steinbach, O. (eds) Advanced Finite Element Methods with Applications. FEM 2017. Lecture Notes in Computational Science and Engineering, vol 128. Springer, Cham. https://doi.org/10.1007/978-3-030-14244-5_5

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