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Adaptive BEM-Based Finite Element Method

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BEM-based Finite Element Approaches on Polytopal Meshes

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 130))

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Abstract

As long as the solutions of boundary value problems are sufficiently regular, the refinement of the mesh size h and the increase of the approximation order k in the discretization space \(V_h^k\) yields an improvement in the accuracy. In particular, this yields optimal convergence rates. But, in most applications the regularity of the solution is restricted due to corners of the domain or jumping physical quantities. Therefore, it is essential to adapt the solution process to the underlying problem in order to retrieve optimal approximation properties. In this chapter, we deal with a posteriori error estimates which can be used to drive an adaptive mesh refinement procedure and we recover optimal rates of convergence for the adaptive methods in the numerical experiments in the presence of singularities. For the error estimation, we cover the classical residual based error estimator as well as goal-oriented techniques on general polytopal meshes. Whereas, we derive reliability and efficiency estimates for the first mentioned estimator, we discuss the benefits and potentials of the second one for general meshes.

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References

  1. Ainsworth, M., Oden, J.T.: A posteriori error estimation in finite element analysis. Comput. Methods Appl. Mech. Eng. 142(1–2), 1–88 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ainsworth, M., Oden, T.J.: A posteriori error estimation in finite element analysis. In: Pure and Applied Mathematics. Wiley, New York (2000)

    Google Scholar 

  3. Babuška, I., Rheinboldt, W.C.: A-posteriori error estimates for the finite element method. Int. J. Numer. Methods Eng. 12(10), 1597–1615 (1978)

    Article  MATH  Google Scholar 

  4. Babuška, I., Rheinboldt, W.C.: Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 15(4), 736–754 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bangerth, W., Rannacher, R.: Adaptive Finite Element Methods for Differential Equations. Lectures in Mathematics, ETH Zürich. Birkhäuser, Basel (2003)

    Book  MATH  Google Scholar 

  6. Bank, R.E., Weiser, A.: Some a posteriori error estimators for elliptic partial differential equations. Math. Comp. 44(170), 283–301 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  7. Becker, R., Rannacher, R.: Weighted a posteriori error control in FE methods. In: Bock, H.G., et al. (eds.) ENUMATH’97, pp. 18–22. World Science Publication, Singapore (1995)

    Google Scholar 

  8. Becker, R., Rannacher, R.: An optimal control approach to a posteriori error estimation in finite element methods. Acta Numer. 10, 1–102 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. Binev, P., Dahmen, W., DeVore, R.: Adaptive finite element methods with convergence rates. Numer. Math. 97(2), 219–268 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Braess, D., Schöberl, J.: Equilibrated residual error estimator for edge elements. Math. Comput. 77(262), 651–672 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Carstensen, C.: Estimation of higher Sobolev norm from lower order approximation. SIAM J. Numer. Anal. 42(5), 2136–2147 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Carstensen, C., Feischl, M., Page, M., Praetorius, D.: Axioms of adaptivity. Comput. Math. Appl. 67(6), 1195–1253 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Carstensen, C., Merdon, C.: Estimator competition for poisson problems. J. Comput. Math. 28(3), 309–330 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Cascon, J.M., Kreuzer, C., Nochetto, R.H., Siebert, K.G.: Quasi-optimal convergence rate for an adaptive finite element method. SIAM J. Numer. Anal. 46(5), 2524–2550 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Deuflhard, P., Leinen, P., Yserentant, H.: Concepts of an adaptive hierarchical finite element code. IMPACT Comput. Sci. Eng. 1, 3–35 (1989)

    Article  MATH  Google Scholar 

  17. Dörfler, W.: A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33(3), 1106–1124 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  18. Eriksson, K., Estep, D., Hansbo, P., Johnson, C.: Introduction to adaptive methods for differential equations. Acta Numer. 4, 105–158 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  19. Giles, M.B., Süli, E.: Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality. Acta Numer. 11, 145–236 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  20. Morin, P., Nochetto, R.H., Siebert, K.S.: Data oscillation and convergence of adaptive FEM. SIAM J. Numer. Anal. 38(2), 466–488 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  21. Morin, P., Siebert, K.G., Veeser, A.: A basic convergence result for conforming adaptive finite elements. Math. Models Methods Appl. Sci. 18(5), 707–737 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Nochetto, R.H., Veeser, A., Verani, M.: A safeguarded dual weighted residual method. IMA J. Numer. Anal. 29(1), 126–140 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Oden, J.T., Prudhomme, S.: On goal-oriented error estimation for elliptic problems: application to the control of pointwise errors. Comput. Methods Appl. Mech. Eng. 176, 313–331 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  24. Peraire, J., Patera, A.T.: Bounds for linear-functional outputs of coercive partial differential equations: local indicators and adaptive refinement. In: Advances in adaptive computational methods in mechanics (Cachan, 1997). Studies in Applied Mechanics, vol. 47, pp. 199–216. Elsevier Science B. V., Amsterdam (1998)

    Chapter  Google Scholar 

  25. Petzoldt, M.: A posteriori error estimators for elliptic equations with discontinuous coefficients. Adv. Comput. Math. 16(1), 47–75 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  26. Rannacher, R., Suttmeier, F.T.: A feed-back approach to error control in finite element methods: application to linear elasticity. Comput. Mech. 19(5), 434–446 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  27. Repin, S.: A Posteriori Estimates for Partial Differential Equations. Radon Series on Computational and Applied Mathematics, vol. 4. Walter de Gruyter GmbH and Co. KG, Berlin (2008)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  29. Scott, R.: Optimal L ∞ estimates for the finite element method on irregular meshes. Math. Comput. 30(136), 681–697 (1976)

    MathSciNet  MATH  Google Scholar 

  30. Verfürth, R.: A Posteriori Error Estimation Techniques for Finite Element Methods. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (2013)

    Book  MATH  Google Scholar 

  31. Weißer, S.: Residual error estimate for BEM-based FEM on polygonal meshes. Numer. Math. 118(4), 765–788 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  32. Weißer, S.: Residual based error estimate and quasi-interpolation on polygonal meshes for high order BEM-based FEM. Comput. Math. Appl. 73(2), 187–202 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  33. Zienkiewicz, O.C., Zhu, J.Z.: A simple error estimator and adaptive procedure for practical engineering analysis. Int. J. Numer. Methods Eng. 24(2), 337–357 (1987)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. FR Mathematik, Universität des Saarlandes, Saarbrücken, Germany

    Steffen Weißer

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  1. Steffen Weißer
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Weißer, S. (2019). Adaptive BEM-Based Finite Element Method. In: BEM-based Finite Element Approaches on Polytopal Meshes. Lecture Notes in Computational Science and Engineering, vol 130. Springer, Cham. https://doi.org/10.1007/978-3-030-20961-2_5

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