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Algebraic and Discretization Error Estimation by Equilibrated Fluxes for Discontinuous Galerkin Methods on Nonmatching Grids

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Abstract

We derive a posteriori error estimates for the discontinuous Galerkin method applied to the Poisson equation. We allow for a variable polynomial degree and simplicial meshes with hanging nodes and propose an approach allowing for simple (nonconforming) flux reconstructions in such a setting. We take into account the algebraic error stemming from the inexact solution of the associated linear systems and propose local stopping criteria for iterative algebraic solvers. An algebraic error flux reconstruction is introduced in this respect. Guaranteed reliability and local efficiency are proven. We next propose an adaptive strategy combining both adaptive mesh refinement and adaptive stopping criteria. At last, we detail a form of the estimates avoiding any practical reconstruction of a flux and only working with the approximate solution, which simplifies greatly their evaluation. Numerical experiments illustrate a tight control of the overall error, good prediction of the distribution of both the discretization and algebraic error components, and efficiency of the adaptive strategy.

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References

  1. Ainsworth, M.: Robust a posteriori error estimation for nonconforming finite element approximation. SIAM J. Numer. Anal. 42(6), 2320–2341 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  2. Ainsworth, M., Rankin, R.: Fully computable error bounds for discontinuous Galerkin finite element approximations on meshes with an arbitrary number of levels of hanging nodes. SIAM J. Numer. Anal. 47(6), 4112–4141 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  3. Ainsworth, M., Rankin, R.: Constant free error bounds for nonuniform order discontinuous Galerkin finite-element approximation on locally refined meshes with hanging nodes. IMA J. Numer. Anal. 31(1), 254–280 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  4. Arioli, M.: A stopping criterion for the conjugate gradient algorithm in a finite element method framework. Numer. Math. 97(1), 1–24 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  5. Arioli, M., Georgoulis, E.H., Loghin, D.: Stopping criteria for adaptive finite element solvers. SIAM J. Sci. Comput. 35(3), A1537–A1559 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  6. Arioli, M., Liesen, J., Międlar, A., Strakoš, Z.: Interplay between discretization and algebraic computation in adaptive numerical solution of elliptic PDE problems. GAMM-Mitt. 36(1), 102–129 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  7. Arioli, M., Loghin, D.: Stopping criteria for mixed finite element problems. Electron. Trans. Numer. Anal. 29, 178–192 (2007/08)

  8. Arioli, M., Loghin, D., Wathen, A.J.: Stopping criteria for iterations in finite element methods. Numer. Math. 99(3), 381–410 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  9. Becker, R., Johnson, C., Rannacher, R.: Adaptive error control for multigrid finite element methods. Computing 55(4), 271–288 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  10. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics, vol. 15. Springer, New York (1991)

    Book  Google Scholar 

  11. Cheddadi, I., Fučík, R., Prieto, M.I., Vohralík, M.: Computable a posteriori error estimates in the finite element method based on its local conservativity: improvements using local minimization. ESAIM Proc. 24, 77–96 (2008)

    Article  MATH  Google Scholar 

  12. Cochez-Dhondt, S., Nicaise, S.: Equilibrated error estimators for discontinuous Galerkin methods. Numer. Method. Part. Differ. Equ. 24(5), 1236–1252 (2008)

  13. Di Pietro, D.A., Ern, A.: Mathematical Aspects of Discontinuous Galerkin Methods, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 69. Springer, Heidelberg (2012)

    Google Scholar 

  14. Drkošová, J., Greenbaum, A., Rozložník, M., Strakoš, Z.: Numerical stability of GMRES. BIT 35(3), 309–330 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  15. Ern, A., Stephansen, A.F., Vohralík, M.: Guaranteed and robust discontinuous Galerkin a posteriori error estimates for convection–diffusion–reaction problems. J. Comput. Appl. Math. 234(1), 114–130 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  16. Ern, A., Vohralík, M.: Flux reconstruction and a posteriori error estimation for discontinuous Galerkin methods on general nonmatching grids. C. R. Math. Acad. Sci. Paris 347(7–8), 441–444 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  17. Ern, A., Vohralík, M.: Adaptive inexact Newton methods with a posteriori stopping criteria for nonlinear diffusion PDEs. SIAM J. Sci. Comput. 35(4), A1761–A1791 (2013)

    Article  MATH  Google Scholar 

  18. Greenbaum, A., Rozložník, M., Strakoš, Z.: Numerical behaviour of the modified Gram–Schmidt GMRES implementation. BIT 37(3), 706–719 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  19. Jiránek, P., Strakoš, Z., Vohralík, M.: A posteriori error estimates including algebraic error and stopping criteria for iterative solvers. SIAM J. Sci. Comput. 32(3), 1567–1590 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  20. Karakashian, O.A., Pascal, F.: A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems. SIAM J. Numer. Anal. 41(6), 2374–2399 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  21. Kim, K.Y.: A posteriori error analysis for locally conservative mixed methods. Math. Comput. 76(257), 43–66 (2007)

    Article  MATH  Google Scholar 

  22. Liesen, J., Strakoš, Z.: Krylov Subspace Methods: Principles and Analysis. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (2013)

    Google Scholar 

  23. Meidner, D., Rannacher, R., Vihharev, J.: Goal-oriented error control of the iterative solution of finite element equations. J. Numer. Math. 17(2), 143–172 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  24. Nicaise, S.: A posteriori error estimations of some cell-centered finite volume methods. SIAM J. Numer. Anal. 43(4), 1481–1503 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  25. Paige, C.C., Rozložník, M., Strakoš, Z.: Modified Gram–Schmidt (MGS), least squares, and backward stability of MGS–GMRES. SIAM J. Matrix Anal. Appl. 28(1), 264–284 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  26. Payne, L.E., Weinberger, H.F.: An optimal Poincaré inequality for convex domains. Arch. Ration. Mech. Anal. 5, 286–292 (1960)

    Article  MATH  MathSciNet  Google Scholar 

  27. Pencheva, G.V., Vohralík, M., Wheeler, M.F., Wildey, T.: Robust a posteriori error control and adaptivity for multiscale, multinumerics, and mortar coupling. SIAM J. Numer. Anal. 51(1), 526–554 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  28. Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations, Springer Series in Computational Mathematics, vol. 23. Springer, Berlin (1994)

    Google Scholar 

  29. Rannacher, R., Westenberger, A., Wollner, W.: Adaptive finite element solution of eigenvalue problems: balancing of discretization and iteration error. J. Numer. Math. 18(4), 303–327 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  30. Rektorys, K.: Variational Methods in Mathematics, Science and Engineering. D. Reidel Publishing Co., Dordrecht (1977). Translated from the Czech by Michael Basch

  31. Saad, Y., Schultz, M.H.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7(3), 856–869 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  32. Silvester, P.: Symmetric quadrature formulae for simplexes. Math. Comput. 24, 95–100 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  33. Strakoš, Z., Tichý, P.: Error estimation in preconditioned conjugate gradients. BIT 45(4), 789–817 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  34. Verfürth, R.: A review of a posteriori error estimation and adaptive mesh–refinement techniques. Teubner-Wiley, Stuttgart (1996)

    MATH  Google Scholar 

  35. Wheeler, M.F., Yotov, I.: A posteriori error estimates for the mortar mixed finite element method. SIAM J. Numer. Anal. 43(3), 1021–1042 (2005)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Faculty of Mathematics and Physics, Charles University in Prague, Sokolovská  83, 186 75, Praha 8, Czech Republic

    Vít Dolejší & Ivana Šebestová

  2. INRIA Paris-Rocquencourt, B.P. 105, 78153, Le Chesnay, France

    Martin Vohralík

Authors
  1. Vít Dolejší
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  2. Ivana Šebestová
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  3. Martin Vohralík
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Correspondence to Ivana Šebestová.

Additional information

This work was supported by the ERC-CZ Project MORE “MOdelling REvisited + MOdel REduction” LL1202. The research of V. Dolejší was supported by the Grant No. 13-00522S of the Czech Science Foundation and the membership in the Nečas Center for Mathematical Modeling (http://ncmm.karlin.mff.cuni.cz). The research of I. Šebestová was supported by the Project MathMAC—University center for mathematical modeling, applied analysis and computational mathematics of the Charles University in Prague.

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Dolejší, V., Šebestová, I. & Vohralík, M. Algebraic and Discretization Error Estimation by Equilibrated Fluxes for Discontinuous Galerkin Methods on Nonmatching Grids. J Sci Comput 64, 1–34 (2015). https://doi.org/10.1007/s10915-014-9921-2

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  • DOI: https://doi.org/10.1007/s10915-014-9921-2

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