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On polynomial robustness of flux reconstructions

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Abstract

We deal with the numerical solution of elliptic not necessarily self-adjoint problems. We derive a posteriori upper bound based on the flux reconstruction that can be directly and cheaply evaluated from the original fluxes and we show for one-dimensional problems that local efficiency of the resulting a posteriori error estimators depends on p1/2 only, where p is the discretization polynomial degree. The theoretical results are verified by numerical experiments.

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References

  1. M. Ainsworth, J.T. Oden: A procedure for a posteriori error estimation for h-p finite element methods. Comput. Methods Appl. Mech. Eng. 101 (1992), 73–96.

    Article  Google Scholar 

  2. M. Ainsworth, J.T. Oden: A Posteriori Error Estimation in Finite Element Analysis. Pure and Applied Mathematics. Wiley-Interscience, New York, 2000.

    Book  Google Scholar 

  3. M. Ainsworth, B. Senior: An adaptive refinement strategy for hp-finite element computations. Appl. Numer. Math. 26 (1998), 165–178.

    Article  MathSciNet  Google Scholar 

  4. I. Babuška, T. Strouboulis: The Finite Element Method and Its Reliability. Numerical Mathematics and Scientific Computation. Clarendon Press, Oxford, 2001.

    MATH  Google Scholar 

  5. I. Babuška, M. Suri: The h-p version of the finite element method with quasi-uniform meshes. RAIRO, Modélisation Math. Anal. Numér. 21 (1987), 199–238.

    Article  MathSciNet  Google Scholar 

  6. D. Boffi, F. Brezzi, M. Fortin: Mixed Finite Element Methods and Applications. Springer Series in Computational Mathematics 44, Springer, Berlin, 2013.

    Book  Google Scholar 

  7. D. Braess, V. Pillwein, J. Schöberl: Equilibrated residual error estimates are p-robust. Comput. Methods Appl. Mech. Eng. 198 (2009), 1189–1197.

    Article  MathSciNet  Google Scholar 

  8. S. Cochez-Dhondt, S. Nicaise, S. Repin: A posteriori error estimates for finite volume approximations. Math. Model. Nat. Phenom. 4 (2009), 106–122.

    Article  MathSciNet  Google Scholar 

  9. K. Eriksson, D. Estep, P. Hansbo, C. Johnson: Computational Differential Equations. Cambridge University Press, Cambridge, 1996.

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  11. A. Ern, M. Vohralík: Polynomial-degree-robust a posteriori estimates in a unified setting for conforming, nonconforming, discontinuous Galerkin, and mixed discretizations. SIAM J. Numer. Anal. 53 (2015), 1058–1081.

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  13. O. Mali, P. Neittaanmäki, S. I. Repin: Accuracy Verification Methods. Theory and Algorithms. Computational Methods in Applied Sciences 32, Springer, Dordrecht, 2014.

    MATH  Google Scholar 

  14. J.M. Melenk, B. Wohlmuth: On residual-based a posteriori error estimation in hp-FEM. Adv. Comput. Math. 150 (2001), 311–331.

    Article  MathSciNet  Google Scholar 

  15. W. Prager, J. L. Synge: Approximations in elasticity based on the concept of function space. Q. Appl. Math. 5 (1947), 241–269.

    Article  MathSciNet  Google Scholar 

  16. S. I. Repin: A posteriori error estimation for variational problems with uniformly convex functionals. Math. Comput. 69 (2000), 481–500.

    Article  MathSciNet  Google Scholar 

  17. S. I. Repin: A Posteriori Estimates for Partial Differential Equations. Radon Series on Computational and Applied Mathematics 4, de Gruyter, Berlin, 2008.

    Google Scholar 

  18. H.-G. Roos, M. Stynes, L. Tobiska: Robust Numerical Methods for Singularly Perturbed Differential Equations. Convection-Diffusion-Reaction and Flow Problems. Springer Series in Computational Mathematics 24, Springer, Berlin, 2008.

    MATH  Google Scholar 

  19. G. Szegő: Orthogonal Polynomials. Colloquium Publication 23, American Mathematical Society, New York, 1939.

    Google Scholar 

  20. S.K. Tomar, S. I. Repin: Efficient computable error bounds for discontinuous Galerkin approximations of elliptic problems. J. Comput. Appl. Math. 226 (2009), 358–369.

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

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Authors and Affiliations

  1. Faculty of Civil Engineering, Czech Technical University, Thákurova 7, 166 29, Praha 6, Czech Republic

    Miloslav Vlasák

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  1. Miloslav Vlasák
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Correspondence to Miloslav Vlasák.

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The work was supported from European Regional Development Fund-Project “Center for Advanced Applied Science” (No. CZ.02.1.01/0.0/0.0/16 019/0000778).

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Vlasák, M. On polynomial robustness of flux reconstructions. Appl Math 65, 153–172 (2020). https://doi.org/10.21136/AM.2020.0152-19

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  • DOI: https://doi.org/10.21136/AM.2020.0152-19

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