On robustness of flux reconstructions - discontinuous Galerkin method
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Abstract
We deal with the numerical solution of the Poisson equation. The equation is discretized with the aid of the incomplete interior penalty discontinuous Galerkin method. Guaranteed a posteriori upper bound based on the flux reconstruction can be derived. The main aim of this paper is to show that the robustness of a certain simple reconstruction depends at most on p^{1/2} in one dimension, where p is the discretization polynomial degree. The theoretical results are verified by numerical experiments.
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How to Cite
Vlasák, M.
(2020).
On robustness of flux reconstructions - discontinuous Galerkin method.
Proceedings Of The Conference Algoritmy, , 240 - 248.
Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/algoritmy/article/view/1580/832
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References
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[17] R. Verfürth: A posteriori error estimation techniques for finite element methods. Numer. Math. Sci. Comput., Oxford University Press, Oxford (2013).
[2] M. Ainsworth, J. T. Oden: A posteriori error estimation in finite element analysis. Pure Appl. Math., Wiley and Sons, New York (2000).
[3] M. Ainsworth, B. Senior: An adaptive refinement strategy for hp-finite-element computations. Appl. Numer. Math. 26 (1998), pp. 165–178.
[4] D. N. Arnold, F. Brezzi, B. Cockburn, and L. D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39 (2002), pp. 1749–1779.
[5] I. Babuška, T. Strouboulis: The finite element method and its reliability. Numer. Math. Sci. Comput., Oxford University Press, New York (2001).
[6] D. Boffi, F. Brezzi, M. Fortin: Mixed finite element methods and applications. Springer Series in Computational Mathematics 44, Berlin: Springer (2013).
[7] D. Braess, V. Pillwine, J. Schöberl: Equilibrated residual error estimates are p-robust. Comput. Methods Appl. Mech. Engrg., 198 (2009), pp. 1189–1197.
[8] V. Dolejšı́ and M. Feistauer, Discontinuous Galerkin method. Analysis and applications to compressible flow., Cham: Springer, 2015.
[9] K. Eriksson, D. Estep, P. Hansbo, C. Jonson: Computational differential equations. Cambridge University Press, Cambridge (1996).
[10] A. Ern, A. F. Stephansen, M. Vohralı́k: Guaranteed and robust discontinuous Galerkin a posteriori error estimates for convection-diffusion-reaction problems. J. Comput. Appl. Math. 234(1) (2010), pp. 114–130.
[11] A. Ern, M. Vohralı́k: Polynomial-degree-robust a posteriori estimates in a unified setting for conforming, nonconforming, discontinuous Galerkin, and mixed discretizations. SIAM J. Numer. Anal. 53(2) (2015), pp. 1058–1081.
[12] J. M. Melenk, B. Wohlmuth: On residual-based a posteriori error estimation in hp–FEM. Advances Comput. Math. 150 (2001), pp. 311–331.
[13] L. Payne and H. Weinberger, An optimal Poincaré inequality for convex domains., Arch. Ration. Mech. Anal., 5 (1960), pp. 286–292.
[14] W. Prager, J. L. Synge: Approximations in elasticity based on the concept of function space, Quart. Appl. Math. 5 (1947), pp. 241–269.
[15] S. I. Repin: A posteriori estimates for partial differential equations. Radon Ser. Comput. Appl. Math., Walter de Gruiter, Berlin (2008).
[16] S. K. Tomar, S. I. Repin: Efficient computable error bounds for discontinuous Galerkin approximations of elliptic problems. J. Comput. Appl. Math. 226(2) (2009), pp. 952–971.
[17] R. Verfürth: A posteriori error estimation techniques for finite element methods. Numer. Math. Sci. Comput., Oxford University Press, Oxford (2013).