On a posteriori error estimates for space--time discontinuous Galerkin method
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Abstract
We deal with nonlinear nonstationary convection--diffusion problem. We discretize this problem by discontinuous Galerkin method in space and in time and, assuming the error is measured as a mesh dependent dual norm of residual, we present a posteriori estimate to this error measure. This a posteriori error estimate is cheap, robust with respect to degeneration to hyperbolic problem and fully computable. Moreover, we present a local asymptotic efficiency of this estimate.
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Dolejší, V., Roskovec, F., & Vlasák, M.
(2016).
On a posteriori error estimates for space--time discontinuous Galerkin method.
Proceedings Of The Conference Algoritmy, , 125-134.
Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/algoritmy/article/view/401/318
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References
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[15] M. Vlasák, V. Dolejší and J. Hájek. A priori error estimates of an extrapolated spacetime discontinuous Galerkin method for nonlinear convection-diffusion problems. Numer. Methods Partial Differ. Equations, 27(6):1456–1482, 2011.
[16] R. Verfurth. Robust a posteriori error estimates for nonstationary convection-diffusion equations. SIAM J. Numer. Anal., 43(4):1793–1802, 2005.
[2] 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(5):1749–1779, 2002.
[3] B. Cockburn, G. E. Karniadakis. and C.-W. Shu. Discontinuous Galerkin methods. In Lecture Notes in Computational Science and Engineering 11. Springer, Berlin, 2000.
[4] V. Dolejší, A. Ern, and M. Vohralík. A framework for robust a posteriori error control in unsteady nonlinear advection-diffusion problems. SIAM J. Numer. Anal., 51(2):773–793, 2013.
[5] V. Dolejší, M. Feistauer. and V. Sobotíková. Analysis of the Discontinuous Galerkin Method for Nonlinear Convection Diffusion Problems. Comput. Methods Appl. Mech. Eng., 194:2709–2733, 2005.
[6] A. Ern, S. Nicaise and M. Vohralík. An accurate H(÷) flux reconstruction for discontinuous Galerkin approximations of elliptic problems. C. R. Math. Acad. Sci. Paris, 345(12):709– 712, 2007.
[7] A. Ern, and M. Vohralík. A posteriori error estimation based on potential and flux reconstruction for the heat equation. SIAM J. Numer. Anal., 48(1):198–223, 2010.
[8] A. Ern, and 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(2):1058–1081, 2015.
[9] E. Hairer, S. P. Norsett, and G. Wanner. Solving ordinary differential equations I, Nonstiff problems. Number 8 in Springer Series in Computational Mathematics. Springer Verlag, 2000.
[10] E. Hairer and G. Wanner. Solving ordinary differential equations II, Stiff and differentialalgebraic problems. Springer Verlag, 2002.
[11] P. Houston, J. Robson and E. Suli. Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal., 39(5):1749–1779, 2002.
[12] P. Jiránek, Z. Strakoš and M. Vohralík. A posteriori error estimates including algebraic error and stopping criteria for iterative solvers. SIAM J. Sci. Comput., 32(3):1567–1590, 2010.
[13] L. E. Payne and H. F. Weinberger. An optimal Poincar ́ e inequality for convex domains Arch. Ration. Mech. Anal, 5:286–292, 1960.
[14] V. Thomee. Galerkin finite element methods for parabolic problems. 2nd revised and expanded ed. Springer, Berlin, 2006.
[15] M. Vlasák, V. Dolejší and J. Hájek. A priori error estimates of an extrapolated spacetime discontinuous Galerkin method for nonlinear convection-diffusion problems. Numer. Methods Partial Differ. Equations, 27(6):1456–1482, 2011.
[16] R. Verfurth. Robust a posteriori error estimates for nonstationary convection-diffusion equations. SIAM J. Numer. Anal., 43(4):1793–1802, 2005.