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Keywords:
discontinuous Galerkin method; Helmholtz decomposition; averaging interpolation operator; Euler backward scheme; residual-based a posteriori error estimate; local cut-off function
Summary:
We deal with the numerical solution of the nonstationary heat conduction equation with mixed Dirichlet/Neumann boundary conditions. The backward Euler method is employed for the time discretization and the interior penalty discontinuous Galerkin method for the space discretization. Assuming shape regularity, local quasi-uniformity, and transition conditions, we derive both a posteriori upper and lower error bounds. The analysis is based on the Helmholtz decomposition, the averaging interpolation operator, and on the use of cut-off functions. Numerical experiments are presented.
References:
[1] Ainsworth, M.: A posteriori error estimation for discontinuous Galerkin finite element approximation. SIAM J. Numer. Anal. 45 (2007), 1777-1798. DOI 10.1137/060665993 | MR 2338409 | Zbl 1151.65083
[2] Arnold, D. N., Brezzi, F., Cockburn, B., Marini, L. D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (2002), 1749-1779. DOI 10.1137/S0036142901384162 | MR 1885715 | Zbl 1008.65080
[3] Becker, R., Hansbo, P., Larson, M. G.: Energy norm a posteriori error estimation for discontinuous Galerkin methods. Comput. Methods Appl. Mech. Eng. 192 (2003), 723-733. DOI 10.1016/S0045-7825(02)00593-5 | MR 1952357 | Zbl 1042.65083
[4] Dari, E., Duran, R., Padra, C., Vampa, V.: A posteriori error estimators for nonconforming finite element methods. RAIRO, Modélisation Math. Anal. Numér. 30 (1996), 385-400. MR 1399496 | Zbl 0853.65110
[5] Ern, A., Vohralík, M.: A posteriori error estimation based on potential and flux reconstruction for the heat equation. SIAM J. Numer. Anal. 48 (2010), 198-223. DOI 10.1137/090759008 | MR 2608366 | Zbl 1215.65152
[6] Feistauer, M., Dolejší, V., Kučera, V., Sobotíková, V.: $L^{\infty}(L^2)$-error estimates for the DGFEM applied to convection-diffusion problems on nonconforming meshes. J. Numer. Math. 17 (2009), 45-65. DOI 10.1515/JNUM.2009.004 | MR 2541520 | Zbl 1171.65064
[7] Girault, V., Raviart, P.-A.: Finite Element Methods for Navier-Stokes Equations. Theory and algorithms. (Extended version of the 1979 publ.). Springer Series in Computational Mathematics 5 Springer, Berlin (1986). MR 0851383 | Zbl 0585.65077
[8] Karakashian, O. A., Pascal, F.: A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems. SIAM J. Numer. Anal. 41 (2003), 2374-2399. DOI 10.1137/S0036142902405217 | MR 2034620 | Zbl 1058.65120
[9] Karakashian, O. A., Pascal, F.: Adaptive discontinuous Galerkin approximations of second-order elliptic problems. European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS 2004 P. Neittaanmäki et al. University of Jyväskylä Jyväskylä (2004).
[10] Karakashian, O. A., Pascal, F.: Convergence of adaptive discontinuous Galerkin approximations of second-order elliptic problems. SIAM J. Numer. Anal. 45 (2007), 641-665. DOI 10.1137/05063979X | MR 2300291 | Zbl 1140.65083
[11] Nečas, J.: Direct methods in the theory of elliptic equations. Academia Prague (1967); Masson et Cie, Paris, 1967, French. Zbl 1225.35003
[12] Nicaise, S., Soualem, N.: A posteriori error estimates for a nonconforming finite element discretization of the heat equation. ESAIM, Math. Model. Numer. Anal. 39 (2005), 319-348. DOI 10.1051/m2an:2005009 | MR 2143951 | Zbl 1078.65079
[13] Repin, S.: Estimates of deviations from exact solutions of initial-boundary value problem for the heat equation. Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 13 (2002), 121-133. MR 1949485 | Zbl 1221.65244
[14] Šebestová, I.: A posteriori error estimates of the discontinuous Galerkin method for convection-diffusion equations. Master Thesis. Charles University in Prague Prague (2009).
[15] Šebestová, I., Dolejší, V.: A posteriori error estimates of the discontinuous Galerkin method for the heat conduction equation. Acta Univ. Carol., Math. Phys. 53 (2012), 77-94. MR 3099403 | Zbl 1280.65098
[16] Verfürth, R.: A posteriori error estimates for finite element discretizations of the heat equation. Calcolo 40 (2003), 195-212. DOI 10.1007/s10092-003-0073-2 | MR 2025602 | Zbl 1168.65418
[17] Verfürth, R.: A review of a posteriori error estimation and adaptive mesh-refinement techniques. Wiley-Teubner Series Advances in Numerical Mathematics John Wiley & Sons, Chichester (1996). Zbl 0853.65108
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