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Algebraic and discretization error estimation by equilibrated fluxes for discontinuous Galerkin methods on nonmatching grids. (English) Zbl 1326.65147
The authors consider the discontinuous Galerkin method applied to the Poisson equation. Some a posteriori error estimates are derived. The results hold in a setting in which a variable polynomial degree and simplicial meshes with hanging nodes are allowed. An approach allowing for simple (nonconforming) flux reconstructions is proposed. An algebraic error flux reconstruction is introduced. Guaranteed reliability and local efficiency are proved and an adaptive strategy combining both adaptive mesh refinement and adaptive stopping criteria is proposed. Numerical experiments illustrate a tight control of the overall error, good prediction of the distribution of the discretization and algebraic error components, and efficiency of the adaptive strategy.

MSC:
65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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References:
[1] Ainsworth, M, Robust a posteriori error estimation for nonconforming finite element approximation, SIAM J. Numer. Anal., 42, 2320-2341, (2005) · Zbl 1085.65102
[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, 4112-4141, (2010) · Zbl 1208.65155
[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, 254-280, (2011) · Zbl 1208.65156
[4] Arioli, M, A stopping criterion for the conjugate gradient algorithm in a finite element method framework, Numer. Math., 97, 1-24, (2004) · Zbl 1048.65029
[5] Arioli, M; Georgoulis, EH; Loghin, D, Stopping criteria for adaptive finite element solvers, SIAM J. Sci. Comput., 35, a1537-a1559, (2013) · Zbl 1276.65077
[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, 102-129, (2013) · Zbl 1279.65130
[7] Arioli, M., Loghin, D.: Stopping criteria for mixed finite element problems. Electron. Trans. Numer. Anal. 29, 178-192 (2007/08) · Zbl 1392.65042
[8] Arioli, M; Loghin, D; Wathen, AJ, Stopping criteria for iterations in finite element methods, Numer. Math., 99, 381-410, (2005) · Zbl 1069.65124
[9] Becker, R; Johnson, C; Rannacher, R, Adaptive error control for multigrid finite element methods, Computing, 55, 271-288, (1995) · Zbl 0848.65074
[10] Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics, vol. 15. Springer, New York (1991) · Zbl 0788.73002
[11] Cheddadi, I; Fučík, R; Prieto, MI; 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) · Zbl 1156.65318
[12] Cochez-Dhondt, S., Nicaise, S.: Equilibrated error estimators for discontinuous Galerkin methods. Numer. Method. Part. Differ. Equ. 24(5), 1236-1252 (2008) · Zbl 1160.65056
[13] Di Pietro, D.A., Ern, A.: Mathematical Aspects of Discontinuous Galerkin Methods, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 69. Springer, Heidelberg (2012) · Zbl 1231.65209
[14] Drkošová, J; Greenbaum, A; Rozložník, M; Strakoš, Z, Numerical stability of GMRES, BIT, 35, 309-330, (1995) · Zbl 0837.65040
[15] Ern, A; Stephansen, AF; Vohralík, M, Guaranteed and robust discontinuous Galerkin a posteriori error estimates for convection-diffusion-reaction problems, J. Comput. Appl. Math., 234, 114-130, (2010) · Zbl 1190.65165
[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, 441-444, (2009) · Zbl 1161.65085
[17] Ern, A; Vohralík, M, Adaptive inexact Newton methods with a posteriori stopping criteria for nonlinear diffusion pdes, SIAM J. Sci. Comput., 35, a1761-a1791, (2013) · Zbl 1362.65056
[18] Greenbaum, A; Rozložník, M; Strakoš, Z, Numerical behaviour of the modified Gram-Schmidt GMRES implementation, BIT, 37, 706-719, (1997) · Zbl 0891.65031
[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, 1567-1590, (2010) · Zbl 1215.65168
[20] Karakashian, OA; Pascal, F, A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems, SIAM J. Numer. Anal., 41, 2374-2399, (2003) · Zbl 1058.65120
[21] Kim, KY, A posteriori error analysis for locally conservative mixed methods, Math. Comput., 76, 43-66, (2007) · Zbl 1121.65112
[22] Liesen, J., Strakoš, Z.: Krylov Subspace Methods: Principles and Analysis. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (2013) · Zbl 1263.65034
[23] Meidner, D; Rannacher, R; Vihharev, J, Goal-oriented error control of the iterative solution of finite element equations, J. Numer. Math., 17, 143-172, (2009) · Zbl 1169.65340
[24] Nicaise, S, A posteriori error estimations of some cell-centered finite volume methods, SIAM J. Numer. Anal., 43, 1481-1503, (2005) · Zbl 1103.65110
[25] Paige, CC; Rozložník, M; Strakoš, Z, Modified Gram-Schmidt (MGS), least squares, and backward stability of MGS-GMRES, SIAM J. Matrix Anal. Appl., 28, 264-284, (2006) · Zbl 1113.65028
[26] Payne, LE; Weinberger, HF, An optimal Poincaré inequality for convex domains, Arch. Ration. Mech. Anal., 5, 286-292, (1960) · Zbl 0099.08402
[27] Pencheva, GV; Vohralík, M; Wheeler, MF; Wildey, T, Robust a posteriori error control and adaptivity for multiscale, multinumerics, and mortar coupling, SIAM J. Numer. Anal., 51, 526-554, (2013) · Zbl 1267.65165
[28] Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations, Springer Series in Computational Mathematics, vol. 23. Springer, Berlin (1994) · Zbl 0803.65088
[29] Rannacher, R; Westenberger, A; Wollner, W, Adaptive finite element solution of eigenvalue problems: balancing of discretization and iteration error, J. Numer. Math., 18, 303-327, (2010) · Zbl 1222.65123
[30] Rektorys, K.: Variational Methods in Mathematics, Science and Engineering. D. Reidel Publishing Co., Dordrecht (1977). Translated from the Czech by Michael Basch · Zbl 1048.65029
[31] Saad, Y; Schultz, MH, GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 7, 856-869, (1986) · Zbl 0599.65018
[32] Silvester, P, Symmetric quadrature formulae for simplexes, Math. Comput., 24, 95-100, (1970) · Zbl 0198.21103
[33] Strakoš, Z; Tichý, P, Error estimation in preconditioned conjugate gradients, BIT, 45, 789-817, (2005) · Zbl 1095.65029
[34] Verfürth, R.: A review of a posteriori error estimation and adaptive mesh-refinement techniques. Teubner-Wiley, Stuttgart (1996) · Zbl 0853.65108
[35] Wheeler, MF; Yotov, I, A posteriori error estimates for the mortar mixed finite element method, SIAM J. Numer. Anal., 43, 1021-1042, (2005) · Zbl 1094.65114
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