Implicit a posteriori error estimation using patch recovery techniques

• Autorzy: Tamás Horváth , Ferenc Izsák
• Języki publikacji: EN

Abstrakty

EN

We develop implicit a posteriori error estimators for elliptic boundary value problems. Local problems are formulated for the error and the corresponding Neumann type boundary conditions are approximated using a new family of gradient averaging procedures. Convergence properties of the implicit error estimator are discussed independently of residual type error estimators, and this gives a freedom in the choice of boundary conditions. General assumptions are elaborated for the gradient averaging which define a family of implicit a posteriori error estimators. We will demonstrate the performance and the favor of the method through numerical experiments.

Słowa kluczowe

EN

Implicit a posteriori error estimation; Finite element method; Gradient recovery

Kategorie tematyczne

• 65Y20: Complexity and performance of numerical algorithms
• 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
• 65N99: None of the above, but in this section

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2012
• Tom: 10
• Numer: 1
• Strony: 55-72

Daty

wydano

2012-02-01

online

2011-12-09

Twórcy

autor

Tamás Horváth

autor

Ferenc Izsák

Bibliografia

• [1] Adams R.A., Fournier J.J.F., Sobolev Spaces, 2nd ed., Pure Appl. Math. (Amst.), 140, Academic Press, Amsterdam, 2003
• [2] Ainsworth M., The influence and selection of subspaces for a posteriori error estimators, Numer. Math., 1996, 73(4), 399–418 http://dx.doi.org/10.1007/s002110050198
• [3] Ainsworth M., Craig A., A posteriori error estimators in the finite element method, Numer. Math., 1992, 60(4), 429–463
• [4] Ainsworth M., Oden J.T., A Posteriori Error Estimation in Finite Element Analysis, Pure Appl. Math. (N.Y.), John Wiley & Sons, New York, 2000 http://dx.doi.org/10.1002/9781118032824
• [5] Ainsworth M., Rankin R., Fully computable bounds for the error in nonconforming finite element approximations of arbitrary order on triangular elements, SIAM J. Numer. Anal., 2008, 46(6), 3207–3232 http://dx.doi.org/10.1137/07070838X
• [6] Babuška I., Rheinboldt W.C., Error estimates for adaptive finite element computations, SIAM J. Numer. Anal., 1978, 15(4), 736–754 http://dx.doi.org/10.1137/0715049
• [7] Bank R.E., Weiser A., Some a posteriori error estimators for elliptic partial differential equations, Math. Comp., 1985, 44(170), 283–301 http://dx.doi.org/10.1090/S0025-5718-1985-0777265-X
• [8] Bank R.E., Xu J., Asymptotically exact a posteriori error estimators I. Grids with superconvergence, SIAM J. Numer. Anal., 2003, 41(6), 2294–2312 http://dx.doi.org/10.1137/S003614290139874X
• [9] Bank R.E., Xu J., Asymptotically exact a posteriori error estimators II. General unstructured grids, SIAM J. Numer. Anal., 2003, 41(6), 2313–2332 http://dx.doi.org/10.1137/S0036142901398751
• [10] Brenner S.C., Scott L.R., The Mathematical Theory of Finite Element Methods, 2nd ed., Texts Appl. Math., 15, Springer, New York, 2002
• [11] Carstensen C., Some remarks on the history and future of averaging techniques in a posteriori finite element error analysis, ZAMM Z. Angew. Math. Mech., 2004, 84(1), 3–21 http://dx.doi.org/10.1002/zamm.200410101
• [12] Carstensen C., A unifying theory of a posteriori finite element error control, Numer. Math., 2005, 100(4), 617–637 http://dx.doi.org/10.1007/s00211-004-0577-y
• [13] Carstensen C., Bartels S., Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids I. Low order conforming, nonconforming, and mixed FEM, Math. Comp., 2002, 71(239), 945–969 http://dx.doi.org/10.1090/S0025-5718-02-01402-3
• [14] Carstensen C., Orlando A., Valdman J., A convergent adaptive finite element method for the primal problem of elastoplasticity, Internat. J. Numer. Methods Engrg., 2006, 67(13), 1851–1887 http://dx.doi.org/10.1002/nme.1686
• [15] Demkowicz L., Computing with hp-Adaptive Finite Elements. I, Chapman Hall/CRC Appl. Math. Nonlinear Sci. Ser., Chapman&Hall/CRC, Boca Raton, 2007
• [16] Ern A., Guermond J.-L., Theory and Practice of Finite Elements, Appl. Math. Sci., 159, Springer, New York, 2004
• [17] Hannukainen A., Korotov S., Křížek M., Nodal O(h 4)-superconvergence in 3D by averaging piecewise linear, bilinear, and trilinear FE approximations, J. Comput. Math., 2010, 28(1), 1–10
• [18] Harutyunyan D., Izsák F., van der Vegt J.J.W., Botchev, M.A., Adaptive finite element techniques for the Maxwell equations using implicit a posteriori error estimates, Comput. Methods Appl. Mech. Engrg., 2008, 197(17–18), 1620–1638 http://dx.doi.org/10.1016/j.cma.2007.12.006
• [19] Hlaváček I., Křížek M., Optimal interior and local error estimates of a recovered gradient of linear elements on nonuniform triangulations, J. Comput. Math., 1996, 14(4), 345–362
• [20] Huang Y., Xu J., Superconvergence of quadratic finite elements on mildly structured grids, Math. Comp., 2008, 77(263), 1253–1268 http://dx.doi.org/10.1090/S0025-5718-08-02051-6
• [21] Izsák F., Harutyunyan D., van der Vegt J.J.W., Implicit a posteriori error estimates for the Maxwell equations, Math. Comp., 2008, 77(263), 1355–1386 http://dx.doi.org/10.1090/S0025-5718-08-02046-2
• [22] Jin H., Prudhomme S., A posteriori error estimation of steady-state finite element solutions of the Navier-Stokes equations by a subdomain residual method, Comput. Methods Appl. Mech. Engrg., 1998, 159(1–2), 19–48 http://dx.doi.org/10.1016/S0045-7825(98)80102-3
• [23] Karátson J., Korotov S., Sharp upper global a posteriori error estimates for nonlinear elliptic variational problems, Appl. Math., 2009, 54(4), 297–336 http://dx.doi.org/10.1007/s10492-009-0020-x
• [24] Korotov S., Neittaanmäki P., Repin S., A posteriori error estimation of goal-oriented quantities by the superconvergence patch recovery, J. Numer. Math., 2003, 11(1), 33–59
• [25] Křížek M., Neittaanmäki P., Superconvergence phenomenon in the finite element method arising from averaging gradients, Numer. Math., 1984, 45(1), 105–116 http://dx.doi.org/10.1007/BF01379664
• [26] Ladevèze P., Leguillon D., Error estimate procedure in the finite element method and applications, SIAM J. Numer. Anal., 1983, 20(3), 485–509 http://dx.doi.org/10.1137/0720033
• [27] McLean W., Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000
• [28] Neittaanmäki P., Repin S., Reliable Methods for Computer Simulation, Stud. Math. Appl., 33, Elsevier Science B.V., Amsterdam, 2004
• [29] Repin S., A Posteriori Estimates for Partial Differential Equations, Radon Ser. Comput. Appl. Math., 4, de Gruyter, Berlin, 2008
• [30] Schöberl J., A posteriori error estimates for Maxwell equations, Math. Comp., 2008, 77(262), 633–649 http://dx.doi.org/10.1090/S0025-5718-07-02030-3
• [31] Schwab Ch., p- and hp-Finite Element Methods, Numer. Math. Sci. Comput., Clarendon Press, Oxford University Press, New York, 1998
• [32] Verfürth R., A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley-Teubner Ser. Adv. Numer. Math., John Wiley & Sons, Teubner, Chichester-Stuttgart, 1996
• [33] Verfürth R., A posteriori error estimators for convection-diffusion equations, Numer. Math., 1998, 80(4), 641–663 http://dx.doi.org/10.1007/s002110050381
• [34] Vohralík M., A posteriori error estimates for lowest-order mixed finite element discretizations of convection-diffusionreaction equations, SIAM J. Numer. Anal., 2007, 45(4), 1570–1599 http://dx.doi.org/10.1137/060653184
• [35] Wahlbin L.B., Superconvergence in Galerkin Finite Element Methods, Lecture Notes in Math., 1605, Springer, Berlin, 1995
• [36] Zienkiewicz O.C., Zhu J.Z., The superconvergent patch recovery and a posteriori error estimates I. The recovery technique, Internat. J. Numer. Methods Engrg., 1992, 33(7), 1331–1364 http://dx.doi.org/10.1002/nme.1620330702
• [37] Zienkiewicz O.C., Zhu J.Z., The superconvergent patch recovery and a posteriori error estimates II. Error estimates and adaptivity, Internat. J. Numer. Methods Engrg., 1992, 33(7), 1365–1382 http://dx.doi.org/10.1002/nme.1620330703

Identyfikatory

DOI

10.2478/s11533-011-0119-7