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Minimization of functional majorant in a posteriori error analysis based on \(H\)(div) multigrid-preconditioned CG method. (English) Zbl 1200.65095

Summary: We consider a Poisson boundary value problem and its functional a posteriori error estimate derived by S. Repin in 1999 [Probl. Mat. Anal. 17, 199–226 (1997; Zbl 0941.65059)]. The estimate majorizes the \(H^{1}\) seminorm of the error of the discrete solution computed by FEM method and contains a free ux variable from the \(H\)(div) space. In order to keep the estimate sharp, a procedure for the minimization of the majorant term with respect to the ux variable is introduced, computing the free ux variable from a global linear system of equations. Since the linear system is symmetric and positive definite, few iterations of a conjugate gradient method with a geometrical multigrid preconditioner are applied. Numerical techniques are demonstrated on one benchmark example with a smooth solution on a unit square domain including the computation of the approximate value of the constant in Friedrichs’ inequality.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs

Citations:

Zbl 0941.65059

Software:

mfem
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Full Text: DOI EuDML

References:

[1] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, Studies in Mathematics and Its Applications, North-Holland, Amsterdam, The Netherlands, 1978. · Zbl 0383.65058
[2] M. Ainsworth and J. T. Oden, A Posteriori Error Estimation in Finite Element Analysis, Pure and Applied Mathematics, John Wiley & Sons, New York, NY, USA, 2000. · Zbl 1049.65135 · doi:10.1137/S1064827599356274
[3] I. Babu\vska and T. Strouboulis, The Finite Element Method and Its Reliability, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, NY, USA, 2001.
[4] W. Bangerth and R. Rannacher, Adaptive Finite Element Methods for Differential Equations, Lectures in Mathematics ETH Zürich, Birkhäuser, Basel, Switzerland, 2003. · Zbl 1020.65058
[5] R. Verfürth, A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, John Wiley & Sons, Teubner, New-York, NY, USA, 1996. · Zbl 0853.65108
[6] I. Babu\vska and W. C. Rheinboldt, “A posteriori error estimates for the finite element method,” International Journal for Numerical Methods in Engineering, vol. 12, pp. 1597-1615, 1978. · Zbl 0396.65068 · doi:10.1002/nme.1620121010
[7] I. Babu\vska and W. C. Rheinboldt, “Error estimates for adaptive finite element computations,” SIAM Journal on Numerical Analysis, vol. 15, no. 4, pp. 736-754, 1978. · Zbl 0398.65069 · doi:10.1137/0715049
[8] C. Carstensen and S. Bartels, “Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. I. Low order conforming, nonconforming, and mixed FEM,” Mathematics of Computation, vol. 71, no. 239, pp. 945-969, 2002. · Zbl 0997.65126 · doi:10.1090/S0025-5718-02-01402-3
[9] S. I. Repin, “A posteriori error estimates for approximate solutions to variational problems with strongly convex functionals,” Journal of Mathematical Sciences, vol. 97, no. 4, pp. 4311-4328, 1999, translated from Problemy Matematicheskogo Analiza, no. 17, pp. 227-237, 1997. · Zbl 0941.65059 · doi:10.1007/BF02365047
[10] S. I. Repin and L. S. Xanthis, “A posteriori error estimation for nonlinear variational problems,” Comptes Rendus de l’Académie des Sciences, vol. 324, no. 10, pp. 1169-1174, 1997. · Zbl 0904.65064 · doi:10.1016/S0764-4442(97)87906-2
[11] S. I. Repin, “A posteriori error estimation for variational problems with uniformly convex functionals,” Mathematics of Computation, vol. 69, no. 230, pp. 481-500, 2000. · Zbl 0949.65070 · doi:10.1090/S0025-5718-99-01190-4
[12] S. I. Repin, “Two-sided estimates of deviation from exact solutions of uniformly elliptic equations,” in Proceedings of the St. Petersburg Mathematical Society, Vol. IX, vol. 209 of American Mathematical Society Translations: Series 2, pp. 143-171, American Mathematical Society, Providence, RI, USA. · Zbl 1039.65076
[13] M. Bildhauer, M. Fuchs, and S. Repin, “A posteriori error estimates for stationary slow flows of power-law fluids,” Journal of Non-Newtonian Fluid Mechanics, vol. 142, no. 1-3, pp. 112-122, 2007. · Zbl 1109.76007 · doi:10.1016/j.jnnfm.2006.06.001
[14] M. Bildhauer, M. Fuchs, and S. Repin, “A functional type a posteriori error analysis for the Ramberg-Osgood model,” Zeitschrift für Angewandte Mathematik und Mechanik, vol. 87, no. 11-12, pp. 860-876, 2007. · Zbl 1128.74006 · doi:10.1002/zamm.200710350
[15] M. Fuchs and S. Repin, “A posteriori error estimates of functional type for variational problems related to generalized Newtonian fluids,” Mathematical Methods in the Applied Sciences, vol. 29, no. 18, pp. 2225-2244, 2006. · Zbl 1105.76049 · doi:10.1002/mma.773
[16] M. E. Frolov, P. Neittaanmäki, and S. I. Repin, “Guaranteed functional error estimates for the Reissner-Mindlin plate problem,” Journal of Mathematical Sciences, vol. 132, no. 4, pp. 553-561, 2006. · Zbl 1125.74028 · doi:10.1007/s10958-005-0515-2
[17] S. Repin and J. Valdman, “Functional a posteriori error estimates for problems with nonlinear boundary conditions,” Journal of Numerical Mathematics, vol. 16, no. 1, pp. 51-81, 2008. · Zbl 1146.65054 · doi:10.1515/JNUM.2008.003
[18] S. Repin, “A posteriori error estimation methods for partial differential equations,” in Lectures on Advanced Computational Methods in Mechanics, J. Kraus and U. Langer, Eds., vol. 1 of Radon Series on Computational and Applied Mathematics, pp. 161-226, Walter de Gruyter, Berlin, Germany, 2007. · Zbl 1132.65101
[19] D. Braess and J. Schöberl, “Equilibrated residual error estimator for edge elements,” Mathematics of Computation, vol. 77, no. 262, pp. 651-672, 2008. · Zbl 1135.65041 · doi:10.1090/S0025-5718-07-02080-7
[20] T. Vejchodský, “Guaranteed and locally computable a posteriori error estimate,” IMA Journal of Numerical Analysis, vol. 26, no. 3, pp. 525-540, 2006. · Zbl 1096.65112 · doi:10.1093/imanum/dri043
[21] M. Vohralík, “A posteriori error estimation in the conforming finite element method based on its local conservativity and using local minimization,” Comptes Rendus Mathématique, vol. 346, no. 11-12, pp. 687-690, 2008. · Zbl 1142.65086 · doi:10.1016/j.crma.2008.03.006
[22] R. Luce and B. I. Wohlmuth, “A local a posteriori error estimator based on equilibrated fluxes,” SIAM Journal on Numerical Analysis, vol. 42, no. 4, pp. 1394-1414, 2004. · Zbl 1078.65097 · doi:10.1137/S0036142903433790
[23] S. Repin, S. Sauter, and A. Smolianski, “Two-sided a posteriori error estimates for mixed formulations of elliptic problems,” SIAM Journal on Numerical Analysis, vol. 45, no. 3, pp. 928-945, 2007. · Zbl 1185.35048 · doi:10.1137/050641533
[24] R. Lazarov, S. Repin , and S. Tomar, “Functional a posteriori error estimates for discontinuous Galerkin approximations of elliptic problems,” Numerical Methods for Partial Differential Equations, vol. 25, no. 4, pp. 952-971, 2008. · Zbl 1167.65451 · doi:10.1002/num.20386
[25] S. Repin, S. Sauter, and A. Smolianski, “A posteriori error estimation for the Dirichlet problem with account of the error in the approximation of boundary conditions,” Computing, vol. 70, no. 3, pp. 205-233, 2003. · Zbl 1128.35319
[26] S. Repin and J. Valdman, “Functional a posteriori error estimates for incremental models in elasto-plasticity,” Central European Journal of Mathematics, vol. 7, no. 3, pp. 506-519, 2009. · Zbl 1269.74202
[27] J. R. Kuttler and V. G. Sigillito, “Eigenvalues of the Laplacian in two dimensions,” SIAM Review, vol. 26, no. 2, pp. 163-193, 1984. · Zbl 0574.65116 · doi:10.1137/1026033
[28] P.-A. Raviart and J. M. Thomas, “A mixed finite element method for 2nd order elliptic problems,” in Mathematical Aspects of Finite Element Methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975), vol. 606 of Lecture Notes in Mathematics, pp. 292-315, Springer, Berlin, Germany, 1977. · Zbl 0362.65089
[29] C. Bahriawati and C. Carstensen, “Three MATLAB implementations of the lowest-order Raviart-Thomas MFEM with a posteriori error control,” Computational Methods in Applied Mathematics, vol. 5, no. 4, pp. 333-361, 2005. · Zbl 1086.65107 · doi:10.2478/cmam-2005-0016
[30] Z. Strako\vs and P. Tichý, “Error estimation in preconditioned conjugate gradients,” BIT Numerical Mathematics, vol. 45, no. 4, pp. 789-817, 2005. · Zbl 1095.65029 · doi:10.1007/s10543-005-0032-1
[31] W. Hackbusch, Multi-Grid Methods and Applications, Springer, Berlin, Germany, 1985. · Zbl 0595.65106
[32] D. N. Arnold, R. S. Falk, and R. Winther, “Preconditioning in H(div) and applications,” Mathematics of Computation, vol. 66, no. 219, pp. 957-984, 1997. · Zbl 0870.65112 · doi:10.1090/S0025-5718-97-00826-0
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