Robust a posteriori error estimation for the nonconforming Fortin–Soulie finite element approximation

Ainsworth, Mark; Rankin, Richard

doi:10.1090/S0025-5718-08-02116-9

Robust a posteriori error estimation for the nonconforming Fortin–Soulie finite element approximation
HTML articles powered by AMS MathViewer

by Mark Ainsworth and Richard Rankin PDF

Math. Comp. 77 (2008), 1917-1939 Request permission

Abstract:

We obtain a computable a posteriori error bound on the broken energy norm of the error in the Fortin–Soulie finite element approximation of a linear second order elliptic problem with variable permeability. This bound is shown to be efficient in the sense that it also provides a lower bound for the broken energy norm of the error up to a constant and higher order data oscillation terms. The estimator is completely free of unknown constants and provides a guaranteed numerical bound on the error.

References

Mark Ainsworth, Robust a posteriori error estimation for nonconforming finite element approximation, SIAM J. Numer. Anal. 42 (2005), no. 6, 2320–2341. MR 2139395, DOI 10.1137/S0036142903425112
Mark Ainsworth, A posteriori error estimation for discontinuous Galerkin finite element approximation, SIAM J. Numer. Anal. 45 (2007), no. 4, 1777–1798. MR 2338409, DOI 10.1137/060665993
Mark Ainsworth and J. Tinsley Oden, A posteriori error estimation in finite element analysis, Pure and Applied Mathematics (New York), Wiley-Interscience [John Wiley & Sons], New York, 2000. MR 1885308, DOI 10.1002/9781118032824
A. Alonso, Error estimators for a mixed method, Numer. Math. 74 (1996), no. 4, 385–395. MR 1414415, DOI 10.1007/s002110050222
Ivo Babuška and Theofanis Strouboulis, The finite element method and its reliability, Numerical Mathematics and Scientific Computation, The Clarendon Press, Oxford University Press, New York, 2001. MR 1857191
C. Bernardi and R. Verfürth, Adaptive finite element methods for elliptic equations with non-smooth coefficients, Numer. Math. 85 (2000), no. 4, 579–608 (English, with English and French summaries). MR 1771781, DOI 10.1007/PL00005393
D. J. Blacker, A robust a posteriori error estimate for the Fortin-Soulie finite-element method, Comput. Math. Appl. 48 (2004), no. 12, 1863–1876. MR 2116962, DOI 10.1016/j.camwa.2004.07.011
Susanne C. Brenner, Kening Wang, and Jie Zhao, Poincaré-Friedrichs inequalities for piecewise $H^2$ functions, Numer. Funct. Anal. Optim. 25 (2004), no. 5-6, 463–478. MR 2106270, DOI 10.1081/NFA-200042165
Carsten Carstensen, A posteriori error estimate for the mixed finite element method, Math. Comp. 66 (1997), no. 218, 465–476. MR 1408371, DOI 10.1090/S0025-5718-97-00837-5
Carsten Carstensen, Sören Bartels, and Stefan Jansche, A posteriori error estimates for nonconforming finite element methods, Numer. Math. 92 (2002), no. 2, 233–256. MR 1922920, DOI 10.1007/s002110100378
E. Dari, R. Duran, C. Padra, and V. Vampa, A posteriori error estimators for nonconforming finite element methods, RAIRO Modél. Math. Anal. Numér. 30 (1996), no. 4, 385–400 (English, with English and French summaries). MR 1399496, DOI 10.1051/m2an/1996300403851
Willy Dörfler, A convergent adaptive algorithm for Poisson’s equation, SIAM J. Numer. Anal. 33 (1996), no. 3, 1106–1124. MR 1393904, DOI 10.1137/0733054
M. Fortin and M. Soulie, A nonconforming piecewise quadratic finite element on triangles, Internat. J. Numer. Methods Engrg. 19 (1983), no. 4, 505–520. MR 702056, DOI 10.1002/nme.1620190405
Ronald H. W. Hoppe and Barbara Wohlmuth, Element-oriented and edge-oriented local error estimators for nonconforming finite element methods, RAIRO Modél. Math. Anal. Numér. 30 (1996), no. 2, 237–263 (English, with English and French summaries). MR 1382112, DOI 10.1051/m2an/1996300202371
Heejeong Lee and Dongwoo Sheen, Basis for the quadratic nonconforming triangular element of Fortin and Soulie, Int. J. Numer. Anal. Model. 2 (2005), no. 4, 409–421. MR 2177630
Luisa Donatella Marini, An inexpensive method for the evaluation of the solution of the lowest order Raviart-Thomas mixed method, SIAM J. Numer. Anal. 22 (1985), no. 3, 493–496. MR 787572, DOI 10.1137/0722029
Pedro Morin, Ricardo H. Nochetto, and Kunibert G. Siebert, Data oscillation and convergence of adaptive FEM, SIAM J. Numer. Anal. 38 (2000), no. 2, 466–488. MR 1770058, DOI 10.1137/S0036142999360044
Serge Nicaise, A posteriori error estimations of some cell-centered finite volume methods, SIAM J. Numer. Anal. 43 (2005), no. 4, 1481–1503. MR 2182137, DOI 10.1137/S0036142903437787
Claudio Padra, A posteriori error estimators for nonconforming approximation of some quasi-Newtonian flows, SIAM J. Numer. Anal. 34 (1997), no. 4, 1600–1615. MR 1461798, DOI 10.1137/S0036142994278322
L. E. Payne and H. F. Weinberger, An optimal Poincaré inequality for convex domains, Arch. Rational Mech. Anal. 5 (1960), 286–292 (1960). MR 117419, DOI 10.1007/BF00252910
Martin Petzoldt, A posteriori error estimators for elliptic equations with discontinuous coefficients, Adv. Comput. Math. 16 (2002), no. 1, 47–75. MR 1888219, DOI 10.1023/A:1014221125034
R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley-Teubner, 1996.