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A posteriori error estimates for finite element exterior calculus: the de Rham complex. (English) Zbl 1308.65187
In a fairly sophisticated framework, i.e., the Hilbert complex structure for finite element exterior calculus, the authors rigorously establish a posteriori upper bounds for errors in approximation to the Hodge-de Rham-Laplace problems. Some elementwise efficiency results are also carried out. Eventually, the authors show how their results apply to several specific examples from the three-dimensional de Rham complex.

MSC:
65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
58A12 de Rham theory in global analysis
58A14 Hodge theory in global analysis
58A15 Exterior differential systems (Cartan theory)
58J05 Elliptic equations on manifolds, general theory
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References:
[1] M. Ainsworth and J. T. Oden, A posteriori error estimation in finite element analysis, Pure and Applied Mathematics (New York), Wiley-Interscience [John Wiley & Sons], New York, 2000. · Zbl 1008.65076
[2] A. Alonso, Error estimators for a mixed method, Numer. Math., 74 (1996), pp. 385-395. · Zbl 0866.65068
[3] D. N. Arnold, Differential complexes and numerical stability, in Proceedings of the International Congress of Mathematicians, Beijing 2002, Volume 1 : Plenary Lectures, 2002.
[4] D. N. Arnold, R. S. Falk, and R. Winther, Finite element exterior calculus, homological techniques, and applications, in Acta Numerica, A. Iserles, ed., vol. 15, Cambridge University Press, 2006, pp. 1-155. · Zbl 1185.65204
[5] DN Arnold, RS Falk, R Winther (2010) Finite element exterior calculus: from Hodge theory to numerical stability. Bull. Amer. Math. Soc. (N.S.) 47:281-354 · Zbl 1207.65134
[6] I. Babuška and W. C. Rheinboldt, Error estimates for adaptive finite element computations, SIAM J. Numer. Anal., 15 (1978), pp. 736-754. · Zbl 0398.65069
[7] W. Bangerth and R. Rannacher, Adaptive finite element methods for differential equations, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2003. · Zbl 1020.65058
[8] R. Beck, R. Hiptmair, R. H. W. Hoppe, and B. Wohlmuth, Residual based a posteriori error estimators for eddy current computation, M2AN Math. Model. Numer. Anal., 34 (2000), pp. 159-182. · Zbl 0949.65113
[9] A. Bossavit, Whitney forms : A class of finite elements for three-dimensional computations in electromagnetism, IEEE Proceedings, 135, Part A (1988), pp. 493-500.
[10] D. Braess and R. Verfürth, A posteriori error estimators for the Raviart-Thomas element, SIAM J. Numer. Anal., 33 (1996), pp. 2431-2444. · Zbl 0866.65071
[11] S. C. Brenner and L. R. Scott, The mathematical theory of finite element methods, vol. 15 of Texts in Applied Mathematics, Springer, New York, third ed., 2008. · Zbl 1135.65042
[12] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991. · Zbl 0788.73002
[13] C. Carstensen, A posteriori error estimate for the mixed finite element method, Math. Comp., 66 (1997), pp. 465-476. · Zbl 0864.65068
[14] J. Chen, Y. Xu, and J. Zou, An adaptive edge element methods and its convergence for a saddle-point problem from magnetostatics, Numer. Methods Partial Differential Equations, 28 (2012), pp. 1643-1666. · Zbl 1259.78039
[15] S. H. Christiansen and R. Winther, Smoothed projections in finite element exterior calculus, Math. Comp., 77 (2008), pp. 813-829. · Zbl 1140.65081
[16] M. Costabel and A. McIntosh, On Bogovskiĭ and regularized Poincaré integral operators for de Rham complexes on Lipschitz domains, Math. Z., 265 (2010), pp. 297-320. · Zbl 1197.35338
[17] J. Dodziuk, Finite-difference approach to the Hodge theory of harmonic forms, Amer. J. Math., 98 (1976), pp. 79-104. · Zbl 0324.58001
[18] R. S. Falk and R. Winther, Local bounded cochain projections, Math. Comp., (To appear). · Zbl 1300.65085
[19] R. Hiptmair, Canonical construction of finite elements, Math. Comp., 68 (1999), pp. 1325-1346. · Zbl 0938.65132
[20] R. Hiptmair, Finite elements in computational electromagnetism, Acta Numer., 11 (2002), pp. 237-339. · Zbl 1123.78320
[21] T. Kato, Perturbation theory for linear operators, Springer-Verlag, Berlin, second ed., 1976. Grundlehren der Mathematischen Wissenschaften, Band 132. · Zbl 0342.47009
[22] M. G. Larson and A. Målqvist, A posteriori error estimates for mixed finite element approximations of elliptic problems, Numer. Math., 108 (2008), pp. 487-500. · Zbl 1136.65101
[23] C. Lovadina and R. Stenberg, Energy norm a posteriori error estimates for mixed finite element methods, Math. Comp., 75 (2006), pp. 1659-1674 (electronic). · Zbl 1119.65110
[24] D. Mitrea, M. Mitrea, and S. Monniaux, The Poisson problem for the exterior derivative operator with Dirichlet boundary condition in nonsmooth domains, Commun. Pure Appl. Anal., 7 (2008), pp. 1295-1333. · Zbl 1143.47031
[25] D. Mitrea, M. Mitrea, and M.-C. Shaw, Traces of differential forms on Lipschitz domains, the boundary de Rham complex, and Hodge decompositions, Indiana Univ. Math. J., 57 (2008), pp. 2061-2095. · Zbl 1167.58001
[26] J. E. Pasciak and J. Zhao, Overlapping Schwarz methods in H(curl) on polyhedral domains, J. Numer. Math., 10 (2002), pp. 221-234. · Zbl 1017.65099
[27] J. Schöberl, Commuting quasi-interpolation operators for mixed finite elements, Tech. Rep. ISC-01-10-MATH, Institute for Scientific Computing, Texas A&M University, 2001.
[28] J. Schöberl, A posteriori error estimates for Maxwell equations, Math. Comp., 77 (2008), pp. 633-649. · Zbl 1136.78016
[29] R. Verfürth, A posteriori error estimators for the Stokes equations, Numer. Math., 55 (1989), pp. 309-325. · Zbl 0674.65092
[30] M. Vohralík, A posteriori error estimates for lowest-order mixed finite element discretizations of convection-diffusion-reaction equations, SIAM J. Numer. Anal., 45 (2007), pp. 1570-1599 (electronic). · Zbl 1151.65084
[31] L. Zhong, L. Chen, S. Shu, G. Wittum, and J. Xu, Convergence and Optimality of adaptive edge finite element methods for time-harmonic Maxwell equations, Math. Comp., (To appear.). · Zbl 1263.78012
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