A simple equilibration procedure leading to polynomial-degree-robust a posteriori error estimators for the curl-curl problem

Chaumont-Frelet, T.

doi:10.1090/mcom/3817

A simple equilibration procedure leading to polynomial-degree-robust a posteriori error estimators for the curl-curl problem
HTML articles powered by AMS MathViewer

by T. Chaumont-Frelet

Math. Comp. 92 (2023), 2413-2437

DOI: https://doi.org/10.1090/mcom/3817

Published electronically: June 20, 2023

HTML | PDF | Request permission

Abstract:

We introduce two a posteriori error estimators for Nédélec finite element discretizations of the curl-curl problem. These estimators pertain to a new Prager–Synge identity and an associated equilibration procedure. They are reliable and efficient, and the error estimates are polynomial-degree-robust. In addition, when the domain is convex, the reliability constants are fully computable. The proposed error estimators are also cheap and easy to implement, as they are computed by solving divergence-constrained minimization problems over edge patches. Numerical examples highlight our key findings, and show that both estimators are suited to drive adaptive refinement algorithms. Besides, these examples seem to indicate that guaranteed upper bounds can be achieved even in non-convex domains.

References

Robert A. Adams and John J. F. Fournier, Sobolev spaces, 2nd ed., Pure and Applied Mathematics (Amsterdam), vol. 140, Elsevier/Academic Press, Amsterdam, 2003. MR 2424078
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
Thomas Apel, Anisotropic finite elements: local estimates and applications, Advances in Numerical Mathematics, B. G. Teubner, Stuttgart, 1999. MR 1716824
I. Babuška and Manil Suri, The $h$-$p$ version of the finite element method with quasi-uniform meshes, RAIRO Modél. Math. Anal. Numér. 21 (1987), no. 2, 199–238 (English, with French summary). MR 896241, DOI 10.1051/m2an/1987210201991
Rudi Beck, Ralf Hiptmair, Ronald H. W. Hoppe, and Barbara Wohlmuth, Residual based a posteriori error estimators for eddy current computation, M2AN Math. Model. Numer. Anal. 34 (2000), no. 1, 159–182 (English, with English and French summaries). MR 1735971, DOI 10.1051/m2an:2000136
Dietrich Braess, Veronika Pillwein, and Joachim Schöberl, Equilibrated residual error estimates are $p$-robust, Comput. Methods Appl. Mech. Engrg. 198 (2009), no. 13-14, 1189–1197. MR 2500243, DOI 10.1016/j.cma.2008.12.010
Dietrich Braess and Joachim Schöberl, Equilibrated residual error estimator for edge elements, Math. Comp. 77 (2008), no. 262, 651–672. MR 2373174, DOI 10.1090/S0025-5718-07-02080-7
F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 8 (1974), no. R-2, 129–151 (English, with French summary). MR 365287
Théophile Chaumont-Frelet, Alexandre Ern, and Martin Vohralík, Polynomial-degree-robust $H(\textrm {curl})$-stability of discrete minimization in a tetrahedron, C. R. Math. Acad. Sci. Paris 358 (2020), no. 9-10, 1101–1110 (English, with English and French summaries). MR 4196782, DOI 10.5802/crmath.133
T. Chaumont-Frelet, A. Ern, and M. Vohralík, Stable broken $\mathbf H(\mathbf {curl})$ polynomial extensions and $p$-robust a posteriori error estimates by broken patchwise equilibration for the curl-curl problem, Math. Comp. 91 (2021), no. 333, 37–74. MR 4350532, DOI 10.1090/mcom/3673
T. Chaumont-Frelet and M. Vohralík, $p$-robust equilibrated flux reconstruction in $H(curl)$ based on local minimizations. Application to a posteriori analysis of the curl-curl problem, preprint, arXiv:2105.07770, 2021.
Philippe G. Ciarlet, The finite element method for elliptic problems, Classics in Applied Mathematics, vol. 40, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. Reprint of the 1978 original [North-Holland, Amsterdam; MR0520174 (58 #25001)]. MR 1930132, DOI 10.1137/1.9780898719208
Martin Costabel, Monique Dauge, and Serge Nicaise, Singularities of Maxwell interface problems, M2AN Math. Model. Numer. Anal. 33 (1999), no. 3, 627–649. MR 1713241, DOI 10.1051/m2an:1999155
Philippe Destuynder and Brigitte Métivet, Explicit error bounds in a conforming finite element method, Math. Comp. 68 (1999), no. 228, 1379–1396. MR 1648383, DOI 10.1090/S0025-5718-99-01093-5
C. Dobrzynski, MMG3D: User guide, Tech. Report 422, Inria, France, 2012.
Alexandre Ern and Martin Vohralík, Polynomial-degree-robust a posteriori estimates in a unified setting for conforming, nonconforming, discontinuous Galerkin, and mixed discretizations, SIAM J. Numer. Anal. 53 (2015), no. 2, 1058–1081. MR 3335498, DOI 10.1137/130950100
Alexandre Ern and Martin Vohralík, Stable broken $H^1$ and $H(\textrm {div})$ polynomial extensions for polynomial-degree-robust potential and flux reconstruction in three space dimensions, Math. Comp. 89 (2020), no. 322, 551–594. MR 4044442, DOI 10.1090/mcom/3482
Paolo Fernandes and Gianni Gilardi, Magnetostatic and electrostatic problems in inhomogeneous anisotropic media with irregular boundary and mixed boundary conditions, Math. Models Methods Appl. Sci. 7 (1997), no. 7, 957–991. MR 1479578, DOI 10.1142/S0218202597000487
Joscha Gedicke, Sjoerd Geevers, and Ilaria Perugia, An equilibrated a posteriori error estimator for arbitrary-order Nédélec elements for magnetostatic problems, J. Sci. Comput. 83 (2020), no. 3, Paper No. 58, 23. MR 4110650, DOI 10.1007/s10915-020-01224-x
Joscha Gedicke, Sjoerd Geevers, Ilaria Perugia, and Joachim Schöberl, A polynomial-degree-robust a posteriori error estimator for Nédélec discretizations of magnetostatic problems, SIAM J. Numer. Anal. 59 (2021), no. 4, 2237–2253. MR 4298538, DOI 10.1137/20M1333365
Christophe Geuzaine and Jean-François Remacle, Gmsh: A 3-D finite element mesh generator with built-in pre- and post-processing facilities, Internat. J. Numer. Methods Engrg. 79 (2009), no. 11, 1309–1331. MR 2566786, DOI 10.1002/nme.2579
Vivette Girault and Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. MR 851383, DOI 10.1007/978-3-642-61623-5
D. J. Griffiths, Introduction to Eelectrodynamics, Prentice Hall, 1999.
Ralf Hiptmair and Clemens Pechstein, Discrete regular decompositions of tetrahedral discrete 1-forms, Maxwell’s equations—analysis and numerics, Radon Ser. Comput. Appl. Math., vol. 24, De Gruyter, Berlin, [2019] ©2019, pp. 199–258. MR 4398318
P. Ladevèze and D. Leguillon, Error estimate procedure in the finite element method and applications, SIAM J. Numer. Anal. 20 (1983), no. 3, 485–509. MR 701093, DOI 10.1137/0720033
R. Luce and B. I. Wohlmuth, A local a posteriori error estimator based on equilibrated fluxes, SIAM J. Numer. Anal. 42 (2004), no. 4, 1394–1414. MR 2114283, DOI 10.1137/S0036142903433790
J. M. Melenk and B. I. Wohlmuth, On residual-based a posteriori error estimation in $hp$-FEM, Adv. Comput. Math. 15 (2001), no. 1-4, 311–331 (2002). A posteriori error estimation and adaptive computational methods. MR 1887738, DOI 10.1023/A:1014268310921
Peter Monk, Finite element methods for Maxwell’s equations, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2003. MR 2059447, DOI 10.1093/acprof:oso/9780198508885.001.0001
J.-C. Nédélec, Mixed finite elements in $\textbf {R}^{3}$, Numer. Math. 35 (1980), no. 3, 315–341. MR 592160, DOI 10.1007/BF01396415
S. Nicaise and E. Creusé, A posteriori error estimation for the heterogeneous Maxwell equations on isotropic and anisotropic meshes, Calcolo 40 (2003), no. 4, 249–271. MR 2025712, DOI 10.1007/s10092-003-0077-y
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
W. Prager and J. L. Synge, Approximations in elasticity based on the concept of function space, Quart. Appl. Math. 5 (1947), 241–269. MR 25902, DOI 10.1090/S0033-569X-1947-25902-8
P.-A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975) Lecture Notes in Math., Vol. 606, Springer, Berlin-New York, 1977, pp. 292–315. MR 483555
Joachim Schöberl, A posteriori error estimates for Maxwell equations, Math. Comp. 77 (2008), no. 262, 633–649. MR 2373173, DOI 10.1090/S0025-5718-07-02030-3