Discrete extension operators for mixed finite element spaces on locally refined meshes

Ainsworth, Mark; Guzmán, Johnny; Sayas, Francisco-Javier

doi:10.1090/mcom/3074

Discrete extension operators for mixed finite element spaces on locally refined meshes
HTML articles powered by AMS MathViewer

by Mark Ainsworth, Johnny Guzmán and Francisco-Javier Sayas;

Math. Comp. 85 (2016), 2639-2650

DOI: https://doi.org/10.1090/mcom/3074

Published electronically: January 14, 2016

PDF | Request permission

Abstract:

The existence of uniformly bounded discrete extension operators is established for conforming Raviart-Thomas and Nédelec discretizations of $H(div)$ and $H(curl)$ on locally refined partitions of a polyhedral domain into tetrahedra.

References

Ana Alonso and Alberto Valli, An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations, Math. Comp. 68 (1999), no. 226, 607–631. MR 1609607, DOI 10.1090/S0025-5718-99-01013-3
Mark Ainsworth, William McLean, and Thanh Tran, The conditioning of boundary element equations on locally refined meshes and preconditioning by diagonal scaling, SIAM J. Numer. Anal. 36 (1999), no. 6, 1901–1932. MR 1712149, DOI 10.1137/S0036142997330809
Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Multigrid in $H(\textrm {div})$ and $H(\textrm {curl})$, Numer. Math. 85 (2000), no. 2, 197–217. MR 1754719, DOI 10.1007/PL00005386
Ivo Babuška and Gabriel N. Gatica, On the mixed finite element method with Lagrange multipliers, Numer. Methods Partial Differential Equations 19 (2003), no. 2, 192–210. MR 1958060, DOI 10.1002/num.10040
Daniele Boffi, Franco Brezzi, and Michel Fortin, Mixed finite element methods and applications, Springer Series in Computational Mathematics, vol. 44, Springer, Heidelberg, 2013. MR 3097958, DOI 10.1007/978-3-642-36519-5
A. Buffa and P. Ciarlet Jr., On traces for functional spaces related to Maxwell’s equations. I. An integration by parts formula in Lipschitz polyhedra, Math. Methods Appl. Sci. 24 (2001), no. 1, 9–30. MR 1809491, DOI 10.1002/1099-1476(20010110)24:1<9::AID-MMA191>3.0.CO;2-2
A. Buffa and P. Ciarlet Jr., On traces for functional spaces related to Maxwell’s equations. II. Hodge decompositions on the boundary of Lipschitz polyhedra and applications, Math. Methods Appl. Sci. 24 (2001), no. 1, 31–48. MR 1809492, DOI 10.1002/1099-1476(20010110)24:1<9::AID-MMA191>3.0.CO;2-2
A. Buffa, M. Costabel, and D. Sheen, On traces for $\textbf {H}(\textbf {curl},\Omega )$ in Lipschitz domains, J. Math. Anal. Appl. 276 (2002), no. 2, 845–867. MR 1944792, DOI 10.1016/S0022-247X(02)00455-9
Franco Brezzi, Jim Douglas Jr., Ricardo Durán, and Michel Fortin, Mixed finite elements for second order elliptic problems in three variables, Numer. Math. 51 (1987), no. 2, 237–250. MR 890035, DOI 10.1007/BF01396752
Snorre H. Christiansen and Ragnar Winther, Smoothed projections in finite element exterior calculus, Math. Comp. 77 (2008), no. 262, 813–829. MR 2373181, DOI 10.1090/S0025-5718-07-02081-9
Monique Dauge, Neumann and mixed problems on curvilinear polyhedra, Integral Equations Operator Theory 15 (1992), no. 2, 227–261. MR 1147281, DOI 10.1007/BF01204238
Leszek Demkowicz, Jayadeep Gopalakrishnan, and Joachim Schöberl, Polynomial extension operators. I, SIAM J. Numer. Anal. 46 (2008), no. 6, 3006–3031. MR 2439500, DOI 10.1137/070698786
Víctor Domínguez and Francisco-Javier Sayas, Stability of discrete liftings, C. R. Math. Acad. Sci. Paris 337 (2003), no. 12, 805–808 (English, with English and French summaries). MR 2033124, DOI 10.1016/j.crma.2003.10.025
Gabriel N. Gatica, Ricardo Oyarzúa, and Francisco-Javier Sayas, Analysis of fully-mixed finite element methods for the Stokes-Darcy coupled problem, Math. Comp. 80 (2011), no. 276, 1911–1948. MR 2813344, DOI 10.1090/S0025-5718-2011-02466-X
Ralf Hiptmair and Shipeng Mao, Stable multilevel splittings of boundary edge element spaces, BIT 52 (2012), no. 3, 661–685. MR 2965296, DOI 10.1007/s10543-012-0369-1
Ralf Hiptmair, Carlos Jerez-Hanckes, and Shipeng Mao, Extension by zero in discrete trace spaces: inverse estimates, Math. Comp. 84 (2015), no. 296, 2589–2615. MR 3378840, DOI 10.1090/mcom/2955
Antonio Márquez, Salim Meddahi, and Francisco-Javier Sayas, Strong coupling of finite element methods for the Stokes-Darcy problem, IMA J. Numer. Anal. 35 (2015), no. 2, 969–988. MR 3335232, DOI 10.1093/imanum/dru023
J.-C. Nédélec, Mixed finite elements in $\textbf {R}^{3}$, Numer. Math. 35 (1980), no. 3, 315–341. MR 592160, DOI 10.1007/BF01396415
J.-C. Nédélec, A new family of mixed finite elements in $\textbf {R}^3$, Numer. Math. 50 (1986), no. 1, 57–81. MR 864305, DOI 10.1007/BF01389668
Francisco Javier Sayas, Infimum-supremum, Bol. Soc. Esp. Mat. Apl. SeMA 41 (2007), 19–40. MR 2388171
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
L. Ridgway Scott and Shangyou Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp. 54 (1990), no. 190, 483–493. MR 1011446, DOI 10.1090/S0025-5718-1990-1011446-7
Jinchao Xu and Ludmil Zikatanov, Some observations on Babuška and Brezzi theories, Numer. Math. 94 (2003), no. 1, 195–202. MR 1971217, DOI 10.1007/s002110100308
O. B. Widlund, An Extension Theorem for Finite Element Spaces with Three Applications, Proceedings of a GAMM Seminar on Numerical Techniques in Continuum Mechanics, held in Kiel, Germany, January 17-19, 1986 (Wolfgang Hackbusch and Kristian Witsch, eds.), Friedr. Vieweg and Sohn, Braunschweig/Wiesbaden, 1987, pp. 110-122.