Extension by zero in discrete trace spaces: Inverse estimates

Hiptmair, Ralf; Jerez-Hanckes, Carlos; Mao, Shipeng

doi:10.1090/mcom/2955

Extension by zero in discrete trace spaces: Inverse estimates
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by Ralf Hiptmair, Carlos Jerez-Hanckes and Shipeng Mao PDF

Math. Comp. 84 (2015), 2589-2615 Request permission

Abstract:

We consider lowest-order ${\boldsymbol H}^{-\frac {1}{2}}(\operatorname {div}_\Gamma , \Gamma )$- and $H^{-\frac {1}{2}}(\Gamma )$-conforming boundary element spaces supported on part of the boundary $\Gamma$ of a Lipschitz polyhedron. Assuming families of triangular meshes created by regular refinement, we prove that on these spaces the norms of the extension by zero operators with respect to (localized) trace norms increase poly-logarithmically with the mesh width. Our approach harnesses multilevel norm equivalences for boundary element spaces, inherited from stable multilevel splittings of finite element spaces.

References

Mark Ainsworth, Johnny Guzmán, and Francisco-Javier Sayas, Discrete extension operators for mixed finite element spaces on locally refined meshes, arXiv:1406.5534v2 [math.NA] (2015).
Mark Ainsworth and William McLean, Multilevel diagonal scaling preconditioners for boundary element equations on locally refined meshes, Numer. Math. 93 (2003), no. 3, 387–413. MR 1953746, DOI 10.1007/s002110100391
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
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
D. N. Arnold, R. S. Falk, and R. Winther, Multigrid preconditioning in $H(\textrm {div})$ on non-convex polygons, Comput. Appl. Math. 17 (1998), no. 3, 303–315. MR 1687885
J. Bey, Tetrahedral grid refinement, Computing 55 (1995), no. 4, 355–378 (English, with English and German summaries). MR 1370107, DOI 10.1007/BF02238487
Folkmar Bornemann and Harry Yserentant, A basic norm equivalence for the theory of multilevel methods, Numer. Math. 64 (1993), no. 4, 455–476. MR 1213412, DOI 10.1007/BF01388699
James H. Bramble, Multigrid methods, Pitman Research Notes in Mathematics Series, vol. 294, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993. MR 1247694
James H. Bramble, Joseph E. Pasciak, and Alfred H. Schatz, The construction of preconditioners for elliptic problems by substructuring. IV, Math. Comp. 53 (1989), no. 187, 1–24. MR 970699, DOI 10.1090/S0025-5718-1989-0970699-3
Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR 1115205, DOI 10.1007/978-1-4612-3172-1
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
Huangxin Chen, Ronald H. W. Hoppe, and Xuejun Xu, Uniform convergence of local multigrid methods for the time-harmonic Maxwell equation, ESAIM Math. Model. Numer. Anal. 47 (2013), no. 1, 125–147. MR 2968698, DOI 10.1051/m2an/2012023
Choi-Hong Lai, Petter E. Bjørstad, Mark Cross, and Olof Widlund (eds.), Eleventh International Conference on Domain Decomposition Methods, DDM.org, Augsburg, 1999. Available electronically at http://www.ddm.org/DD11/index.html. MR 1827403
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
Clark R. Dohrmann and Olof B. Widlund, An iterative substructuring algorithm for two-dimensional problems in $H(\textrm {curl})$, SIAM J. Numer. Anal. 50 (2012), no. 3, 1004–1028. MR 2970732, DOI 10.1137/100818145
C. Dohrmann and O. Widlund, Some recent tools and a BDDC algorithm for 3D problems in $\mathbb {H}(\mathrm {curl})$, in Domain Decomposition Methods in Science and Engineering XX, R. Bank, M. Holst, O. Widlund, and J. Xu, eds., vol. 91 of Lecture Notes in Computational Science and Engineering, Springer, Berlin, Heidelberg, 2013, pp. 15–25.
M. Dryja, A method of domain decomposition for three-dimensional finite element elliptic problems, First International Symposium on Domain Decomposition Methods for Partial Differential Equations (Paris, 1987) SIAM, Philadelphia, PA, 1988, pp. 43–61. MR 972511
Alexandre Ern and Jean-Luc Guermond, Theory and practice of finite elements, Applied Mathematical Sciences, vol. 159, Springer-Verlag, New York, 2004. MR 2050138, DOI 10.1007/978-1-4757-4355-5
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
P. Grisvard, Singularities in boundary value problems, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 22, Masson, Paris; Springer-Verlag, Berlin, 1992. MR 1173209
Norbert Heuer, Ernst P. Stephan, and Thanh Tran, Multilevel additive Schwarz method for the $h$-$p$ version of the Galerkin boundary element method, Math. Comp. 67 (1998), no. 222, 501–518. MR 1451325, DOI 10.1090/S0025-5718-98-00926-0
R. Hiptmair, Multigrid method for $\mathbf H(\textrm {div})$ in three dimensions, Electron. Trans. Numer. Anal. 6 (1997), no. Dec., 133–152. Special issue on multilevel methods (Copper Mountain, CO, 1997). MR 1615161
R. Hiptmair, Multigrid method for Maxwell’s equations, SIAM J. Numer. Anal. 36 (1999), no. 1, 204–225. MR 1654571, DOI 10.1137/S0036142997326203
R. Hiptmair, Finite elements in computational electromagnetism, Acta Numer. 11 (2002), 237–339. MR 2009375, DOI 10.1017/S0962492902000041
R. Hiptmair, Analysis of multilevel methods for eddy current problems, Math. Comp. 72 (2003), no. 243, 1281–1303. MR 1972736, DOI 10.1090/S0025-5718-02-01468-0
R. Hiptmair and C. Jerez-Hanckes, Multiple traces boundary integral formulation for Helmholtz transmission problems, Adv. Comput. Math. 37 (2012), no. 1, 39–91. MR 2927645, DOI 10.1007/s10444-011-9194-3
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
R. Hiptmair and W.-Y. Zheng, Local multigrid in $\mathbf {H}(\mathbf {curl})$, Tech. Rep. 2007-03, SAM, ETH Zürich, Switzerland, March 2007. http://arxiv.org/abs/0901.0764.
Ralf Hiptmair and Weiying Zheng, Local multigrid in $\textbf {H}(\bf {curl})$, J. Comput. Math. 27 (2009), no. 5, 573–603. MR 2536903, DOI 10.4208/jcm.2009.27.5.012
Q. Hu and J. Zou, A non-overlapping domain decomposition method for Maxwell’s equation in three dimensions, SIAM J. Numer. Anal., 41 (2003), pp. 1682–1708.
Qiya Hu and Jun Zou, Substructuring preconditioners for saddle-point problems arising from Maxwell’s equations in three dimensions, Math. Comp. 73 (2004), no. 245, 35–61. MR 2034110, DOI 10.1090/S0025-5718-03-01541-2
Igor Kossaczký, A recursive approach to local mesh refinement in two and three dimensions, J. Comput. Appl. Math. 55 (1994), no. 3, 275–288. MR 1329875, DOI 10.1016/0377-0427(94)90034-5
J. Lions and F. Magenes, Nonhomogeneous boundary value problems and applications, Springer–Verlag, Berlin, 1972.
William McLean, Strongly elliptic systems and boundary integral equations, Cambridge University Press, Cambridge, 2000. MR 1742312
W. McLean and O. Steinbach, Boundary element preconditioners for a hypersingular integral equation on an interval, Adv. Comput. Math. 11 (1999), no. 4, 271–286. MR 1732138, DOI 10.1023/A:1018944530343
J.-C. Nédélec, Mixed finite elements in $\textbf {R}^{3}$, Numer. Math. 35 (1980), no. 3, 315–341. MR 592160, DOI 10.1007/BF01396415
P. Oswald, On function spaces related to finite element approximation theory, Z. Anal. Anwendungen 9 (1990), no. 1, 43–64 (English, with German and Russian summaries). MR 1063242, DOI 10.4171/ZAA/380
P. Oswald, On discrete norm estimates related to multilevel preconditioners in the finite element method, in Constructive Theory of Functions, Proc. Int. Conf. Varna 1991, K. Ivanov, P. Petrushev, and B. Sendov, eds., Bulg. Acad. Sci., 1992, pp. 203–214.
Peter Oswald, Multilevel finite element approximation, Teubner Skripten zur Numerik. [Teubner Scripts on Numerical Mathematics], B. G. Teubner, Stuttgart, 1994. Theory and applications. MR 1312165, DOI 10.1007/978-3-322-91215-2
Stefan A. Sauter and Christoph Schwab, Boundary element methods, Springer Series in Computational Mathematics, vol. 39, Springer-Verlag, Berlin, 2011. Translated and expanded from the 2004 German original. MR 2743235, DOI 10.1007/978-3-540-68093-2
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
Luc Tartar, An introduction to Sobolev spaces and interpolation spaces, Lecture Notes of the Unione Matematica Italiana, vol. 3, Springer, Berlin; UMI, Bologna, 2007. MR 2328004
Andrea Toselli and Olof Widlund, Domain decomposition methods—algorithms and theory, Springer Series in Computational Mathematics, vol. 34, Springer-Verlag, Berlin, 2005. MR 2104179, DOI 10.1007/b137868
Jinchao Xu, Iterative methods by space decomposition and subspace correction, SIAM Rev. 34 (1992), no. 4, 581–613. MR 1193013, DOI 10.1137/1034116
Jinchao Xu and Jun Zou, Some nonoverlapping domain decomposition methods, SIAM Rev. 40 (1998), no. 4, 857–914. MR 1659681, DOI 10.1137/S0036144596306800
Jinchao Xu, Long Chen, and Ricardo H. Nochetto, Optimal multilevel methods for $H(\textrm {grad})$, $H(\textrm {curl})$, and $H(\textrm {div})$ systems on graded and unstructured grids, Multiscale, nonlinear and adaptive approximation, Springer, Berlin, 2009, pp. 599–659. MR 2648382, DOI 10.1007/978-3-642-03413-8_{1}4
Xuejun Zhang, Multilevel Schwarz methods, Numer. Math. 63 (1992), no. 4, 521–539. MR 1189535, DOI 10.1007/BF01385873