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# Extension by zero in discrete trace spaces: Inverse estimates

MATHEMATICS OF COMPUTATION, no. 296 (2015): 2589-2615

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Abstract

We consider lowest-order H-1/2 (div(Gamma), Gamma)- and H-1/2 (Gamma)-conforming boundary element spaces supported on part of the boundary G 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)...More

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Introduction

- The authors consider a bounded Lipschitz polyhedron Ω ⊂ R3 with trivial topology and set Γ := ∂Ω to be its compact boundary composed of a small number of flat faces.
- Inverse estimates, multilevel norm equivalences.
- On Γh the authors consider the finite element space S1(Γh) ⊂ H1(Γ) of Γh-piecewise linear, globally continuous functions on Γ; see [42, Sect.
- Inverse inequalities with logarithmic blow-up in terms of the mesh width are typical for such borderline cases and piecewise polynomial finite element spaces of fixed degree [20, Lemma 1.142].

Highlights

- We consider a bounded Lipschitz polyhedron Ω ⊂ R3 with trivial topology and set Γ := ∂Ω to be its compact boundary composed of a small number of flat faces
- Inverse inequalities with logarithmic blow-up in terms of the mesh width are typical for such borderline cases and piecewise polynomial finite element spaces of fixed degree [20, Lemma 1.142]
- We summarize the main results that extend (1.3) to the boundary element spaces RT 0(Γ+l ) and Q0(Γ+) and the corresponding trace norms
- (4.2) V − ΠRl t V L2(Ω) hl V H (Ω), ∀ V ∈ {V ∈ (H (Ω))3 : div V = 0}. This estimate is instrumental in constructing various discrete extension operators for both boundary element spaces Q0(Γl) and RT 0(Γl)
- We find c0 ∈ RT 0(Γ+0 ) such that wL · n∂Γ+ dl =

Results

- The authors summarize the main results that extend (1.3) to the boundary element spaces RT 0(Γ+l ) and Q0(Γ+) and the corresponding trace norms.
- Similar norm equivalences hold for boundary element spaces and involve the multilevel norms
- 1 2 (Γ)-conforming boundary element spaces the authors have the following equivalent norms: (3.5) (3.6)
- The localized trace norm of boundary element functions allows an equivalent characterization through multilevel splittings: (3.9)
- The stable discrete extension from Lemma 3.1 played a key role in the proof of the multilevel decomposition
- This estimate is instrumental in constructing various discrete extension operators for both boundary element spaces Q0(Γl) and RT 0(Γl).
- Similar norm equivalences hold for the edge boundary element spaces RT 0(ΓL) relying on the multilevel norms
- The authors use the extension operators provided by Lemmas 4.1, 4.5, and 3.2, 3.1 together with Theorem 5.1 in the very same fashion as in the proof of Theorem 3.4; see [29, Sect.
- Norm equivalences as expressed in Theorem 5.2 hold for spaces of piecewise constant functions on ΓL.
- In order to prove the “ ” estimate of (5.18), by Lemma 4.4, the trace inequality and Theorem 5.4 the authors have to establish

Conclusion

- To show the estimate “ ” of (5.18) the authors again apply the trivial local extension operators of Lemma 4.2 to the terms of a multilevel splitting of φL.
- In analogy to the second part of the proof of Theorem 5.2, the authors use the continuity of the trace operator and the norm equivalence from Theorem 5.4.
- The authors follow exactly the reasoning of the proof of Theorem 5.3, this time using the discrete extension operator E2l provided by Lemma 4.4.
- The authors just follow Part of the proof of Theorem 2.1, as in (6.20) and (6.21) modify the ql in order to enforce zero normal components on ∂Γ+, pay with another factor of L2 in the estimate, as in (6.22), and the authors are done.

Funding

- The work of the second author was funded by FONDECYT 11121166 and CONICYT project Anillo ACT1118 (ANANUM)
- The work of the third author was partly supported by Thales SA under contract “Preconditioned Boundary Element Methods for Electromagnetic Scattering at Dielectric Objects” and NSFC 11101414, 11101386, 11471329

Reference

- Mark Ainsworth, Johnny Guzman, and Francisco-Javier Sayas, Discrete extension operators for mixed finite element spaces on locally refined meshes, arXiv:1406.5534v2 [math.NA] (2015).
- M. Ainsworth and W. McLean, Multilevel diagonal scaling preconditioners for boundary element equations on locally refined meshes, Numer. Math. 93 (2003), no. 3, 387–413, DOI 10.1007/s002110100391. MR1953746 (2004i:65130)
- A. Alonso and A. Valli, An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations, Math. Comp. 68 (1999), no. 226, 607–631, DOI 10.1090/S0025-5718-99-01013-MR1609607 (99i:78002)
- D. N. Arnold, R. S. Falk, and R. Winther, Multigrid in H(div) and H(curl), Numer. Math. 85 (2000), no. 2, 197–217, DOI 10.1007/PL00005386. MR1754719 (2001d:65161)
- D. N. Arnold, R. S. Falk, and R. Winther, Multigrid preconditioning in H(div) on non-convex polygons, Comput. Appl. Math. 17 (1998), no. 3, 303–31MR1687885 (2000k:65224)
- J. Bey, Tetrahedral grid refinement (English, with English and German summaries), Computing 55 (1995), no. 4, 355–378, DOI 10.1007/BF02238487. MR1370107 (96i:65105)
- F. Bornemann and H. Yserentant, A basic norm equivalence for the theory of multilevel methods, Numer. Math. 64 (1993), no. 4, 455–476, DOI 10.1007/BF01388699. MR1213412 (94b:65155)
- J. 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. MR1247694 (95b:65002)
- J. H. Bramble, J. E. Pasciak, and A. H. Schatz, The construction of preconditioners for elliptic problems by substructuring. IV, Math. Comp. 53 (1989), no. 187, 1–24, DOI 10.2307/2008346. MR970699 (89m:65098)
- F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR1115205 (92d:65187)
- 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, DOI 10.1002/1099-1476(20010110)24:1 9::AID-MMA191 3.0.CO;2-2. MR1809491 (2002b:78024)
- 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, DOI 10.1002/1099-1476(20010110)24:1¡9::AIDMMA191¿3.0.CO;2-2. MR1809492 (2002b:78025)
- A. Buffa, M. Costabel, and D. Sheen, On traces for H(curl, Ω) in Lipschitz domains, J. Math. Anal. Appl. 276 (2002), no. 2, 845–867, DOI 10.1016/S0022-247X(02)00455-9. MR1944792 (2004i:35045)
- H. Chen, R. H. W. Hoppe, and X. Xu, Uniform convergence of local multigrid methods for the time-harmonic Maxwell equation, ESAIM Math. Model. Numer. Anal. 47 (2013), no. 1, 125–147, DOI 10.1051/m2an/2012023. MR2968698
- 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. MR1827403
- S. H. Christiansen and R. Winther, Smoothed projections in finite element exterior calculus, Math. Comp. 77 (2008), no. 262, 813–829, DOI 10.1090/S0025-5718-07-02081-9. MR2373181 (2009a:65310)
- C. R. Dohrmann and O. B. Widlund, An iterative substructuring algorithm for twodimensional problems in H(curl), SIAM J. Numer. Anal. 50 (2012), no. 3, 1004–1028, DOI 10.1137/100818145. MR2970732
- C. Dohrmann and O. Widlund, Some recent tools and a BDDC algorithm for 3D problems in H(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. MR972511 (90b:65200)
- A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Applied Mathematical Sciences, vol. 159, Springer-Verlag, New York, 2004. MR2050138 (2005d:65002)
- V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. MR851383 (88b:65129)
- P. Grisvard, Singularities in Boundary Value Problems, Recherches en Mathematiques Appliquees [Research in Applied Mathematics], vol. 22, Masson, Paris; Springer-Verlag, Berlin, 1992. MR1173209 (93h:35004)
- N. Heuer, E. P. Stephan, and T. Tran, Multilevel additive Schwarz method for the h-p version of the Galerkin boundary element method, Math. Comp. 67 (1998), no. 222, 501–518, DOI 10.1090/S0025-5718-98-00926-0. MR1451325 (98g:65108)
- R. Hiptmair, Multigrid method for H(div) in three dimensions, Electron. Trans. Numer. Anal. 6 (1997), no. Dec., 133–152. Special issue on multilevel methods (Copper Mountain, CO, 1997). MR1615161 (99c:65232)
- R. Hiptmair, Multigrid method for Maxwell’s equations, SIAM J. Numer. Anal. 36 (1999), no. 1, 204–225, DOI 10.1137/S0036142997326203. MR1654571 (99j:65229)
- R. Hiptmair, Finite elements in computational electromagnetism, Acta Numer. 11 (2002), 237–339, DOI 10.1017/S0962492902000041. MR2009375 (2004k:78028)
- R. Hiptmair, Analysis of multilevel methods for eddy current problems, Math. Comp. 72 (2003), no. 243, 1281–1303, DOI 10.1090/S0025-5718-02-01468-0. MR1972736 (2004c:78033)
- R. Hiptmair and C. Jerez-Hanckes, Multiple traces boundary integral formulation for Helmholtz transmission problems, Adv. Comput. Math. 37 (2012), no. 1, 39–91, DOI 10.1007/s10444-011-9194-3. MR2927645
- R. Hiptmair and S. Mao, Stable multilevel splittings of boundary edge element spaces, BIT 52 (2012), no. 3, 661–685, DOI 10.1007/s10543-012-0369-1. MR2965296
- R. Hiptmair and W.-Y. Zheng, Local multigrid in H(curl), Tech. Rep. 2007-03, SAM, ETH Zurich, Switzerland, March 2007. http://arxiv.org/abs/0901.0764.
- R. Hiptmair and W. Zheng, Local multigrid in H(curl), J. Comput. Math. 27 (2009), no. 5, 573–603, DOI 10.4208/jcm.2009.27.5.012. MR2536903 (2010h:65249)
- 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.
- Q. Hu and J. Zou, Substructuring preconditioners for saddle-point problems arising from Maxwell’s equations in three dimensions, Math. Comp. 73 (2004), no. 245, 35–61, DOI 10.1090/S0025-5718-03-01541-2. MR2034110 (2004m:65197)
- I. Kossaczky, A recursive approach to local mesh refinement in two and three dimensions, J. Comput. Appl. Math. 55 (1994), no. 3, 275–288, DOI 10.1016/0377-0427(94)90034-5. MR1329875 (95m:65207)
- J. Lions and F. Magenes, Nonhomogeneous boundary value problems and applications, Springer–Verlag, Berlin, 1972.
- W. McLean, Strongly elliptic systems and boundary integral equations, Cambridge University Press, Cambridge, 2000. MR1742312 (2001a:35051)
- 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, DOI 10.1023/A:1018944530343. MR1732138 (2000k:65236)
- J.-C. Nedelec, Mixed finite elements in R3, Numer. Math. 35 (1980), no. 3, 315–341, DOI 10.1007/BF01396415. MR592160 (81k:65125)
- P. Oswald, On function spaces related to finite element approximation theory (English, with German and Russian summaries), Z. Anal. Anwendungen 9 (1990), no. 1, 43–64. MR1063242 (91g:65246)
- 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.
- P. Oswald, Multilevel finite element approximation, Teubner Skripten zur Numerik. [Teubner Scripts on Numerical Mathematics], B. G. Teubner, Stuttgart, 1994. Theory and applications. MR1312165 (95k:65110)
- S. A. Sauter and C. Schwab, Boundary Element Methods, Springer Series in Computational Mathematics, vol. 39, Springer-Verlag, Berlin, 2011. Translated and expanded from the 2004 German original. MR2743235 (2011i:65003)
- L. R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp. 54 (1990), no. 190, 483–493, DOI 10.2307/2008497. MR1011446 (90j:65021)
- L. Tartar, An Introduction to Sobolev Spaces and Interpolation Spaces, Lecture Notes of the Unione Matematica Italiana, vol. 3, Springer, Berlin; UMI, Bologna, 2007. MR2328004 (2008g:46055)
- A. Toselli and O. Widlund, Domain Decomposition Methods—Algorithms and Theory, Springer Series in Computational Mathematics, vol. 34, Springer-Verlag, Berlin, 2005. MR2104179 (2005g:65006)
- J. Xu, Iterative methods by space decomposition and subspace correction, SIAM Rev. 34 (1992), no. 4, 581–613, DOI 10.1137/1034116. MR1193013 (93k:65029)
- J. Xu and J. Zou, Some nonoverlapping domain decomposition methods, SIAM Rev. 40 (1998), no. 4, 857–914, DOI 10.1137/S0036144596306800. MR1659681 (99m:65241)
- J. Xu, L. Chen, and R. H. Nochetto, Optimal multilevel methods for H(grad), H(curl), and H(div) systems on graded and unstructured grids, Multiscale, nonlinear and adaptive approximation, Springer, Berlin, 2009, pp. 599–659, DOI 10.1007/978-3-642-03413-8 14. MR2648382 (2011k:65178)
- X. Zhang, Multilevel Schwarz methods, Numer. Math. 63 (1992), no. 4, 521–539, DOI 10.1007/BF01385873. MR1189535 (93h:65047)

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