Polynomial extension operators. Part III
HTML articles powered by AMS MathViewer
- by L. Demkowicz, J. Gopalakrishnan and J. Schöberl PDF
- Math. Comp. 81 (2012), 1289-1326 Request permission
Abstract:
In this concluding part of a series of papers on tetrahedral polynomial extension operators, the existence of a polynomial extension operator in the Sobolev space $\boldsymbol {H}(\mathrm {div})$ is proven constructively. Specifically, on any tetrahedron $K$, given a function $w$ on the boundary $\partial K$ that is a polynomial on each face, the extension operator applied to $w$ gives a vector function whose components are polynomials of at most the same degree in the tetrahedron. The vector function is an extension in the sense that the trace of its normal component on the boundary $\partial K$ coincides with $w$. Furthermore, the extension operator is continuous from $H^{-1/2}(\partial K)$ into $\boldsymbol {H}(\mathrm {div},K)$. The main application of this result and the results of this series of papers is the existence of commuting projectors with good $hp$-approximation properties.References
- Mark Ainsworth and Leszek Demkowicz, Explicit polynomial preserving trace liftings on a triangle, Math. Nachr. 282 (2009), no. 5, 640–658. MR 2523203, DOI 10.1002/mana.200610762
- 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
- William M. Boothby, An introduction to differentiable manifolds and Riemannian geometry, 2nd ed., Pure and Applied Mathematics, vol. 120, Academic Press, Inc., Orlando, FL, 1986. MR 861409
- Franco Brezzi, Jim Douglas Jr., and L. D. Marini, Two families of mixed finite elements for second order elliptic problems, Numer. Math. 47 (1985), no. 2, 217–235. MR 799685, DOI 10.1007/BF01389710
- 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
- Michel Cessenat, Mathematical methods in electromagnetism, Series on Advances in Mathematics for Applied Sciences, vol. 41, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. Linear theory and applications. MR 1409140, DOI 10.1142/2938
- M. Costabel, M. Dauge, and L. Demkowicz, Polynomial extension operators for $H^1$, $H(\rm curl)$ and $H(\rm div)$-spaces on a cube, Math. Comp. 77 (2008), no. 264, 1967–1999. MR 2429871, DOI 10.1090/S0025-5718-08-02108-X
- Daniele Boffi, Franco Brezzi, Leszek F. Demkowicz, Ricardo G. Durán, Richard S. Falk, and Michel Fortin, Mixed finite elements, compatibility conditions, and applications, Lecture Notes in Mathematics, vol. 1939, Springer-Verlag, Berlin; Fondazione C.I.M.E., Florence, 2008. Lectures given at the C.I.M.E. Summer School held in Cetraro, June 26–July 1, 2006; Edited by Boffi and Lucia Gastaldi. MR 2459075, DOI 10.1007/978-3-540-78319-0
- L. Demkowicz and I. Babuška, $p$ interpolation error estimates for edge finite elements of variable order in two dimensions, SIAM J. Numer. Anal. 41 (2003), no. 4, 1195–1208. MR 2034876, DOI 10.1137/S0036142901387932
- L. Demkowicz and A. Buffa, $H^1$, $H(\textrm {curl})$ and $H(\textrm {div})$-conforming projection-based interpolation in three dimensions. Quasi-optimal $p$-interpolation estimates, Comput. Methods Appl. Mech. Engrg. 194 (2005), no. 2-5, 267–296. MR 2105164, DOI 10.1016/j.cma.2004.07.007
- 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
- Leszek Demkowicz, Jayadeep Gopalakrishnan, and Joachim Schöberl, Polynomial extension operators. II, SIAM J. Numer. Anal. 47 (2009), no. 5, 3293–3324. MR 2551195, DOI 10.1137/070698798
- 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
- Jayadeep Gopalakrishnan and Leszek F. Demkowicz, Quasioptimality of some spectral mixed methods, J. Comput. Appl. Math. 167 (2004), no. 1, 163–182. MR 2059719, DOI 10.1016/j.cam.2003.10.001
- Benqi Guo and Jianming Zhang, Stable and compatible polynomial extensions in three dimensions and applications to the $p$ and $h$-$p$ finite element method, SIAM J. Numer. Anal. 47 (2009), no. 2, 1195–1225. MR 2485450, DOI 10.1137/070688006
- M. R. Hestenes, Extension of the range of a differentiable function, Duke Math. J. 8 (1941), 183–192. MR 3434
- R. Hiptmair, Canonical construction of finite elements, Math. Comp. 68 (1999), no. 228, 1325–1346. MR 1665954, DOI 10.1090/S0025-5718-99-01166-7
- 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
- Rafael Muñoz-Sola, Polynomial liftings on a tetrahedron and applications to the $h$-$p$ version of the finite element method in three dimensions, SIAM J. Numer. Anal. 34 (1997), no. 1, 282–314. MR 1445738, DOI 10.1137/S0036142994267552
- P.-A. Raviart and J. M. Thomas, Primal hybrid finite element methods for $2$nd order elliptic equations, Math. Comp. 31 (1977), no. 138, 391–413. MR 431752, DOI 10.1090/S0025-5718-1977-0431752-8
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
Additional Information
- L. Demkowicz
- Affiliation: Institute of Computational Engineering and Sciences, 1 University Station, C0200, The University of Texas at Austin, Texas 78712
- Email: leszek@ices.utexas.edu
- J. Gopalakrishnan
- Affiliation: University of Florida, Department of Mathematics, Gainesville, Florida 32611–8105
- MR Author ID: 661361
- Email: jayg@ufl.edu
- J. Schöberl
- Affiliation: Technische Universität Wein, Wiedner Hauptstrasse 8-10, Wein 1040, Austria
- Email: joachim.schoeberl@tuwien.ac.at
- Received by editor(s): January 3, 2010
- Received by editor(s) in revised form: February 23, 2011
- Published electronically: September 20, 2011
- Additional Notes: This work was supported in part by the National Science Foundation under grants DMS-1014817, the Johann Radon Institute for Computational and Applied Mathematics (RICAM), and the FWF-Start-Project Y-192 “hp-FEM”
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 81 (2012), 1289-1326
- MSC (2010): Primary 46E35, 46E40; Secondary 41A10, 65D05, 65L60
- DOI: https://doi.org/10.1090/S0025-5718-2011-02536-6
- MathSciNet review: 2904580