A new H(div)-conforming p-interpolation operator in two dimensions

Alexei Bespalov; Norbert Heuer

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2011)

  • Volume: 45, Issue: 2, page 255-275
  • ISSN: 0764-583X

Abstract

top
In this paper we construct a new H(div)-conforming projection-based p-interpolation operator that assumes only Hr(K) ∩ 𝐇 ˜ -1/2(div, K)-regularity (r > 0) on the reference element (either triangle or square) K. We show that this operator is stable with respect to polynomial degrees and satisfies the commuting diagram property. We also establish an estimate for the interpolation error in the norm of the space 𝐇 ˜ -1/2(div, K), which is closely related to the energy spaces for boundary integral formulations of time-harmonic problems of electromagnetics in three dimensions.

How to cite

top

Bespalov, Alexei, and Heuer, Norbert. "A new H(div)-conforming p-interpolation operator in two dimensions." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 45.2 (2011): 255-275. <http://eudml.org/doc/273207>.

@article{Bespalov2011,
abstract = {In this paper we construct a new H(div)-conforming projection-based p-interpolation operator that assumes only Hr(K) $\cap $$\{\bf \tilde\{H\}\}$-1/2(div, K)-regularity (r &gt; 0) on the reference element (either triangle or square) K. We show that this operator is stable with respect to polynomial degrees and satisfies the commuting diagram property. We also establish an estimate for the interpolation error in the norm of the space $\{\bf \tilde\{H\}\}$-1/2(div, K), which is closely related to the energy spaces for boundary integral formulations of time-harmonic problems of electromagnetics in three dimensions.},
author = {Bespalov, Alexei, Heuer, Norbert},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {p-interpolation; error estimation; maxwell's equations; boundary element method; -interpolation; Maxwell's equations},
language = {eng},
number = {2},
pages = {255-275},
publisher = {EDP-Sciences},
title = {A new H(div)-conforming p-interpolation operator in two dimensions},
url = {http://eudml.org/doc/273207},
volume = {45},
year = {2011},
}

TY - JOUR
AU - Bespalov, Alexei
AU - Heuer, Norbert
TI - A new H(div)-conforming p-interpolation operator in two dimensions
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2011
PB - EDP-Sciences
VL - 45
IS - 2
SP - 255
EP - 275
AB - In this paper we construct a new H(div)-conforming projection-based p-interpolation operator that assumes only Hr(K) $\cap $${\bf \tilde{H}}$-1/2(div, K)-regularity (r &gt; 0) on the reference element (either triangle or square) K. We show that this operator is stable with respect to polynomial degrees and satisfies the commuting diagram property. We also establish an estimate for the interpolation error in the norm of the space ${\bf \tilde{H}}$-1/2(div, K), which is closely related to the energy spaces for boundary integral formulations of time-harmonic problems of electromagnetics in three dimensions.
LA - eng
KW - p-interpolation; error estimation; maxwell's equations; boundary element method; -interpolation; Maxwell's equations
UR - http://eudml.org/doc/273207
ER -

References

top
  1. [1] I. Babuška and M. Suri, The h – p version of the finite element method with quasiuniform meshes. RAIRO Modél. Math. Anal. Numér.21 (1987) 199–238. Zbl0623.65113MR896241
  2. [2] I. Babuška, A. Craig, J. Mandel and J. Pitkäranta, Efficient preconditioning for the p-version finite element method in two dimensions. SIAM J. Numer. Anal.28 (1991) 624–661. Zbl0754.65083MR1098410
  3. [3] A. Bespalov and N. Heuer, Optimal error estimation for H(curl)-conforming p-interpolation in two dimensions. SIAM J. Numer. Anal.47 (2009) 3977–3989. Zbl1203.65229MR2576528
  4. [4] A. Bespalov and N. Heuer, Natural p-BEM for the electric field integral equation on screens. IMA J. Numer. Anal. (2010) DOI:10.1093/imanum/drn072. Zbl1231.78001MR2670107
  5. [5] A. Bespalov and N. Heuer, Thehp – BEMS with quasi-uniform meshes for the electric field integral equation on polyhedral surfaces: a priori error analysis. Appl. Numer. Math.60 (2010) 705–718. Zbl1204.78030MR2646471
  6. [6] A. Bespalov, N. Heuer and R. Hiptmair, Convergence of the natural hp-BEM for the electric field integral equation on polyhedral surfaces. arXiv:0907.5231 (2009). Zbl1223.65083MR2684345
  7. [7] D. Boffi, L. Demkowicz and M. Costabel, Discrete compactness for the p and hp 2D edge finite elements. Math. Models Methods Appl. Sci.13 (2003) 1673–1687. Zbl1056.65108MR2024467
  8. [8] D. Boffi, M. Costabel, M. Dauge and L. Demkowicz, Discrete compactness for the hp version of rectangular edge finite elements. SIAM J. Numer. Anal.44 (2006) 979–1004. Zbl1122.65110MR2231852
  9. [9] D. Boffi, M. Costabel, M. Dauge, L. Demkowicz and R. Hiptmair, Discrete compactness for the p -version of discrete differential forms. arXiv:0909.5079 (2009). Zbl1230.78025MR2764424
  10. [10] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics 15. Springer-Verlag, New York (1991). Zbl0788.73002MR1115205
  11. [11] A. Buffa, Remarks on the discretization of some noncoercive operator with applications to heterogeneous Maxwell equations. SIAM J. Numer. Anal.43 (2005) 1–18. Zbl1128.78010MR2177953
  12. [12] A. Buffa and S.H. Christiansen, The electric field integral equation on Lipschitz screens: definitions and numerical approximation. Numer. Math.94 (2003) 229–267. Zbl1027.65188MR1974555
  13. [13] A. Buffa and P. Ciarlet, Jr., On traces for functional spaces related to Maxwell's equations, Part II: Hodge decompositions on the boundary of Lipschitz polyhedra and applications. Math. Methods Appl. Sci. 24 (2001) 31–48. Zbl0976.46023MR1809492
  14. [14] A. Buffa, R. Hiptmair, T. von Petersdorff and C. Schwab, Boundary element methods for Maxwell transmission problems in Lipschitz domains. Numer. Math.95 (2003) 459–485. Zbl1071.65160MR2012928
  15. [15] M. Costabel and M. Dauge, Singularities of electromagnetic fields in polyhedral domains. Arch. Rational Mech. Anal.151 (2000) 221–276. Zbl0968.35113MR1753704
  16. [16] M. Costabel and A. McIntosh, On Bogovskiĭ and regularized Poincaré integral operators for de Rham complexes on Lipschitz domains. Math. Z.265 (2010) 297–320. Zbl1197.35338MR2609313
  17. [17] M. Costabel, M. Dauge and L. Demkowicz, Polynomial extension operators for H1, H(curl) and H(div)-spaces on a cube. Math. Comp. 77 (2008) 1967–1999. Zbl1198.65235MR2429871
  18. [18] L. Demkowicz, Polynomial exact sequences and projection-based interpolation with applications to Maxwell equations, in Mixed Finite Elements, Compatibility Conditions and Applications, D. Boffi, F. Brezzi, L. Demkowicz, R. Duran, R. Falk and M. Fortin Eds., Lect. Notes in Mathematics 1939, Springer-Verlag, Berlin (2008) 101–158. Zbl1143.78366
  19. [19] 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) 1195–1208. Zbl1067.78016MR2034876
  20. [20] M.R. Dorr, The approximation theory for the p-version of the finite element method. SIAM J. Numer. Anal.21 (1984) 1180–1207. Zbl0572.65074MR765514
  21. [21] N. Heuer, Additive Schwarz method for the p-version of the boundary element method for the single layer potential operator on a plane screen. Numer. Math.88 (2001) 485–511. Zbl1003.65137MR1835468
  22. [22] R. Hiptmair, Discrete compactness for the p-version of tetrahedral edge elements. Seminar for Applied Mathematics, ETH Zürich, Switzerland (2008) arXiv:0901.0761. 
  23. [23] J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications I. Springer-Verlag, New York (1972). Zbl0223.35039MR350177
  24. [24] S.E. Mikhailov, About traces, extensions and co-normal derivative operators on Lipschitz domains, in Integral Methods in Science and Engineering: Techniques and Applications, C. Constanda and S. Potapenko Eds., Birkhäuser, Boston (2008) 149–160. Zbl1217.35053MR2389522
  25. [25] R.E. Roberts and J.-M. Thomas, Mixed and hybrid methods, in Handbook of Numerical Analysis II, P.G. Ciarlet and J.-L. Lions Eds., Amsterdam, North-Holland (1991) 523–639. Zbl0875.65090MR1115239

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.