A new H(div)-conforming p-interpolation operator in two dimensions
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 45 (2011) no. 2, pp. 255-275.

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.

DOI : 10.1051/m2an/2010039
Classification : 65N15, 41A10, 65N38
Mots clés : p-interpolation, error estimation, maxwell's equations, boundary element method
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     title = {A new {H(div)-conforming} $p$-interpolation operator in two dimensions},
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Bespalov, Alexei; Heuer, Norbert. A new H(div)-conforming $p$-interpolation operator in two dimensions. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 45 (2011) no. 2, pp. 255-275. doi : 10.1051/m2an/2010039. http://www.numdam.org/articles/10.1051/m2an/2010039/

[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. | Numdam | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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). | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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). | MR | Zbl

[10] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics 15. Springer-Verlag, New York (1991). | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[15] M. Costabel and M. Dauge, Singularities of electromagnetic fields in polyhedral domains. Arch. Rational Mech. Anal. 151 (2000) 221-276. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | Zbl

[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. | MR | Zbl

[20] M.R. Dorr, The approximation theory for the p-version of the finite element method. SIAM J. Numer. Anal. 21 (1984) 1180-1207. | MR | Zbl

[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. | MR | Zbl

[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] J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications I. Springer-Verlag, New York (1972). | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

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