The lifting of polynomial traces revisited. (English) Zbl 1227.46026

Let \(\Theta\) be the rectangle \((-1,1)\times(0,1)\) and let \(H^1(\Theta)\) be the Sobolev space of order 1 on \(\Theta\). Let \(\Lambda\) be the lower edge \((-1,1)\times\{0\}\) of \(\Theta\) and let \(H^{1/2}(\Lambda)\) be the Sobolev space of order \(1/2\) on \(\Lambda\). The authors describe how to lift continuously elements of \(H^{1/2}(\Lambda)\) (“traces”) to elements of \(H^1(\Theta)\) in a polynomial preserving way: polynomial functions on \(\Lambda\) with degree \(N\) are lifted to polynomial functions on \(\Theta\) with degree at most \(N\) in each variable. The main tool is the Lebesgue differentiation theorem that yields a lifting from \(\Lambda\) into the equilateral triangle \(\mathcal T\) with base \(\Lambda\), combined with a homography that maps \(\mathcal T\cap\Theta\) onto \(\Theta\).
Then the authors describe how to lift continuously elements of \(H^{1/2}(\Lambda)\) that vanish in the endpoints of \(\Lambda\) (“flat” traces) to elements of \(H^1(\Theta)\) that vanish on the vertical edges of \(\Theta\) in a polynomial preserving way. Their method, a division by \((1-x)(1+x)\) followed by a lifting and by a multiplication with \((1-x)(1+x)\), requires the use of weighted Sobolev spaces; it has already been outlined in their note [C. R. Acad.Sci., Paris, Sér.I 315, No.3, 333–338 (1992; Zbl 0755.65103)]; see M. Ainsworth and L. Demkowicz [Math.Nachr.282, No.5, 640–658 (2009; Zbl 1175.46019)] for another method.
The article ends with two numerical illustrations. (1) The evaluation of the \(H^{1/2}(\Lambda)\)-norm of Legendre polynomials \(L_n\), which is known to grow like \(\sqrt{\log n}\). (2) The evaluation of the norm of the orthogonal projection \(\pi_N\) from \(L^2(\Lambda)\) onto the space of polynomial functions of degree at most \(N\) when acting on \(H^{1/2}(\Lambda)\): interpolation yields that it is bounded by \(CN^{1/4}\); numerical evidence leads the authors to conjecture that it grows in fact like \(\log N\).


46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
47B38 Linear operators on function spaces (general)
46B70 Interpolation between normed linear spaces
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[1] Yves Achdou, Yvon Maday, and Olof B. Widlund, Iterative substructuring preconditioners for mortar element methods in two dimensions, SIAM J. Numer. Anal. 36 (1999), no. 2, 551 – 580. · Zbl 0931.65110 · doi:10.1137/S0036142997321005
[2] Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. · Zbl 0314.46030
[3] M. Ainsworth, L. Demkowicz, Explicit polynomial preserving trace liftings on a triangle, Math. Nachr. 282 (2009), 640-658. · Zbl 1175.46019
[4] I. Babuška and Manil Suri, The \?-\? 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). · Zbl 0623.65113
[5] Faker Ben Belgacem, Polynomial extensions of compatible polynomial traces in three dimensions, Comput. Methods Appl. Mech. Engrg. 116 (1994), no. 1-4, 235 – 241. ICOSAHOM’92 (Montpellier, 1992). · Zbl 0826.65097 · doi:10.1016/S0045-7825(94)80028-6
[6] C. Bernardi, T. Chacón Rebollo, E. Chacón Vera, D. Franco Coronil, A posteriori error analysis for two non-overlapping domain decomposition techniques, Appl. Num. Math. 59 (2009), 1214-1236. · Zbl 1166.65053
[7] Christine Bernardi, Monique Dauge, and Yvon Maday, Relèvements de traces préservant les polynômes, C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), no. 3, 333 – 338 (French, with English and French summaries). · Zbl 0755.65103
[8] C. Bernardi, M. Dauge, Y. Maday, Polynomials in Sobolev Spaces and Application to the Mortar Spectral Element Method, in preparation. · Zbl 0802.41002
[9] Christine Bernardi and Yvon Maday, Relèvement polynomial de traces et applications, RAIRO Modél. Math. Anal. Numér. 24 (1990), no. 5, 557 – 611 (French, with English summary). · Zbl 0707.65077
[10] Christine Bernardi and Yvon Maday, Approximations spectrales de problèmes aux limites elliptiques, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 10, Springer-Verlag, Paris, 1992 (French, with French summary). · Zbl 0773.47032
[11] C. Canuto and D. Funaro, The Schwarz algorithm for spectral methods, SIAM J. Numer. Anal. 25 (1988), no. 1, 24 – 40. · Zbl 0642.65076 · doi:10.1137/0725003
[13] Ben Qi Guo, The \?-\? version of the finite element method for elliptic equations of order 2\?, Numer. Math. 53 (1988), no. 1-2, 199 – 224. · Zbl 0628.65102 · doi:10.1007/BF01395885
[14] J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968 (French). · Zbl 0212.43801
[15] Yvon Maday, Relèvements de traces polynomiales et interpolations hilbertiennes entre espaces de polynômes, C. R. Acad. Sci. Paris Sér. I Math. 309 (1989), no. 7, 463 – 468 (French, with English summary). · Zbl 0678.46027
[16] Rafael Muñoz-Sola, Polynomial liftings on a tetrahedron and applications to the \?-\? version of the finite element method in three dimensions, SIAM J. Numer. Anal. 34 (1997), no. 1, 282 – 314. · Zbl 0871.46016 · doi:10.1137/S0036142994267552
[17] Luca F. Pavarino and Olof B. Widlund, A polylogarithmic bound for an iterative substructuring method for spectral elements in three dimensions, SIAM J. Numer. Anal. 33 (1996), no. 4, 1303 – 1335. · Zbl 0856.41007 · doi:10.1137/S0036142994265176
[18] Alfio Quarteroni and Alberto Valli, Domain decomposition methods for partial differential equations, Numerical Mathematics and Scientific Computation, The Clarendon Press, Oxford University Press, New York, 1999. Oxford Science Publications. · Zbl 0931.65118
[19] Andrea Toselli and Olof Widlund, Domain decomposition methods — algorithms and theory, Springer Series in Computational Mathematics, vol. 34, Springer-Verlag, Berlin, 2005. · Zbl 1069.65138
[20] H. Triebel, Interpolation theory, function spaces, differential operators, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978. Hans Triebel, Interpolation theory, function spaces, differential operators, North-Holland Mathematical Library, vol. 18, North-Holland Publishing Co., Amsterdam-New York, 1978.
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