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On the constants in $$hp$$-finite element trace inverse inequalities. (English) Zbl 1038.65116
From the authors’ abstract: We derive inverse trace inequalities for $$hp$$-finite elements. Utilizing orthogonal polynomials, we show how to derive explicit expressions for the constants when considering triangular and tetrahedral elements. We also discuss how to generalize this technique to the general $$d$$-simplex.

##### MSC:
 65N15 Error bounds for boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
##### Keywords:
$$hp$$-finite element; trace inverse inequality
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##### References:
 [1] Ciarlet, P.G., The finite-element method for elliptic problems, (1978), North-Holland The Netherlands · Zbl 0445.73043 [2] Dubiner, M., Spectral methods on triangles and other domains, J. sci. comput., 6, 345-390, (1991) · Zbl 0742.76059 [3] Harari, I.; Hughes, T.J.R., What are C and h?: inequalities for the analysis and design of finite element methods, Comput. methods appl. mech. engrg., 97, 157-192, (1992) · Zbl 0764.73083 [4] Koornwinder, T., Two-variable analogues of the classical orthogonal polynomials, (), 435-495 [5] Owens, R.G., Spectral approximations on the triangle, Proc. roy. soc. lond. A, 454, 857-872, (1998) · Zbl 0915.35077 [6] Proriol, J., Sur une famille de polynomes à deux variables orthogonaux dans un triangle, C. R. acad. sci. Paris, 257, 2459-2461, (1957) · Zbl 0080.05204 [7] Schwab, C., p- and hp-finite element methods. theory and applications in solid and fluid mechanics, Numerical mathematics and scientific computation, (1998), Clarendon Press Oxford · Zbl 0910.73003 [8] S.J. Sherwin, Triangular and tetrahedral spectral/hp element methods for fluid dynamics, Ph.D. thesis, Princeton University, 1995 [9] E. Suli, C. Schwab, P. Houston, hp-DGFEM for partial differential equations with nonnegative characteristic form, in: B. Cockburn, G.E. Karniadakis, C.W. Shu (Eds.), Discontinuous Galerkin Methods. Theory, Computation and Applications, in: Lecture Notes in Computational Science and Engineering, vol. 11, Springer-Verlag, Berlin, 2000, pp. 221-230 · Zbl 0946.65102 [10] Szegö, G., Orthogonal polynomials, Colloquium publications, vol. 23, (1939), American Mathematical Society Providence, RI · JFM 65.0278.03 [11] R. Verfürth, On the constants in some inverse inequalities for finite element functions, Ruhr-Universität Bochum, preprint
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