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de Rham diagram for $$hp$$ finite element spaces. (English) Zbl 0955.65084
Summary: We prove that the $$hp$$ finite elements for $${\mathbf H}(\text{curl})$$ spaces, introduced by L. Demkowicz and L. Vardapetyan [Comput. Methods Appl. Mech. Eng. 152, No. 1-2, 103-124 (1998; Zbl 0994.78011)], fit into a general de Rahm diagram involving $$hp$$ approximations. The corresponding interpolation operators generalize the notion of $$hp$$ interpolation introduced by J. T. Oden, L. Demkowicz, W. Rachowicz and T. A. Westermann [ibid. 77, No. 1/2, 113-180 (1989; Zbl 0723.73075)] and are different from the classical operators of J. C. Nedelec [Numer. Math. 50, 57-81 (1986; Zbl 0625.65107)] and of P. A. Raviart and J. M. Thomas [Math. Comput. 31, 391-413 (1977; Zbl 0364.65082)].

##### MSC:
 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations
HP90; 2Dhp90
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##### References:
 [1] Demkowicz, L.; Vardapetyan, L., Modeling of electromagnetic absorption/scattering problems using hp-adaptive finite elements, Computer methods in applied mechanics and engineering, 152, 1/2, 103-124, (1998) · Zbl 0994.78011 [2] Oden, J.T.; Demkowicz, L.; Rachowicz, W.; Westermann, T.A., Towards a universal h-p adaptive finite element strategy, part 2. A posteriori error estimation, Comp. meth. appl. meth. eng., 77, 113-180, (1989) · Zbl 0723.73075 [3] Brezzi, F.; Fortin, M., Mixed and hybrid finite element methods, (1991), Springer-Verlag Berlin · Zbl 0788.73002 [4] Bossavit, A., Un nouveau point de vue sur LES éléments finis mixtes, Matapli (bulletin de la société de mathématiques appliquëes et industrielles), 23-35, (1989) [5] Demkowicz, L.; Gerdes, K.; Schwab, C.; Bajer, A.; Walsh, T., HP90: A general and flexible Fortran 90 hp-FE code, Computing and visualization in science, 1, 145-163, (1998) · Zbl 0912.68014 [6] L. Demkowicz, T. Walsh, K. Gerdes and A. Bajer, 2D hp-adaptive finite element package. Fortran 90 implementation (2Dhp90), TICAM Report 98-14, The University of Texas at Austin, Austin, TX 78712. · Zbl 0912.68014 [7] Rachowicz, W.; Demkowicz, L., A two-dimensional hp-adaptive finite element package for electromagnetics, Computer methods in applied mechanics and engineering, TICAM report 98-15, (July 1998), (to appear) [8] W. Rachowicz and L. Demkowicz, A three-dimensional hp-adaptive finite element package for electromagnetics, (in preparation). · Zbl 0994.78012 [9] Nedelec, J.C., A new family of mixed finite elements in $$R$$^{3}, Numerische Mathematik, 50, 57-81, (1986) · Zbl 0625.65107 [10] Nedelec, J.C., Mixed finite elements in $$R$$^{3}, Numerische Mathematik, 35, 315-341, (1980) · Zbl 0419.65069 [11] Vardapetyan, L.; Demkowicz, L., hp-adaptive finite elements in electromagnetics, Computers methods in applied mechanics and engineering, 169, 331-344, (1999) · Zbl 0956.78013 [12] Raviart, P.A.; Thomas, J.M., Primal hybrid finite element methods for 2^{nd} order elliptic equations, Mathematics of computation, 31, 138, 391-413, (1977) · Zbl 0364.65082
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