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finite element; triangular elements; superconvergence; post-processing; averaged gradient; elliptic systems
A simple superconvergent scheme for the derivatives of finite element solution is presented, when linear triangular elements are employed to solve second order elliptic systems with boundary conditions of Newton's or Neumann's type. For bounded plane domains with smooth boundary the local $O(h^{3/2})$-superconvergence of the derivatives in the $L^2$-norm is proved. The paper is a direct continuations of [2], where an analogous problem with Dirichlet's boundary conditions is treated.
[1] P. G. Ciarlet: The finite element method for elliptic problems. North-Holland, Amsterdam, New York, Oxford, 1978. MR 0520174 | Zbl 0383.65058
[2] I. Hlaváček M. Křížek: On a superconvergent finite element scheme for elliptic systems, I. Dirichlet boundary conditions. Apl. Mat. 32 (1987), 131 -154. MR 0885758
[3] I. Hlaváček J. Nečas: On inequalities of Korn's type. Arch. Rational Mech. Anal. 36 (1970), 305-311, 312-334. DOI 10.1007/BF00249518
[4] M. Křížek P. Neittaanmäki: Superconvergence phenomenon in the finite element method arising from averaging gradients. Numer. Math. 45 (1984), 105-116. DOI 10.1007/BF01379664 | MR 0761883
[5] L. A. Oganesjan L. A. Ruchovec: Variational-difference methods for the solution of elliptic equations. Izd. Akad. Nauk Armjanskoi SSR, Jerevan, 1979.
[6] J. Nečas: Les méthodes directes en théorie des équations elliptiques. Academia, Prague, 1967. MR 0227584
[7] M. Zlámal: Some superconvergence results in the finite element method. Mathematical Aspects of Finite Element Methods (Proc. Conf., Rome, 1975). Springer-Verlag, Berlin, Heidelberg, New York, 1977, 353-362. MR 0488863
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