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Keywords:
post-processing; averaged gradient; Galerkin method; system; finite elements; superconvergence; global estimate; elliptic systems
post-processing; averaged gradient; Galerkin method; system; finite elements; superconvergence; global estimate; elliptic systems
Summary:
Second order elliptic systems with Dirichlet boundary conditions are solved by means of affine finite elements on regular uniform triangulations. A simple averagign scheme is proposed, which implies a superconvergence of the gradient. For domains with enough smooth boundary, a global estimate $O(h^{3/2})$ is proved in the $L^2$-norm. For a class of polygonal domains the global estimate $O(h^2)$ can be proven.
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[1] S. Agmon A. Douglis L. Nirenberg: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II. Comm. Pure Appl. Math. 17 (1964), 35-92. DOI 10.1002/cpa.3160170104 | MR 0162050
[2] A. B. Andreev: Superconvergence of the gradient for linear triangle elements for elliptic and parabolic equations. C. R. Acad. Bulgare Sci. 37 (1984), 293 - 296. MR 0758156 | Zbl 0575.65106
[3] I. Babuška A. Miller: The post-processing technique in the finite element method. Parts I-III, Internat. J. Numer. Methods Engrg. 20 (1984), 1085-1109, 1111-1129.
[4] C. M. Chen: Optimal points of the stresses for triangular linear element. Numer. Math. J. Chinese Univ. 2 (1980), 12-20. MR 0619174 | Zbl 0534.73057
[5] C. M. Chen: $W^{1,\infty}$-interior estimates for finite element method on regular mesh. J. Comput. Math. 3 (1985), 1-7. MR 0815405 | Zbl 0603.34024
[6] P. G. Ciarlet: The finite element method for elliptic problems. North-Holland, Amsterdam, New York, Oxford, 1978. MR 0520174 | Zbl 0383.65058
[7] I. Hlaváček M. Hlaváček: On the existence and uniqueness of solutions and some variational principles in linear theories of elasticity with couple-stresses. Apl. Mat. 14 (1969), 387-410. MR 0250537
[8] V. P. Iljin: Svojstva někotorych klassov differenciruemych funkcij mnogich peremennych, zadannych v n-mernoj oblasti. Trudy Mat. Inst. Steklov. 66 (1962), 227-363.
[9] 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
[10] M. Křížek P. Neittaanmäki: On Superconvergence techniques. Preprint n. 34, Univ. of Jyväskylä, 1984, 1 - 43 (to appear in Acta Appl. Math.). MR 0900263
[11] N. Levine: Superconvergent recovery of the gradient from piecewise linear finite element approximations. IMA J. Numer. Anal. 5 (1985), 407-427. DOI 10.1093/imanum/5.4.407 | MR 0816065 | Zbl 0584.65067
[12] Q. Lin J. Ch. Xu: Linear finite elements with high accuracy. J. Comput. Math. 3 (1985), 115-133. MR 0854355 | Zbl 0577.65094
[13] A. Louis: Acceleration of convergence for finite element solutions of the Poisson equation. Numer. Math. 33 (1979), 43-53. DOI 10.1007/BF01396494 | MR 0545741 | Zbl 0435.65090
[14] L. A. Oganesjan V. J. Rivkind L. A. Ruchovec: Variational-difference methods for the solution of elliptic equations. Part I. (Proc. Sem., Issue 5, Vilnius, 1973), Inst. of Phys. and Math., Vilnius, 1973, 3-389.
[15] L. A. Oganesjan L. A. Ruchovec: An investigation of the rate of convergence of variational-difference schemes for second order elliptic equations in a two-dimensional regions with smooth boundary. Ž. Vyčisl. Mat. i Mat. Fiz. 9 (1969), 1102-1120. MR 0295599
[16] L. A. Oganesjan L. A. Ruchovec: Variational-difference methods for the solution of elliptic equations. Izd. Akad. Nauk Armjanskoi SSR, Jerevan, 1979.
[17] J. Nečas: Les méthodes directes en théorie des équations elliptiques. Academia, Prague, 1967. MR 0227584
[18] J. Nečas I. Hlaváček: On inequalities of Korn's type. Arch. Rational Mech. Anal. 36 (1970), 305-334. DOI 10.1007/BF00249518 | MR 0252844
[19] J. Nečas I. Hlaváček: Mathematical theory of elastic and elasto-plastic bodies: an introduction. Elsevier, Amsterdam, Oxford, New York, 1981. MR 0600655
[20] V. Thomée: High order local approximations to derivatives in the finite element method. Math. Соmр. 31 (1977), 652-660. MR 0438664
[21] B. Westergren: Interior estimates for elliptic systems of difference equations. (Thesis). Univ. of Goteborg, 1982.
[22] Q. D. Zhu: Natural inner Superconvergence for the finite element method. (Proc. China-France Sympos. on the Finite Element method, Beijing, 1982), Science Press, Beijing, Gordon and Breach, New York, 1983, 935-960. MR 0754041
[23] M. Zlámal: Superconvergence and reduced integration in the finite element method. Math. Соmр. 32 (1978), 663-685. MR 0495027
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