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On a superconvergent finite element scheme for elliptic systems. I. Dirichlet boundary condition. (English) Zbl 0622.65097
A system of linear second order elliptic equations with Dirichlet boundary conditions in a bounded plane region is approximated by an averaging scheme based on linear finite elements. The scheme guarantees superconvergence of the derivatives of the Galerkin solution. For domains with enough smoothness on the boundary a global estimate \(O(h^{3/2})\) is established in the \(L^ 2\) norm. For a class of polygonal domains the global estimate \(O(h^ 2)\) is found.
Reviewer: W.Ames

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
35J55 Systems of elliptic equations, boundary value problems (MSC2000)
74S05 Finite element methods applied to problems in solid mechanics
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References:
[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. · Zbl 0123.28706 · doi:10.1002/cpa.3160170104
[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. · 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. · Zbl 0535.73052
[4] C. M. Chen: Optimal points of the stresses for triangular linear element. Numer. Math. J. Chinese Univ. 2 (1980), 12-20. · 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. · Zbl 0603.34024
[6] P. G. Ciarlet: The finite element method for elliptic problems. North-Holland, Amsterdam, New York, Oxford, 1978. · 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.
[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. · Zbl 0575.65104 · doi:10.1007/BF01379664 · eudml:132955
[10] M. Křížek P. Neittaanmäki: On Superconvergence techniques. Preprint n. 34, Univ. of Jyväskylä, 1984, 1 - 43
[11] N. Levine: Superconvergent recovery of the gradient from piecewise linear finite element approximations. IMA J. Numer. Anal. 5 (1985), 407-427. · Zbl 0584.65067 · doi:10.1093/imanum/5.4.407
[12] Q. Lin J. Ch. Xu: Linear finite elements with high accuracy. J. Comput. Math. 3 (1985), 115-133. · Zbl 0577.65094
[13] A. Louis: Acceleration of convergence for finite element solutions of the Poisson equation. Numer. Math. 33 (1979), 43-53. · Zbl 0435.65090 · doi:10.1007/BF01396494 · eudml:132627
[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. · Zbl 0234.65093
[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. · Zbl 1225.35003
[18] J. Nečas I. Hlaváček: On inequalities of Korn’s type. Arch. Rational Mech. Anal. 36 (1970), 305-334. · Zbl 0193.39001 · doi:10.1007/BF00249518
[19] J. Nečas I. Hlaváček: Mathematical theory of elastic and elasto-plastic bodies: an introduction. Elsevier, Amsterdam, Oxford, New York, 1981.
[20] V. Thomée: High order local approximations to derivatives in the finite element method. Math. Comp. 31 (1977), 652-660.
[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.
[23] M. Zlámal: Superconvergence and reduced integration in the finite element method. Math. Comp. 32 (1978), 663-685. · Zbl 0448.65068 · doi:10.2307/2006479
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