On a global superconvergence of the gradient of linear triangular elements. (English) Zbl 0602.65084
We study a simple superconvergent scheme which recovers the gradient when solving a second-order elliptic problem in the plane by the usual linear elements. The recovered gradient globally approximates the true gradient even by one order of accuracy higher in the \(L^ 2\)-norm than the piecewise constant gradient of the Ritz-Galerkin solution. A superconvergent approximation to the boundary flux is presented as well.
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 |
Keywords:
Ritz-Galerkin; global superconvergence for the gradient; post-processing; error estimates; boundary flux; displacement finite element methodReferences:
| [1] | Andreev, A. B., Superconvergence of the gradient for linear triangle elements for elliptic and parabolic equations, C.R. Acad. Bulgare Sci., 37, 293-296 (1984) · Zbl 0575.65106 |
| [2] | Bramble, J.; Hilbert, S. R., Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation, SIAM J. Numer. Anal., 7, 112-124 (1970) · Zbl 0201.07803 |
| [3] | Ciarlet, P. G., The Finite Element Method for Elliptic Problems (1978), North-Holland: North-Holland Amsterdam/New York/Oxford · Zbl 0445.73043 |
| [4] | Chen, C. M., Optimal points of the stresses for triangular linear element, Numer. Math. J. Chinese Univ., 2, 12-20 (1980) · Zbl 0534.73057 |
| [5] | Chen, C. M., \(W^{1,∞}\)-interior estimates for finite element method on regular mesh, J. Comp. Math., 3, 1-7 (1985) · Zbl 0573.65084 |
| [6] | Chen, C. M.; Liu, J., (Superconvergence of the gradient of triangular linear element in general domain (1984), Xiangtan Univ), 1-19, Preprint |
| [7] | Douglas, J.; Dupont, T.; Wheeler, M. F., A Galerkin procedure for approximation the flux on the boundary for elliptic and parabolic boundary value problems, RAIRO Anal. Numér., 8, 47-59 (1974) · Zbl 0315.65063 |
| [8] | Glowinski, R., Numerical Methods for Nonlinear Variational Problems (1984), Springer: Springer New York, Springer Series in Comp. Physics · Zbl 0575.65123 |
| [9] | Grisvard, P., Elliptic Problems in Nonsmooth Domains, 24 (1985), Pitman: Pitman Boston, MA, Monographs and Studies in Mathematics · Zbl 0695.35060 |
| [10] | Kondratěv, V. A., Boundary problems for elliptic equations in domains with conical or angular points, Trudy Moskov. Mat. Obšč., 16, 209-292 (1967) · Zbl 0162.16301 |
| [11] | Křížek, M.; Neittaanmäki, P., Superconvergence phenomenon in the finite element method arising from averaging gradients, Numer. Math., 45, 105-116 (1984) · Zbl 0575.65104 |
| [12] | Křížek, M.; Neittaanmäki, P., On superconvergence techniques, Acta Appl. Math. (1987), to appear · Zbl 0602.65084 |
| [13] | Levine, N., Superconvergent estimation of the gradient from linear finite element approximations on triangular elements, (Numerical analysis report 3/85 (1985), Univ. of Reading), 1-202 |
| [14] | Lin, Q.; Lü, T., Asymptotic expansions for finite element approximation of elliptic problem on polygonal domains, (Computing Methods in Applied Sciences and Engineering (1984), North-Holland: North-Holland Amsterdam), 317-321, Proc. Sixth Int. Conf., Versailles, 1983 · Zbl 0561.65080 |
| [15] | Lin, Q.; Xu, J. Ch., Linear finite elements with high accuracy, J. Comp. Math., 3, 115-133 (1985) · Zbl 0577.65094 |
| [16] | Marchuk, G. I.; Shaidurov, V. V., Difference Methods and Their Extrapolations (1983), Springer: Springer New York/Berlin/Heidelberg/Tokyo · Zbl 0511.65076 |
| [17] | Oganesjan, L. A.; Ruhovec, L. A., Variational-Difference Methods for the Solution of Elliptic Equations (1979), Izd. Akad. Nauk Armajanskoi SSR: Izd. Akad. Nauk Armajanskoi SSR Jerevan · Zbl 0496.65053 |
| [18] | Oganesjan, L. A.; Ruhovec, L. A., An investigation of the rate of convergence of variational-difference schemes for second order elliptic equations in a two-dimensional region with smooth boundary, Ž. Vyčisl. Mat. i Mat. Fiz., 9, 1102-1120 (1969) · Zbl 0234.65093 |
| [19] | Thomée, V., High order local approximations to derivatives in the finite element method, Math. Comp., 31, 652-660 (1977) · Zbl 0367.65055 |
| [20] | Volkov, E. A., Solution of boundary value problems for Poisson’s equation in a rectangle, Dokl. Akad. Nauk SSSR, 147, 13-16 (1962) · Zbl 0146.35402 |
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