Geodesic
Superconvergence of the gradient for cubic triangular finite elements
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 8 (2005) no. 2, pp. 89-100.

Voir la notice de l'article provenant de la source Math-Net.Ru

Superconvergence of the gradient of approximate solutions to second order elliptic equations is analyzed and justified for the 10-node cubic triangular elements. The existence of superconvergent points is proved. A recovery gradient technique in a subdomain is presented. The superclose property is proved. A rigorous proof of the superconvergent error estimate in a recovered gradient function is obtained. Numerical experiments supporting the theory under study are presented. 
@article{SJVM_2005_8_2_a0,
     author = {A. B. Andreev and T. J. Todorov},
     title = {Superconvergence of the gradient for cubic triangular finite elements},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {89--100},
     publisher = {mathdoc},
     volume = {8},
     number = {2},
     year = {2005},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2005_8_2_a0/}
}
TY  - JOUR
AU  - A. B. Andreev
AU  - T. J. Todorov
TI  - Superconvergence of the gradient for cubic triangular finite elements
JO  - Sibirskij žurnal vyčislitelʹnoj matematiki
PY  - 2005
SP  - 89
EP  - 100
VL  - 8
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SJVM_2005_8_2_a0/
LA  - en
ID  - SJVM_2005_8_2_a0
ER  - 
%0 Journal Article
%A A. B. Andreev
%A T. J. Todorov
%T Superconvergence of the gradient for cubic triangular finite elements
%J Sibirskij žurnal vyčislitelʹnoj matematiki
%D 2005
%P 89-100
%V 8
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SJVM_2005_8_2_a0/
%G en
%F SJVM_2005_8_2_a0
A. B. Andreev; T. J. Todorov. Superconvergence of the gradient for cubic triangular finite elements. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 8 (2005) no. 2, pp. 89-100. http://geodesic.mathdoc.fr/item/SJVM_2005_8_2_a0/

[1] Andreev A. B., Lazarov R. D., “Superconvergence of the gradient for quadratic triangular finite element”, Numerical Methods for Partial Differential Equations, 4 (1988), 15–32 | DOI | MR | Zbl

[2] Andreev A. B., Kascieva V. A., Vanmaele M., “Some results in lumped mass finite element approximation of eigenvalue problems using numerical quadrature formulas”, J. Comp. Appl. Math., 43 (1992), 291–311 | DOI | MR | Zbl

[3] Babuška I., Strouboulis T., Upadhyay C. S., Gangaraj S. K., “Validation of recipes for the recovery of stresses and derivatives by a computer-based approach”, Math. Comput. Model., 20:6 (1994), 45–89 | DOI | MR | Zbl

[4] Babuška I., Strouboulis T., Upadhyay S. C., Gangaraj S. K., “Computer-based proof of the existence of superconvergence points in the finite element method; superconvergence of the derivatives in finite element solutions of Laplace's, Poisson's and elasticity equations”, Numer. Methods for PDEs, 12 (1996), 347–392 | MR | Zbl

[5] Brandts J., Křžek M., History and Future of Superconvergence in Three-Dimensional Finite Element Methods, Report Universiteit Utrecht; 1155, 2001

[6] Ciarlet P. G., The Finite Element Method for Elliptic Problems, North-Nolland, Amsterdam, 1978 | MR | Zbl

[7] Goodsell G., Whiteman J. R., “Superconvergence of recovered gradients of piecewise quadratic finite element approximations. Part I: $L_2$-error estimates”, Numer. Methods for PDEs, 1 (1991), 61–83 | MR | Zbl

[8] Grisvard P., Elliptic Problems in Nonsmooth Domains, Pitman Advanced Publishing Program, London, 1985 | MR | Zbl

[9] Lesaint P., Zlamal M., “Superconvergence of the gradient of finite element method”, RAIRO Anal. Numer., 13 (1979), 139–166 | MR | Zbl

[10] Nitsche J. A., Schatz A. H., “Interior estimates for Ritz–Galerkin methods”, Math. Comp., 28 (1974), 937–958 | DOI | MR | Zbl

[11] Wahlbin L. B., Superconvergence in Galerkin FEM, Springer, Berlin, 1995 | MR | Zbl

[12] Whiteman J. R., Goodsell G., “A survey of gradient superconvergence for finite element approximations to second order elliptic problems on triangular and tetrahedral meshes”, The Mathematics of Finite Elements and Applications VII, Academic Press, 1991, 55–74 | MR

[13] Zhang Z., Yan N., Sun T., “Superconvergent derivative recovery for the intermediate finite element family of the second type”, IMA J. Numer. Anal., 21 (2001), 643–665 | DOI | MR | Zbl

[14] Zienkiewicz O. C., Zhu J. Z., “The superconvergence patch-recovery (SPR) and finite element refinement”, Comp. Methods Appl. Mech. Eng., 101 (1992), 207–224 | DOI | MR | Zbl