Superconvergence Recovery for the Gradient of the Trilinear Finite Element

Abstract:

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For a second-order elliptic boundary value problem in three dimensions, we use an interpolation postprocessing technique to obtain recovered gradients of tri- linear elements over regular meshes. Further, superconvergence of these gradients are proved.

Info:

Periodical:

Advanced Materials Research (Volumes 268-270)

Edited by:

Feng Xiong

Pages:

1021-1024

Citation:

J. H. Liu and D. C. Yin, "Superconvergence Recovery for the Gradient of the Trilinear Finite Element", Advanced Materials Research, Vols. 268-270, pp. 1021-1024, 2011

Online since:

July 2011

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$38.00

[1] J. H. Brandts and M. Krizek: History and future of superconvergence in three dimensional finite element methods, Proceedings of the Conference on Finite Element Methods: Three-dimensional Problems, GAKUTO International Series Mathematics Science Application, Gakkotosho, Tokyo, 15 (2001).

[2] J. H. Brandts and M. Krizek: Gradient superconvergence on uniform simplicial partitions of polytopes, IMA J. Numer. Anal. 23 (2003), 489–505.

DOI: https://doi.org/10.1093/imanum/23.3.489

[3] J. H. Brandts and M. Krizek: Superconvergence of tetrahedral quadratic finite elements, J. Comput. Math. 23 (2005), 27–36.

[4] C. M. Chen: Optimal points of stresses for the linear tetrahedral element (in Chinese), Nat. Sci. J. Xiangtan Univ. 3 (1980), 16–24.

[5] C. M. Chen: Construction theory of superconvergence of finite elements (in Chinese), Hunan Science and Technology Press, Changsha, China, (2001).

[6] L. Chen: Superconvergence of tetrahedral linear finite elements, Internat. J. Numer. Anal. Model. 3 (2006), 273–282.

[7] G. Goodsell: Gradient superconvergence for piecewise linear tetrahedral finite el- ements, Technical Report RAL-90-031, Science and Engineering Research Council, Rutherford Appleton Laboratory, (1990).

[8] G. Goodsell: Pointwise superconvergence of the gradient for the linear tetrahedral element, Numer. Methods Partial Differential Equations 10 (1994), 651–666.

DOI: https://doi.org/10.1002/num.1690100511

[9] V. Kantchev and R. D. Lazarov: Superconvergence of the gradient of linear finite elements for 3D Poisson equation, B. Sendov, editor, Proceedings of the Conference on Optimal Algorithms, Bulgarian Academy of Sciences, Sofia, 1986, p.172–182.

[10] Q. Lin and N. N. Yan: Construction and analysis of high efficient finite elements (in Chinese), Hebei University Press, Baoding, China, (1996).

[11] R. C. Lin and Z. M. Zhang: Natural superconvergent points in 3D finite elements, SIAM J. Numer. Anal. 46 (2008), 1281–1297.

[12] J. H. Liu and Q. D. Zhu: Uniform superapproximation of the derivative of tetra- hedral quadratic finite element approximation, J. Comput. Math. 23 (2005), 75–82.

[13] J. H. Liu and Q. D. Zhu: Maximum-norm superapproximation of the gradient for the trilinear block finite element, Numer. Methods Partial Differential Equations 23 (2007), 1501–1508.

DOI: https://doi.org/10.1002/num.20237

[14] J. H. Liu and Q. D. Zhu: Pointwise supercloseness of tensor-product block finite elements, Numer. Methods Partial Differential Equations 25 (2009), 990–1008.

DOI: https://doi.org/10.1002/num.20384

[15] J. H. Liu and Q. D. Zhu: Pointwise supercloseness of pentahedral finite elements, Numer. Methods Partial Differential Equations 26 (2010), 1572–1580.

DOI: https://doi.org/10.1002/num.20510

[16] A. Pehlivanov: Superconvergence of the gradient for quadratic 3D simplex finite elements, Proceedings of the Conference on Numerical Methods and Application, Bulgarian Academy of Sciences, Sofia, 1989, p.362–366.

[17] A. H. Schatz, I. H. Sloan and L. B. Wahlbin, Superconvergence in finite element methods and meshes that are locally symmetric with respect to a point, SIAM J. Numer. Anal. 33 (1996), 505–521.

DOI: https://doi.org/10.1137/0733027

[18] Z. M. Zhang and R. C. Lin: Locating natural superconvergent points of finite element methods in 3D, Internat. J. Numer. Anal. Model. 2 (2005), 19–30.

[19] M. Zlamal: Superconvergence and reduced integration in the finite element method, Math. Comp. 32 (1978), 663–685.

[20] J. H. Liu, D. C. Yin and Q. D. Zhu: A note on superconvergence of recovered gradients of tensor-product linear pentahedral finite element approximations. Proceedings of the 3rd International Conference on Computational Intelligence and Industrial Application 3 (2010).

DOI: https://doi.org/10.1109/icicis.2011.65