High accuracy analysis of tensor-product linear pentahedral finite elements for variable coefficient elliptic equations

J. H. Brandts and M. Křížek, 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, 2001, 15: 22–33.

J. H. Brandts and M. Křížek, Gradient superconvergence on uniform simplicial partitions of polytopes, IMA J. Numer. Anal., 2003, 23: 489–505.

J. H. Brandts and M. Křížek, Superconvergence of tetrahedral quadratic finite elements, J. Comp. Math., 2005, 23: 27–36.

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

C. M. Chen, Construction Theory of Superconvergence of Finite Elements, Hunan Science and Technology Press, Changsha, 2001 (in Chinese).

L. Chen, Superconvergence of tetrahedral linear finite elements, Inter. J. Numer. Anal. Model., 2006, 3: 273–282.

G. Goodsell, Gradient superconvergence for piecewise linear tetrahedral finite elements, Technical Report RAL-90-031, Science and Engineering Research Council, Rutherford Appleton Laboratory, 1990.

G. Goodsell, Pointwise superconvergence of the gradient for the linear tetrahedral element, Numer. Meth. Part. Diff. Equa., 1994, 10: 651–666.

V. Kantchev and R. D. Lazarov, Superconvergence of the gradient of linear finite elements for 3D Poisson equation, Proceedings of the Conference on Optimal Algorithms (ed. by B. Sendov), Bulgarian Academy of Sciences, Sofia, 1986: 172–182.

Q. Lin and N. N. Yan, Construction and Analysis of High Efficient Finite Elements, Hebei University Press, Baoding, 1996 (in Chinese).

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

J. H. Liu and Q. D. Zhu, Uniform superapproximation of the derivative of tetrahedral quadratic finite element approximation, J. Comp. Math., 2005, 23: 75–82.

J. H. Liu and Q. D. Zhu, Maximum-norm superapproximation of the gradient for the trilinear block finite element, Numer. Meth. Part. Diff. Equa., 2007, 23: 1501–1508.

J. H. Liu and Q. D. Zhu, Pointwise supercloseness of tensor-product block finite elements, Numer. Meth. Part. Diff. Equa., 2009, 25: 990–1008.

J. H. Liu and Q. D. Zhu, Pointwise supercloseness of pentahedral finite elements, Numer. Meth. Part. Diff. Equa., 2010, 26: 1572–1580.

J. H. Liu and Q. D. Zhu, The estimate for the W 1,1-seminorm of discrete derivative Green’s function in three dimensions, J. Hunan Univ. Art. Sci., 2004, 16: 1–3 (in Chinese).

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: 362–366.

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., 1996, 33: 505–521.

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

M. Zlámal, Superconvergence and reduced integration in the finite element method, Math. Comp., 1978, 32: 663–685.

Q. D. Zhu and Q. Lin, Superconvergence Theory of the Finite Element Methods, Hunan Science and Technology Press, Changsha, 1989 (in Chinese).