Superconvergence patch recovery for the gradient of the tensor-product linear triangular prism element
© Liu and Jia; licensee Springer. 2014
Received: 29 July 2013
Accepted: 6 December 2013
Published: 2 January 2014
In this article, we study superconvergence of the finite element approximation to the solution of a general second-order elliptic boundary value problem in three dimensions over a fully uniform mesh of piecewise tensor-product linear triangular prism elements. First, we give the superclose property of the gradient between the finite element solution and the interpolant Πu. Second, we introduce a superconvergence recovery scheme for the gradient of the finite element solution. Finally, superconvergence of the recovered gradient is derived.
Superconvergence of the gradient for the finite element approximation is a phenomenon whereby the convergent order of the derivatives of the finite element solutions exceeds the optimal global rate. Up to now, superconvergence is still an active research topic (see [1–6]). Recently, we studied the superconvergence patch recovery (SPR) technique introduced by Zienkiewicz and Zhu [7–9] for the linear tetrahedral element and proved pointwise superconvergent property of the recovered gradient by SPR. For the linear tetrahedral element, Chen and Wang  also discussed superconvergent properties of the gradients by SPR and obtained superconvergence results of the recovered gradients in the average sense of the -norm. In addition, Chen  and Goodsell  derived superconvergence estimates of the recovered gradient by the -projection technique and the average technique, respectively. This article will use the SPR technique to obtain a superconvergence estimate for the gradient of the tensor-product linear triangular prism element. In this article, we shall use the letter C to denote a generic constant which may not be the same in each occurrence and also use the standard notations for the Sobolev spaces and their norms.
2 General elliptic boundary value problem and finite element discretization
Here is a rectangular block with boundary ∂ Ω consisting of faces parallel to the x-, y-, and z-axes. We also assume that the given functions , , and . In addition, we write , , and , which are usual partial derivatives.
Moreover, from the definitions of and , we may define a global tensor-product linear interpolation operator . Here . As for this interpolation operator, the following Lemma 2.1 holds (see ).
3 Gradient recovery and superconvergence
For , we consider a SPR scheme of ∇v. We denote by the SPR-recovery operator for ∇v and begin by defining the point values of at the element nodes. After the recovered values at all nodes are obtained, we construct a tensor-product linear interpolation by using these values, namely SPR-recovery gradient . Obviously, .
where , and the high-order term satisfies .
Combining (3.3) and (3.5), we obtain the result (3.2). □
which completes the proof of the result (3.6). Finally, we give the main result in this article. □
which combined with (2.3) and (3.6) completes the proof of the result (3.7). □
This work is supported by the National Natural Science Foundation of China Grant 11161039, the Zhejiang Provincial Natural Science Foundation of China Grant LY13A010007, and the Natural Science Foundation of Ningbo City Grant 2013A610104.
- Babus̆ka I, Strouboulis T: The Finite Element Method and Its Reliability, Numerical Mathematics and Scientific Computation. Oxford Science Publications, Oxford; 2001.Google Scholar
- Chen CM, Huang YQ: High Accuracy Theory of Finite Element Methods. Hunan Science and Technology Press, Changsha; 1995.Google Scholar
- Chen CM: Construction Theory of Superconvergence of Finite Elements. Hunan Science and Technology Press, Changsha; 2001.Google Scholar
- Lin Q, Yan NN: Construction and Analysis of High Efficient Finite Elements. Hebei University Press, Baoding; 1996.Google Scholar
- Wahlbin L: Superconvergence in Galerkin Finite Element Methods. Springer, Berlin; 1995.Google Scholar
- Zhu QD, Lin Q: Superconvergence Theory of the Finite Element Methods. Hunan Science and Technology Press, Changsha; 1989.Google Scholar
- Zienkiewicz OC, Zhu JZ: A simple estimator and adaptive procedure for practical engineering analysis. Int. J. Numer. Methods Eng. 1987, 24: 337-357. 10.1002/nme.1620240206MathSciNetView ArticleGoogle Scholar
- Zienkiewicz OC, Zhu JZ: The superconvergence patch recovery and a posteriori error estimates. I. The recovery techniques. Int. J. Numer. Methods Eng. 1992, 33: 1331-1364. 10.1002/nme.1620330702MathSciNetView ArticleGoogle Scholar
- Zienkiewicz OC, Zhu JZ: The superconvergence patch recovery and a posteriori error estimates. II. Error estimates and adaptivity. Int. J. Numer. Methods Eng. 1992, 33: 1365-1382. 10.1002/nme.1620330703MathSciNetView ArticleGoogle Scholar
- Chen J, Wang DS: Three-dimensional finite element superconvergent gradient recovery on Par6 patterns. Numer. Math., Theory Methods Appl. 2010, 3: 178-194.MathSciNetGoogle Scholar
- Chen L: Superconvergence of tetrahedral linear finite elements. Int. J. Numer. Anal. Model. 2006, 3: 273-282.MathSciNetGoogle Scholar
- Goodsell G: Pointwise superconvergence of the gradient for the linear tetrahedral element. Numer. Methods Partial Differ. Equ. 1994, 10: 651-666. 10.1002/num.1690100511MathSciNetView ArticleGoogle Scholar
- Liu JH, Deng YJ, Zhu QD: High accuracy analysis of tensor-product linear pentahedral finite elements for variable coefficient elliptic equations. J. Syst. Sci. Complex. 2012, 25: 410-416. 10.1007/s11424-011-0045-6MathSciNetView ArticleGoogle Scholar
- Bramble JH, Hilbert SR: Estimation of linear functionals on Sobolev spaces with applications to Fourier transforms and spline interpolation. SIAM J. Numer. Anal. 1970, 7: 112-118. 10.1137/0707006MathSciNetView ArticleGoogle Scholar
- Adams RA: Sobolev Spaces. Academic Press, New York; 1975.Google Scholar
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