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This article is part of the series Recent Advances in Operator Equations, Boundary Value Problems, Fixed Point Theory and Applications, and General Inequalities.

Open Access Research

Superconvergence patch recovery for the gradient of the tensor-product linear triangular prism element

Jinghong Liu1* and Yinsuo Jia2

Author Affiliations

1 Department of Fundamental Courses, Ningbo Institute of Technology, Zhejiang University, Qianhu South Road, Ningbo, China

2 School of Mathematics and Computer Science, Shangrao Normal University, Zhimin Road, Shangrao, 334001, China

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Boundary Value Problems 2014, 2014:1  doi:10.1186/1687-2770-2014-1


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/1


Received:29 July 2013
Accepted:6 December 2013
Published:2 January 2014

© 2014 Liu and Jia; licensee Springer.

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M1">View MathML</a> 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.

Keywords:
superconvergence patch recovery; superclose property; triangular prism element

1 Introduction

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 [10] also discussed superconvergent properties of the gradients by SPR and obtained superconvergence results of the recovered gradients in the average sense of the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M2">View MathML</a>-norm. In addition, Chen [11] and Goodsell [12] derived superconvergence estimates of the recovered gradient by the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M2">View MathML</a>-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

We consider the model problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M4">View MathML</a>

(2.1)

Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M5">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M7">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M8">View MathML</a>. In addition, we write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M10">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M11">View MathML</a>, which are usual partial derivatives.

To discretize the problem, one proceeds as follows. The domain Ω is firstly partitioned into subcubes of side h, and each of these is then subdivided into two triangular prisms. We denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M12">View MathML</a> these uniform partitions as above. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M13">View MathML</a>. Obviously, we can write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M14">View MathML</a> (see Figure 1), where D and L represent a triangle parallel to the xy-plane and a one-dimensional interval parallel to the z-axes, respectively.

thumbnailFigure 1. Triangular prisms partition. This figure gives how to partition the domain Ω. The domain Ω is firstly partitioned into subcubes of side h, and each of these is then subdivided into two triangular prisms.

We introduce a tensor-product linear polynomial space denoted by , that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M16">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M18">View MathML</a> stands for the linear polynomial space with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M19">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M20">View MathML</a> is the linear polynomial space with respect to z. The indexing set ℐ satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M21">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M22">View MathML</a> be the linear interpolation operator with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M23">View MathML</a>, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M24">View MathML</a> be the linear interpolation operator with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M25">View MathML</a>. Thus we may define the tensor-product linear interpolation operator by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M26">View MathML</a>. Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M27">View MathML</a>. In addition, the weak form of problem (2.1) is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M28">View MathML</a>

(2.2)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M29">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M30">View MathML</a>

Define the tensor-product linear triangular prism finite element space by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M31">View MathML</a>

Thus, the finite element method of problem (2.2) is to find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M32">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M33">View MathML</a>

Moreover, from the definitions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M34">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M35">View MathML</a>, we may define a global tensor-product linear interpolation operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M36">View MathML</a>. Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M37">View MathML</a>. As for this interpolation operator, the following Lemma 2.1 holds (see [13]).

Lemma 2.1For Πuand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M1">View MathML</a>, the tensor-product linear interpolant and the tensor-product linear triangular prism finite element approximation tou, respectively, we have the supercloseness estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M39">View MathML</a>

(2.3)

3 Gradient recovery and superconvergence

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M40">View MathML</a>, we consider a SPR scheme of ∇v. We denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M41">View MathML</a> the SPR-recovery operator for ∇v and begin by defining the point values of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M42">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M42">View MathML</a>. Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M44">View MathML</a>.

Let us first assume that N is an interior node of the partition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M45">View MathML</a>, and denote by ω the element patch around N containing 12 triangular prisms. Under the local coordinate system centered N, we let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M46">View MathML</a> be the barycenter of a triangular prism <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M47">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M48">View MathML</a>. Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M49">View MathML</a>. SPR uses the discrete least-squares fitting to seek linear vector function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M50">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M51">View MathML</a>

(3.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M40">View MathML</a>. We define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M53">View MathML</a>. If N is a boundary node, we calculate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M54">View MathML</a> by linear extrapolation from the values of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M42">View MathML</a> already obtained at two neighboring interior nodes, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M56">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M57">View MathML</a> (with diagonal directions being used for edge nodes and corner nodes) (see Figure 2). Namely,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M58">View MathML</a>

thumbnailFigure 2. N, a boundary node.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M56">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M57">View MathML</a> are interior nodes. We can calculate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M54">View MathML</a> by linear extrapolation from the values of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M42">View MathML</a> already obtained at two neighboring interior nodes, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M56">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M57">View MathML</a>.

Lemma 3.1Letωbe the element patch around an interior nodeN, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M65">View MathML</a>. For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M66">View MathML</a>the interpolant tou, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M67">View MathML</a>

(3.2)

Proof Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M68">View MathML</a> and set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M69">View MathML</a> in (3.1) to obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M70">View MathML</a>. Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M71">View MathML</a>

That is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M72">View MathML</a>

(3.3)

Further,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M73">View MathML</a>

(3.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M74">View MathML</a>, and the high-order term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M75">View MathML</a> satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M76">View MathML</a>.

In (3.4), we write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M77">View MathML</a>. For every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M78">View MathML</a>, it is not difficult to verify <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M79">View MathML</a>. Thus, by the Bramble-Hilbert lemma [14], we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M80">View MathML</a>. Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M81">View MathML</a>

(3.5)

Combining (3.3) and (3.5), we obtain the result (3.2). □

Lemma 3.2For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M66">View MathML</a>the tensor-product linear interpolant tou, the solution of (2.2), and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M41">View MathML</a>the SPR recovery operator, we have the superconvergent estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M84">View MathML</a>

(3.6)

Proof Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M85">View MathML</a> an affine transformation. Obviously, there exists an element <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M86">View MathML</a>, using the triangle inequality and the Sobolev embedding theorem [15], and (3.3), such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M87">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M88">View MathML</a> is a small patch of elements surrounding the triangular prism, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M89">View MathML</a>. Due to the fact that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M90">View MathML</a> quadratic over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M88">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M92">View MathML</a>

so, from the Bramble-Hilbert lemma [14],

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M93">View MathML</a>

which completes the proof of the result (3.6). Finally, we give the main result in this article. □

Theorem 3.1For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M32">View MathML</a>the tensor-product linear triangular prism finite element approximation tou, the solution of (2.2), and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M41">View MathML</a>the SPR recovery operator, we have the superconvergence estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M96">View MathML</a>

(3.7)

Proof Using the triangle inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/1/mathml/M97">View MathML</a>

which combined with (2.3) and (3.6) completes the proof of the result (3.7). □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The first author gave the idea of this article and the methods of proving the main results. He also proved Lemmas 3.1 and 3.2. The second author provided the proof of Theorem 3.1 and the correction of the English language.

Acknowledgements

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.

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