Abstract
In this article, we study superconvergence of the finite element approximation to
the solution of a general secondorder elliptic boundary value problem in three dimensions
over a fully uniform mesh of piecewise tensorproduct linear triangular prism elements.
First, we give the superclose property of the gradient between the finite element
solution
Keywords:
superconvergence patch recovery; superclose property; triangular prism element1 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 [16]). Recently, we studied the superconvergence patch recovery (SPR) technique introduced
by Zienkiewicz and Zhu [79] 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
2 General elliptic boundary value problem and finite element discretization
We consider the model problem
Here
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
Figure 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 tensorproduct linear polynomial space denoted by , that is,
where
where
and
Define the tensorproduct linear triangular prism finite element space by
Thus, the finite element method of problem (2.2) is to find
Moreover, from the definitions of
Lemma 2.1For Πuand
3 Gradient recovery and superconvergence
For
Let us first assume that N is an interior node of the partition
where
Figure 2. N, a boundary node.
Lemma 3.1Letωbe the element patch around an interior nodeN, and
Proof Choose
That is,
Further,
where
In (3.4), we write
Combining (3.3) and (3.5), we obtain the result (3.2). □
Lemma 3.2For
Proof Denote by
where
so, from the BrambleHilbert lemma [14],
which completes the proof of the result (3.6). Finally, we give the main result in this article. □
Theorem 3.1For
Proof Using the triangle inequality, we have
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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