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, .
Let us first assume that N is an interior node of the partition , and denote by ω the element patch around N containing 12 triangular prisms. Under the local coordinate system centered N, we let be the barycenter of a triangular prism , . Obviously, . SPR uses the discrete least-squares fitting to seek linear vector function such that
(3.1)
where . We define . If N is a boundary node, we calculate by linear extrapolation from the values of already obtained at two neighboring interior nodes, and (with diagonal directions being used for edge nodes and corner nodes) (see Figure 2). Namely,
Lemma 3.1 Let ω be the element patch around an interior node N, and . For the interpolant to u, we have
(3.2)
Proof Choose and set in (3.1) to obtain . Therefore,
That is,
(3.3)
Further,
(3.4)
where , and the high-order term satisfies .
In (3.4), we write . For every , it is not difficult to verify . Thus, by the Bramble-Hilbert lemma [14], we have . Therefore,
(3.5)
Combining (3.3) and (3.5), we obtain the result (3.2). □
Lemma 3.2 For the tensor-product linear interpolant to u, the solution of (2.2), and the SPR recovery operator, we have the superconvergent estimate
(3.6)
Proof Denote by an affine transformation. Obviously, there exists an element , using the triangle inequality and the Sobolev embedding theorem [15], and (3.3), such that
where is a small patch of elements surrounding the triangular prism, . Due to the fact that for quadratic over ,
so, from the Bramble-Hilbert lemma [14],
which completes the proof of the result (3.6). Finally, we give the main result in this article. □
Theorem 3.1 For the tensor-product linear triangular prism finite element approximation to u, the solution of (2.2), and the SPR recovery operator, we have the superconvergence estimate
(3.7)
Proof Using the triangle inequality, we have
which combined with (2.3) and (3.6) completes the proof of the result (3.7). □