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Superconvergence of the function value for pentahedral finite elements for an elliptic equation with varying coefficients

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Abstract

In this article, for an elliptic equation with varying coefficients, we first derive an interpolation fundamental estimate for the \(\mathcal{P}_{2}(x,y)\otimes \mathcal{P}_{2}(z)\) pentahedral finite element over uniform partitions of the domain. Then combined with the estimate for the \(W^{2,1}\)-seminorm of the discrete Green function, superconvergence of the function value between the finite element approximation and the corresponding interpolant to the true solution is given.

Introduction and preliminaries

Superconvergence is a phenomenon in numerical methods that refers to faster than normal convergence for the approximate solutions arising from numerical procedures, and it was first addressed in [1]. The term “superconvergence” was first used in [2]. Since then, it has become an actively researched topic in the domain of finite element methods. So far, numerous studies on superconvergence have been published. For one- and two-dimensions, superconvergence has been extensively investigated. For three and more dimensions, studies on superconvergence are progressing at a slow rate. Recently, we focused on superconvergence of the finite element method for three-dimensional problems, and we found that there have been some studies concerning it. Some books and survey papers have also been published. We refer to [325] and the references therein. In general, according to the domain partition, there usually exist three types of finite elements for three-dimensional problems, namely tetrahedral elements, pentahedral elements, and block elements. In this paper, we only consider the pentahedral elements. To the best of our knowledge, superconvergence of pentahedral elements (or prismatic elements) has been investigated in [7, 14, 15, 20, 24]. Of these studies, [7] considered superconvergence of pentahedral elements for the elliptic equation with constant coefficients. The study [15] is concerned with superconvergence for the Poisson equation, and demonstrated accuracy of the order \(\mathcal{O}(h^{4}| \ln h|^{\frac{2}{3}})\) in terms of \(L^{\infty }\)-norm for the value of the function between the \(\mathcal{P}_{2}(x,y)\otimes \mathcal{P}_{2}(z)\) pentahedral finite element approximation and the corresponding interpolant. In this paper, we will generalize the results in [7] and [15] to general elliptic equations with varying coefficients.

Additionally, we will use the symbol C to denote a generic constant, which is independent of the discretization parameters \(h_{xy}\) and \(h_{z}\) and which may not be the same for each occurrence. We will also use the standard notations for the Sobolev spaces and their norms.

The model problem considered in the article is as follows:

$$ \mathcal{L} u\equiv -\sum^{3}_{i,j=1} \partial _{j}(a_{ij}\partial _{i}u)+ \sum ^{3}_{i=1}a_{i}\partial _{i}u+a_{0}u=f \quad \text{in } \varOmega , u=0 \text{ on } \partial \varOmega . $$
(1.1)

Here, \(\varOmega =\varOmega _{xy}\times \varOmega _{z}\equiv (0,1)^{2}\times (0,1) \subset {\mathcal{R}}^{3}\) is the unit cube with boundary, ∂Ω, comprising faces parallel to the x-, y-, and z-axes. The diffusion coefficients \(a_{ij}\) satisfy the following condition:

There exists a positive constant C such that, for all \(X\in \varOmega \), we have

$$ \sum_{i,j=1}^{3}a_{ij}(X)\eta _{i}\eta _{j}\geq C\sum_{i=1}^{3} \eta _{i} ^{2} \quad \forall \eta =(\eta _{1},\eta _{2},\eta _{3})^{\top }\in {\mathcal{R}} ^{3}. $$

In addition, we also assume \(a_{ij}, a_{i}\in W^{1,\infty }(\varOmega )\), \(a_{0}\in L^{\infty }(\varOmega )\), \(f\in L^{2}(\varOmega )\), \(a_{0}\geq 0\), and write \(\partial _{1}u=\frac{\partial u}{\partial x}\), \(\partial _{2}u=\frac{ \partial u}{\partial y}\), and \(\partial _{3}u=\frac{\partial u}{\partial z}\).

Thus, the weak formulation of (1.1) is as follows:

$$ \textstyle\begin{cases} \text{Find } u\in H^{1}_{0}(\varOmega ) \text{ satisfying} \\ a(u,v)=(f,v)\quad \forall v\in H^{1}_{0}(\varOmega ), \end{cases} $$
(1.2)

where

$$ a(u,v)= \int _{\varOmega }\Biggl(\sum^{3}_{i,j=1}a_{ij} \partial _{i}u\partial _{j}v+ \sum ^{3}_{i=1}a_{i}\partial _{i}uv+a_{0}uv\Biggr)\,dx\,dy\,dz $$

and

$$ (f,v)= \int _{\varOmega }fv\,dx\,dy\,dz. $$

To provide the discrete formulation of (1.2), we should first partition the domain Ω. Denote by \(\{{\mathcal{T}}^{h}\}\) a uniform family of pentahedral partitions, and thus, \(\bar{\varOmega }= \bigcup_{e\in {\mathcal{T}}^{h}}\bar{e}\). Therefore, we can write \(\bar{e}=D\times L\) (see Fig. 1), where D and L are closed, and denote an isosceles right triangle with legs \(h_{xy}\) parallel to the xy-plane and a one-dimensional interval with length \(h_{z}\) parallel to the z-axis, respectively. We assume that there exist two positive constants \(C_{1}\) and \(C_{2}\) such that \(C_{1}\leq \frac{h_{z}}{h_{xy}}\leq C_{2}\).

Figure 1
figure1

An \(\mathcal{P}_{2}(x,y)\otimes \mathcal{P}_{2}(z)\) pentahedral element and interpolation nodes

We introduce an \(\mathcal{P}_{2}(x,y)\otimes \mathcal{P}_{2}(z)\) polynomial space denoted by \(\mathcal{P}\), that is,

$$ q(x,y,z)=\sum_{(i,j,k)\in \mathcal{I}}a_{ijk}x^{i}y^{j}z^{k}, \quad a_{ijk}\in \mathcal{R}, q\in \mathcal{P}\equiv \mathcal{P}_{2}(x,y)\otimes \mathcal{P}_{2}(z), $$

where \(\mathcal{P}_{2}(x,y)\) denotes the quadratic polynomial space with respect to \((x,y)\), and \(\mathcal{P}_{2}(z)\) is the quadratic polynomial space with respect to z. The indexing set \(\mathcal{I}\) satisfies

$$ \mathcal{I}=\bigl\{ (i,j,k)|i,j,k\geq 0, i+j\leq 2, k\leq 2\bigr\} . $$

An \(\mathcal{P}_{2}(x,y)\otimes \mathcal{P}_{2}(z)\) interpolation operator is defined by \(\varPi ^{e}: H^{1}(\bar{e})\cap C( \bar{e})\rightarrow \mathcal{P}(\bar{e})\). Obviously,

$$ \varPi ^{e}=\varPi ^{e}_{xy}\otimes \varPi ^{e}_{z}, $$

where \(\varPi ^{e}_{xy}\) stands for the Lagrange quadratic interpolation operator with respect to \((x,y)\in D\), and \(\varPi ^{e}_{z}\) stands for the Lagrange quadratic interpolation operator or the quadratic interpolation operator of projection type with respect to \(z\in L\).

Furthermore, the \(\mathcal{P}_{2}(x,y)\otimes \mathcal{P}_{2}(z)\) pentahedral finite element space is defined as follows:

$$ S^{h}_{0}(\varOmega )= \bigl\{ v\in H^{1}_{0}(\varOmega )\cap C(\varOmega ): v|_{e} \in \mathcal{P}(e) \ \forall e\in {\mathcal{T}}^{h} \bigr\} . $$

Thus, the finite element method of (1.2) is

$$ \textstyle\begin{cases} \text{Find } u_{h}\in S^{h}_{0}(\varOmega ) \text{ satisfying} \\ a(u_{h},v)=(f,v)\quad \forall v\in S^{h}_{0}(\varOmega ). \end{cases} $$
(1.3)

From (1.2) and (1.3), the following Galerkin orthogonal relation holds:

$$ a(u-u_{h},v)=0\quad \forall v\in S^{h}_{0}( \varOmega ). $$
(1.4)

In addition, from the definitions of \(\varPi ^{e}\) and \(S^{h}_{0}(\varOmega )\), we can define a global \(\mathcal{P}_{2}(x,y)\otimes \mathcal{P} _{2}(z)\) interpolation operator \(\varPi : H^{1}_{0}(\varOmega ) \cap C(\varOmega )\rightarrow S^{h}_{0}(\varOmega )\) such that \((\varPi u)|_{e}= \varPi ^{e} u\). In next section, we will bound the term \(a(u-\varPi u,v)\).

An important interpolation fundamental estimate

Lemma 2.1

Let\(\{\mathcal{T}^{h}\}\)be a uniform family of pentahedral partitions ofΩ, \(u\in W^{5, \infty }( \varOmega )\cap H^{1}_{0}(\varOmega )\), and\(v\in S_{0}^{h}(\varOmega )\). Subsequently, the interpolation operatorΠsatisfies the following interpolation fundamental estimate:

$$ \bigl\vert a(u-\varPi u,v) \bigr\vert \leq C\bigl(h_{xy}^{4}+h_{z}^{4} \bigr) \Vert u \Vert _{5, \infty ,\varOmega } \vert v \vert ^{h}_{2,1,\varOmega }, $$
(2.1)

where\(|v|^{h}_{2,1,\varOmega }=\sum_{e\in {\mathcal{T}}^{h}}|v|_{2,1,e}\).

Proof

Clearly, the interpolation remainder is

$$ \begin{aligned}[b] u-\varPi u&= (u-\varPi _{xy}u)+(u- \varPi _{z}u)+\bigl(\varPi _{xy}(u-\varPi _{z}u)-(u-\varPi _{z}u)\bigr) \\ &= R_{xy}+R_{z}+R^{*}, \end{aligned} $$
(2.2)

where \((\varPi _{xy}u)|_{e}=\varPi ^{e}_{xy}u\), \((\varPi _{z}u)|_{e}=\varPi ^{e}_{z}u\), and \(R^{*}\) is a high-order term. Thus, it suffices to analyze \(R_{xy}\) and \(R_{z}\). We first have the bound

$$ a(R_{xy},v)= \int _{\varOmega }\Biggl(\sum^{3}_{i,j=1}a_{ij} \partial _{i}R_{xy} \partial _{j}v+\sum ^{3}_{i=1}a_{i}\partial _{i}R_{xy}v+a_{0}R_{xy}v \Biggr)\,dx\,dy\,dz. $$
(2.3)

We set

$$\begin{aligned}& I_{1}= \int _{\varOmega }\Biggl(\sum^{2}_{i,j=1}a_{ij} \partial _{i}R_{xy} \partial _{j}v+\sum ^{2}_{i=1}a_{i}\partial _{i}R_{xy}v+a_{0}R_{xy}v \Biggr)\,dx\,dy\,dz, \end{aligned}$$
(2.4)
$$\begin{aligned}& I_{2}= \int _{\varOmega }\Biggl(\sum^{2}_{j=1}a_{3j} \partial _{3}R_{xy} \partial _{j}v+a_{3} \partial _{3}R_{xy}v\Biggr)\,dx\,dy\,dz, \end{aligned}$$
(2.5)
$$\begin{aligned}& I_{3}= \int _{\varOmega }\sum^{2}_{i=1}a_{i3} \partial _{i}R_{xy} \partial _{3}v\,dx\,dy\,dz, \end{aligned}$$
(2.6)
$$\begin{aligned}& I_{4}= \int _{\varOmega }a_{33}\partial _{3}R_{xy} \partial _{3}v\,dx\,dy\,dz. \end{aligned}$$
(2.7)

Clearly,

$$ a(R_{xy},v)=I_{1}+I_{2}+I_{3}+I_{4}. $$
(2.8)

By the two-dimensional interpolation fundamental estimate of triangular quadratic elements [26], we have

$$ \vert I_{1} \vert \leq Ch_{xy}^{4} \int _{\varOmega _{z}} \Vert u \Vert _{4,\infty ,\varOmega _{xy}} \vert v \vert ^{h} _{2,1,\varOmega _{xy}}\,dz\leq Ch_{xy}^{4} \Vert u \Vert _{4,\infty ,\varOmega } \vert v \vert ^{h} _{2,1,\varOmega }. $$
(2.9)

As for \(I_{2}\), by Green’s formula, we have

$$ I_{2}= \int _{\varOmega }\Biggl(-\sum^{2}_{j=1}a_{3j} \partial _{j}\partial _{3}R _{xy}v+(a_{3}- \partial _{1}a_{31}-\partial _{2}a_{32}) \partial _{3}R_{xy}v\Biggr)\,dx\,dy\,dz. $$

Obviously, \(\partial _{3}R_{xy}=\partial _{3}u-\varPi _{xy}\partial _{3}u\). Thus, by the two-dimensional interpolation fundamental estimate of triangular quadratic elements [26], we have

$$ \vert I_{2} \vert \leq Ch_{xy}^{4} \int _{\varOmega _{z}} \Vert \partial _{3}u \Vert _{4, \infty , \varOmega _{xy}} \vert v \vert ^{h}_{2,1,\varOmega _{xy}}\,dz\leq Ch_{xy}^{4} \Vert u \Vert _{5, \infty ,\varOmega } \vert v \vert ^{h}_{2,1,\varOmega }. $$
(2.10)

As for \(I_{3}\), we first bound the integral

$$ \int _{\varOmega }a_{13}\partial _{1}R_{xy} \partial _{3}v\,dx\,dy\,dz. $$

By Green’s formula and \(v=0\) on ∂Ω, we get

$$\begin{aligned}& \int _{\varOmega }a_{13}\partial _{1}R_{xy} \partial _{3}v\,dx\,dy\,dz\\& \quad = \int _{\varOmega _{z}}\sum_{D}\biggl( \int _{D}a_{13}\partial _{1}R_{xy} \partial _{3}v\,dx\,dy\biggr)\,dz \\& \quad = \int _{\varOmega _{z}}\sum_{D}\biggl( \int _{\partial D}a_{13}R_{xy}\partial _{3}v\,dy\biggr)\,dz- \int _{\varOmega _{z}}\sum_{D}\biggl( \int _{D}R_{xy}\partial _{1}(a_{13} \partial _{3}v)\,dx\,dy\biggr)\,dz \\& \quad = \int _{\varOmega _{z}\times \partial \varOmega _{xy}}a_{13}R_{xy}\partial _{3}v\,dy\,dz- \int _{\varOmega }R_{xy}\partial _{1}(a_{13} \partial _{3}v)\,dx\,dy\,dz \\& \quad =- \int _{\varOmega }\partial _{1}a_{13}R_{xy} \partial _{3}v\,dx\,dy\,dz- \int _{\varOmega }a_{13}R_{xy}\partial _{1}\partial _{3}v\,dx\,dy\,dz \\& \quad =K_{1}+K_{2}. \end{aligned}$$

Let \(S_{0,2}^{h}(\varOmega _{xy})\) be the triangular quadratic finite element space in the domain \(\varOmega _{xy}\), and \(\{\psi _{j}\}\) be the basis of this space. Obviously, the support \(S_{j}\) of \(\psi _{j}\) is a patch of elements that share an internal edge or internal node. Moreover, because the partition of the domain is uniform, each \(S_{j}\) is point-symmetric. Subsequently, for all cubic polynomials \(p_{3}\) on \(S_{j}\), we have

$$ \int _{S_{j}}(p_{3}-\varPi _{xy}p_{3}) \psi _{j}\,dx\,dy=0. $$
(2.11)

The proof of (2.11) is similar to Lemma 3.2 in [5].

As \(v\in S_{0}^{h}(\varOmega )\), \(\partial _{3}v\in S_{0,2}^{h}(\varOmega _{xy})\). Thus, \(\partial _{3}v=\sum_{j}\alpha _{j}(z)\psi _{j}(x,y) \equiv \sum_{j}\alpha _{j}\psi _{j}\). To bound the term \(K_{1}\), we also assume \(\partial _{1}a_{13}\in W^{1,\,\infty }(\varOmega )\). Then

$$ \partial _{1}a_{13}(Q)=\partial _{1}a_{13}(Q_{0})+ \mathcal{O}(h_{xy}) \equiv a^{0}_{13}+ \mathcal{O}(h_{xy}) \quad \forall Q\in S_{j}, $$
(2.12)

where \(Q_{0}\) is the center of \(S_{j}\). Thus, by (2.11) and (2.12), we have

$$\begin{aligned} \vert K_{1} \vert =& \biggl\vert \int _{\varOmega }\partial _{1}a_{13}(u- \varPi _{xy}u)\partial _{3}v\,dx\,dy\,dz \biggr\vert \\ \leq & \int _{\varOmega _{z}}\sum_{j} \vert \alpha _{j} \vert \biggl\vert \int _{S_{j}}\bigl(a^{0} _{13}+ \mathcal{O}(h_{xy})\bigr) (u-\varPi _{xy}u)\psi _{j}\,dx\,dy \biggr\vert \,dz \\ \leq & \int _{\varOmega _{z}}\sum_{j} \vert \alpha _{j} \vert \biggl\vert \int _{S_{j}}a^{0} _{13} \bigl(u-p_{3}-\varPi _{xy}(u-p_{3})\bigr)\psi _{j}\,dx\,dy \biggr\vert \, dz \\ &{}+ \int _{\varOmega _{z}}\sum_{j} \vert \alpha _{j} \vert \biggl\vert \int _{S_{j}} \mathcal{O}(h_{xy}) (u-\varPi _{xy}u)\psi _{j}\,dx\,dy \biggr\vert \,dz \\ =&K'_{1}+K''_{1}. \end{aligned}$$

Similar to the arguments in [15], we may obtain \(\sum_{j}| \alpha _{j}|\leq C(z)h^{-2}_{xy} \vert v \vert ^{h}_{2,1,\varOmega _{xy}}\). Therefore, we obtain

$$ K'_{1}\leq Ch_{xy}^{4} \Vert u \Vert _{4,\infty ,\varOmega } \vert v \vert ^{h}_{2,1,\varOmega }. $$
(2.13)

Furthermore, we easily obtain

$$ K''_{1}\leq Ch_{xy}^{4} \Vert u \Vert _{3,\infty ,\varOmega } \vert v \vert ^{h}_{2,1,\varOmega }. $$
(2.14)

From (2.13) and (2.14),

$$ \vert K_{1} \vert \leq Ch_{xy}^{4} \Vert u \Vert _{4,\infty ,\varOmega } \vert v \vert ^{h}_{2,1,\varOmega }. $$
(2.15)

Let \(S_{0,2}^{h}(\varOmega _{z})\) be the quadratic finite element space in \(\varOmega _{z}\), and \(\{\phi _{i}\}\) be basis of this space. Clearly, \(S_{0}^{h}(\varOmega )=S_{0,2}^{h}(\varOmega _{z})\otimes S_{0,2}^{h}(\varOmega _{xy})\). Thus, for \(v\in S_{0}^{h}(\varOmega )\), we have \(v=\sum_{i,j}v(x_{j},y_{j},z_{i})\phi _{i}(z)\times\psi _{j}(x,y)\equiv \sum_{i,j}v_{ij}\phi _{i}\psi _{j}\), and \(\partial _{1}\partial _{3}v=\sum_{i,j}v_{ij}\partial _{3}\phi _{i}\partial _{1}\psi _{j}\). Note that the support \(S_{ij}\) of \(\phi _{i}\psi _{j}\) is a patch of elements that share an internal node, an edge, or a face. Moreover, as the partition of the domain is uniform, each \(S_{ij}\) is point-symmetric. Thus, similar to (2.11), we have for all cubic polynomials \(\tilde{p}_{3}\) on \(S_{ij}\)

$$ \int _{S_{ij}}(\tilde{p}_{3}-\varPi _{xy} \tilde{p}_{3})\partial _{3}\phi _{i}\partial _{1}\psi _{j}\,dx\,dy\,dz=0. $$
(2.16)

Similar to (2.12), we have

$$ a_{13}\bigl(Q^{*}\bigr)=a_{13} \bigl(Q^{*}_{0}\bigr)+\mathcal{O}(h_{xy}) \equiv a'_{13}+ \mathcal{O}(h_{xy}) \quad \forall Q^{*}\in S_{ij}, $$
(2.17)

where \(Q^{*}_{0}\) is the center of \(S_{ij}\). Hence, by (2.16) and (2.17), we get

$$\begin{aligned} \vert K_{2} \vert =& \biggl\vert \int _{\varOmega }a_{13}(u-\varPi _{xy}u) \partial _{1}\partial _{3}v\,dx\,dy\,dz \biggr\vert \\ \leq &\sum_{i,j} \vert v_{ij} \vert \biggl\vert \int _{S_{ij}}\bigl(a'_{13}+ \mathcal{O}(h _{xy})\bigr) (u-\varPi _{xy}u)\partial _{3}\phi _{i}\partial _{1}\psi _{j}\,dx\,dy\,dz \biggr\vert \\ \leq &\sum_{i,j} \vert v_{ij} \vert \biggl\vert \int _{S_{ij}}a'_{13}\bigl(u- \tilde{p}_{3}- \varPi _{xy}(u-\tilde{p}_{3}) \bigr)\partial _{3}\phi _{i}\partial _{1} \psi _{j}\,dx\,dy\,dz \biggr\vert \\ &+\sum_{i,j} \vert v_{ij} \vert \biggl\vert \int _{S_{ij}}\mathcal{O}(h_{xy}) (u-\varPi _{xy}u)\partial _{3}\phi _{i}\partial _{1}\psi _{j}\,dx\,dy\,dz \biggr\vert \\ =&K'_{2}+K''_{2}. \end{aligned}$$

For simplicity, we write

$$ M_{ij}= \biggl\vert \int _{S_{ij}}a'_{13}\bigl(u- \tilde{p}_{3}-\varPi _{xy}(u- \tilde{p}_{3}) \bigr)\partial _{3}\phi _{i}\partial _{1} \psi _{j}\,dx\,dy\,dz \biggr\vert . $$
(2.18)

Thus,

$$ K'_{2}=\sum_{i,j} \vert v_{ij} \vert \vert \partial _{3}\phi _{i}\partial _{1}\psi _{j} \vert _{0,1,\varOmega } M_{ij} \vert \partial _{3}\phi _{i}\partial _{1}\psi _{j} \vert ^{-1}_{0,1,\varOmega }. $$
(2.19)

Clearly,

$$ M_{ij}\leq Ch^{3}_{xy} \Vert u- \tilde{p}_{3} \Vert _{3,\infty ,S_{ij}} \vert \partial _{3}\phi _{i}\partial _{1}\psi _{j} \vert _{0,1, \varOmega }. $$
(2.20)

Taking \(\tilde{p}_{3}\) a three-degree interpolant to u on \(S_{ij}\) in (2.20), we have

$$ M_{ij}\leq Ch^{4}_{xy} \Vert u \Vert _{4,\infty ,\varOmega } \vert \partial _{3}\phi _{i}\partial _{1}\psi _{j} \vert _{0,1,\varOmega }. $$
(2.21)

To obtain the desired result, we need to introduce an affine transformation defined by \(F:\hat{P}\in \hat{e}\longrightarrow P=B \hat{P}+b\in e\) such that \(e=F(\hat{e})\), where \(B=(b_{ij})\) is a matrix of order \(3\times 3\). For all \(\varphi \in L^{2}(e)\), we write \(\hat{\varphi }(\hat{P})=\varphi (F\hat{P})\). The usual transformation rules between the element e and the reference element ê (see [5, 26], and [27]) tell us that there exists a constant C independent of the mesh parameters such that

$$ \vert \hat{\varphi } \vert _{0,1,\hat{e}}\leq C \vert \operatorname{det}B \vert ^{-1} \vert \varphi \vert _{0,1,e} \quad \text{and} \quad \vert \varphi \vert _{0,1,e}\leq C \vert \operatorname{det}B \vert \vert \hat{\varphi } \vert _{0,1,\hat{e}}, $$
(2.22)

In addition, we set \(w=\partial _{1}\partial _{3}v=\sum_{i,j}v _{ij}\partial _{3}\phi _{i}\partial _{1}\psi _{j}\). It is easy to prove that

$$ \sum_{i=1}^{3}\sum _{j=1}^{6} \vert v_{ij} \vert \vert \partial _{3} \phi _{i}\partial _{1} \psi _{j} \vert _{0,1,e} $$

is a seminorm of w on e. Using the rightmost rule from (2.22), we find that

$$ \sum_{i=1}^{3}\sum _{j=1}^{6} \vert v_{ij} \vert \vert \partial _{3} \phi _{i}\partial _{1} \psi _{j} \vert _{0,1,e} \leq C \vert \operatorname{det}B \vert \sum_{i=1}^{3}\sum _{j=1}^{6} \vert v_{ij} \vert \vert \hat{\partial _{3}\phi _{i}} \hat{\partial _{1}\psi _{j}} \vert _{0,1,\hat{e}}. $$
(2.23)

By the equivalence of norms in the finite-dimensional space, there also exists a constant C, depending only on the reference element ê, such that

$$ \sum_{i=1}^{3}\sum _{j=1}^{6} \vert v_{ij} \vert \vert \hat{\partial _{3}\phi _{i}} \hat{\partial _{1}\psi _{j}} \vert _{0,1,\hat{e}} \leq C \vert \hat{w} \vert _{0,1,\hat{e}}. $$
(2.24)

Using the left rule from (2.22), we get

$$ \vert \hat{w} \vert _{0,1,\hat{e}}\leq C \vert \operatorname{det}B \vert ^{-1} \vert w \vert _{0,1,e}. $$
(2.25)

Combining (2.23)–(2.25) yields

$$ \sum_{i=1}^{3}\sum _{j=1}^{6} \vert v_{ij} \vert \vert \partial _{3} \phi _{i}\partial _{1} \psi _{j} \vert _{0,1,e} \leq C \vert w \vert _{0,1,e}. $$

Summing over all e in \(\mathcal{T}^{h}\) results in

$$ \sum_{i,j} \vert v_{ij} \vert \vert \partial _{3}\phi _{i}\partial _{1} \psi _{j} \vert _{0,1,\varOmega }\leq C \vert w \vert ^{h}_{0,1,\varOmega }\leq C \vert v \vert ^{h}_{2,1,\varOmega }. $$
(2.26)

Combining (2.19), (2.21), and (2.26) yields

$$ K'_{2}\leq Ch^{4}_{xy} \Vert u \Vert _{4,\infty ,\varOmega } \vert v \vert ^{h}_{2,1,\varOmega }. $$
(2.27)

Similar to the arguments mentioned above, we also get

$$ K''_{2}\leq Ch^{4}_{xy} \Vert u \Vert _{3,\infty ,\varOmega } \vert v \vert ^{h}_{2,1,\varOmega }. $$
(2.28)

From (2.27) and (2.28),

$$ \vert K_{2} \vert \leq Ch^{4}_{xy} \Vert u \Vert _{4,\infty ,\varOmega } \vert v \vert ^{h}_{2,1,\varOmega }. $$
(2.29)

Thus, by (2.15) and (2.29), we have

$$ \biggl\vert \int _{\varOmega }a_{13}\partial _{1}R_{xy} \partial _{3}v\,dx\,dy\,dz \biggr\vert \leq \vert K_{1} \vert + \vert K_{2} \vert \leq Ch^{4}_{xy} \Vert u \Vert _{4,\infty ,\varOmega } \vert v \vert ^{h}_{2,1,\varOmega }. $$
(2.30)

Similar to the proof of (2.30), we have

$$ \biggl\vert \int _{\varOmega }a_{23}\partial _{2}R_{xy} \partial _{3}v\,dx\,dy\,dz \biggr\vert \leq Ch^{4}_{xy} \Vert u \Vert _{4,\infty ,\varOmega } \vert v \vert ^{h}_{2,1,\varOmega }. $$
(2.31)

Combining (2.6), (2.30), and (2.31), we get

$$ \vert I_{3} \vert \leq Ch^{4}_{xy} \Vert u \Vert _{4,\infty ,\varOmega } \vert v \vert ^{h}_{2,1,\varOmega }. $$
(2.32)

As for \(I_{4}\), we write \(a_{33}(Q)=a_{33}(Q_{0})+\mathcal{O}(h_{xy}) \equiv a^{0}_{33}+\mathcal{O}(h_{xy})\,\,\forall Q\in S_{j}\), and \(\partial _{3}v=\sum_{j}\alpha _{j}\psi _{j}\). Thus,

$$\begin{aligned} \vert I_{4} \vert =& \biggl\vert \int _{\varOmega }a_{33}\partial _{3}R_{xy} \sum_{j}\alpha _{j}\psi _{j}\,dx\,dy\,dz \biggr\vert \\ \leq & \int _{\varOmega _{z}}\sum_{j} \vert \alpha _{j} \vert \biggl\vert \int _{S_{j}}a _{33}\partial _{3}R_{xy} \psi _{j}\,dx\,dy \biggr\vert \,dz \\ \leq & \int _{\varOmega _{z}}\sum_{j} \vert \alpha _{j} \vert \biggl\vert \int _{S_{j}}a ^{0}_{33}\bigl(\partial _{3}u-p_{3}-\varPi _{xy}(\partial _{3}u-p_{3})\bigr)\psi _{j}\,dx\,dy \biggr\vert \,dz \\ &+ \int _{\varOmega _{z}}\sum_{j} \vert \alpha _{j} \vert \biggl\vert \int _{S_{j}} \mathcal{O}(h_{xy}) \bigl(\partial _{3}u-\varPi _{xy}(\partial _{3}u)\bigr)\psi _{j}\,dx\,dy \biggr\vert \,dz \\ =&K_{3}+K_{4}. \end{aligned}$$

Similar to the proof of (2.13), we have

$$ K_{3}\leq Ch_{xy}^{4} \Vert u \Vert _{5,\infty ,\varOmega } \vert v \vert ^{h}_{2,1,\varOmega }. $$
(2.33)

Clearly,

$$ K_{4}\leq Ch_{xy}^{4} \Vert u \Vert _{4,\infty ,\varOmega } \vert v \vert ^{h}_{2,1,\varOmega }. $$
(2.34)

Combining (2.33) and (2.34) yields

$$ \vert I_{4} \vert \leq Ch_{xy}^{4} \Vert u \Vert _{5,\infty ,\varOmega } \vert v \vert ^{h}_{2,1,\varOmega }. $$
(2.35)

From (2.8)–(2.10), (2.32), and (2.35),

$$ \bigl\vert a(R_{xy},v) \bigr\vert \leq Ch_{xy}^{4} \Vert u \Vert _{5,\infty ,\varOmega } \vert v \vert ^{h} _{2,1,\varOmega }. $$
(2.36)

Now, we can bound the term

$$ a(R_{z},v)= \int _{\varOmega }\Biggl(\sum^{3}_{i,j=1}a_{ij} \partial _{i}R_{z} \partial _{j}v+\sum ^{3}_{i=1}a_{i}\partial _{i}R_{z}v+a_{0}R_{z}v \Biggr)\,dx\,dy\,dz. $$
(2.37)

Additionally, we set

$$\begin{aligned}& J_{1}= \int _{\varOmega }\Biggl(\sum^{2}_{i,j=1}a_{ij} \partial _{i}R_{z} \partial _{j}v+\sum ^{2}_{i=1}a_{i}\partial _{i}R_{z}v+a_{0}R_{z}v \Biggr)\,dx\,dy\,dz, \end{aligned}$$
(2.38)
$$\begin{aligned}& J_{2}= \int _{\varOmega }\Biggl(\sum^{2}_{j=1}a_{3j} \partial _{3}R_{z} \partial _{j}v+a_{3} \partial _{3}R_{z}v\Biggr)\,dx\,dy\,dz, \end{aligned}$$
(2.39)
$$\begin{aligned}& J_{3}= \int _{\varOmega }\sum^{2}_{i=1}a_{i3} \partial _{i}R_{z} \partial _{3}v\,dx\,dy\,dz, \end{aligned}$$
(2.40)
$$\begin{aligned}& J_{4}= \int _{\varOmega }a_{33}\partial _{3}R_{z} \partial _{3}v\,dx\,dy\,dz. \end{aligned}$$
(2.41)

Clearly,

$$ a(R_{z},v)=J_{1}+J_{2}+J_{3}+J_{4}. $$
(2.42)

To simply bound the aforementioned terms, we may use the so-called interpolation operator of projection type (see [15]).

Let \(\{l_{j}(z)\}^{\infty }_{j=0}\) be the normalized orthogonal Legendre polynomial system from the space \(\mathcal{L}^{2}(L)\), and \(\partial _{z}u\in \mathcal{L}^{2}(L)\). For a fixed point \((x,y)\in D\), we have the following expansion:

$$ u(x,y,z)=\sum^{\infty }_{j=0}\beta _{j}(x,y)\omega _{j}(z),\quad (x,y,z) \in \bar{e}=D\times L, $$

where

$$ \omega _{0}(z)=1,\qquad \omega _{j+1}(z)= \int ^{z}_{z_{i-1}}l_{j}(\xi )\,d \xi = \mathcal{O}\bigl(h^{\frac{1}{2}}_{z}\bigr),\qquad l_{j}(z)= \mathcal{O}\bigl(h^{- \frac{1}{2}}_{z}\bigr), \quad j\geq 0. $$
(2.43)

The coefficients \(\beta _{j}(x,y)\) satisfy \(\beta _{0}(x,y)=u(x,y,z_{i-1})\), and for \(j\geq 1\),

$$ \beta _{j}(x,y)= \int _{L}\partial _{z}ul_{j-1}(z)\,dz =\mathcal{O}\bigl(h^{j- \frac{1}{2}}_{z}\bigr). $$
(2.44)

Let \(\varPi _{z}^{e}\) be the quadratic interpolation operator of projection type with respect to z defined by

$$ \varPi _{z}^{e}u=\sum^{2}_{j=0} \beta _{j}(x,y)\omega _{j}(z),\quad (x,y,z) \in \bar{e}=D \times L. $$

Thus, the interpolation remainder is

$$ R_{z}=u-\varPi _{z}^{e}u=\sum _{j=3}^{\infty }\beta _{j}(x,y)\omega _{j}(z), \quad (x,y,z)\in \bar{e}. $$
(2.45)

The above-mentioned statements are presented in [15]. Obviously, we only need to consider the main term \(r_{3}=\beta _{3}(x,y) \omega _{3}(z)\) in (2.45). As for \(J_{1}\), we first bound

$$ \int _{\varOmega }a_{11}\partial _{1}r_{3} \partial _{1}v\,dx\,dy\,dz. $$

By integration by parts, the Poincaré inequality, (2.43), and (2.44), we get

$$\begin{aligned}& \biggl\vert \int _{\varOmega }a_{11}\partial _{1}r_{3} \partial _{1}v\,dx\,dy\,dz \biggr\vert \\& \quad \leq \sum _{e} \biggl\vert \int _{e} a_{11}\partial _{1}r_{3} \partial _{1}v\,dx\,dy\,dz \biggr\vert \\& \quad \leq \sum_{e} \biggl\vert \int _{e} a^{0}_{11}\partial _{1}\beta _{3}(x,y) \omega _{3}(z)\partial _{1}v\,dx\,dy\,dz \biggr\vert \\& \qquad {}+\sum_{e} \biggl\vert \int _{e}\mathcal{O}(h_{z})\partial _{1}\beta _{3}(x,y) \omega _{3}(z)\partial _{1}v\,dx\,dy\,dz \biggr\vert \\& \quad \leq \sum_{e} \biggl\vert \int _{e} a^{0}_{11}\partial _{1}\beta _{3}(x,y) \tilde{D}^{-1}\omega _{3}(z)\partial _{3}\partial _{1}v\,dx\,dy\,dz \biggr\vert \\& \qquad {}+\sum_{e} \biggl\vert \int _{e}\mathcal{O}(h_{z})\partial _{1}\beta _{3}(x,y) \omega _{3}(z)\partial _{1}v\,dx\,dy\,dz \biggr\vert \\& \quad \leq Ch^{4}_{z} \Vert u \Vert _{4,\infty ,\varOmega } \vert v \vert ^{h}_{2,1,\varOmega } +Ch ^{4}_{z} \Vert u \Vert _{4,\infty ,\varOmega }\sum_{e} \int _{e} \vert \partial _{1}v \vert \,dx\,dy\,dz \\& \quad \leq Ch^{4}_{z} \Vert u \Vert _{4,\infty ,\varOmega } \vert v \vert ^{h}_{2,1,\varOmega } +Ch ^{4}_{z} \Vert u \Vert _{4,\infty ,\varOmega } \int _{\varOmega _{xy}} \vert \partial _{1}v \vert _{1,1,\varOmega _{z}}\,dx\,dy \\& \quad \leq Ch^{4}_{z} \Vert u \Vert _{4,\infty ,\varOmega } \vert v \vert ^{h}_{2,1,\varOmega }, \end{aligned}$$

where \(\frac{d(\tilde{D}^{-1}\omega _{3}(z))}{dz}=\omega _{3}(z)\), \(\tilde{D}^{-1}\omega _{3}=\mathcal{O}(h^{1.5} _{z})\), \(a_{11}(N)=a_{11}(N_{0})+\mathcal{O}(h_{z})\equiv a^{0}_{11}+ \mathcal{O}(h_{z})\) for every \(N\in \bar{e}\), and \(N_{0}\) is the center of ē.

Similarly, for the rightmost term from (2.38), we can easily obtain

$$ \biggl\vert \int _{\varOmega }a_{0}r_{3}v\,dx\,dy\,dz \biggr\vert \leq Ch^{4}_{z} \Vert u \Vert _{3,\infty ,\varOmega } \vert v \vert ^{h}_{2,1,\varOmega }. $$

As for the other terms from (2.38), using arguments similar to the ones mentioned above, we derive their bounds as follows:

$$ Ch^{4}_{z} \Vert u \Vert _{4,\infty ,\varOmega } \vert v \vert ^{h}_{2,1,\varOmega }. $$

Thus, we have

$$ \vert J_{1} \vert \leq Ch^{4}_{z} \Vert u \Vert _{4,\infty ,\varOmega } \vert v \vert ^{h}_{2,1,\varOmega }. $$
(2.46)

As for \(J_{2}\), we first analyze the case of \(j=1\). By Green’s formula, we get

$$ \int _{\varOmega }a_{31}\partial _{3}r_{3} \partial _{1}v\,dx\,dy\,dz =-\sum_{e} \int _{e}\partial _{1}\bigl(a_{31} \beta _{3}(x,y)\bigr)l_{2}(z)v\,dx\,dy\,dz. $$
(2.47)

For the right term from (2.47), integration by parts yields

$$\begin{aligned} \int _{\varOmega }a_{31}\partial _{3}r_{3} \partial _{1}v\,dx\,dy\,dz =&-\sum_{e} \int _{e}\tilde{D}^{-2}l_{2}\bigl( \partial _{3}\partial _{3}v\partial _{1}(a _{31}\beta _{3}) \\ &{}+2\partial _{3}v\partial _{3}\partial _{1}(a_{31}\beta _{3})+v\partial _{3}\partial _{3}\partial _{1}(a_{31} \beta _{3})\bigr)\,dx\,dy\,dz, \end{aligned}$$

where \(\frac{d^{2}(\tilde{D}^{-2}l_{2}(z))}{dz^{2}}=l _{2}(z)\). From (2.43) and (2.44),

$$ \tilde{D}^{-2}l_{2}=\mathcal{O}\bigl(h^{1.5}_{z} \bigr),\qquad \vert \beta _{3} \vert \leq Ch^{2.5}_{z} \Vert u \Vert _{3,\infty ,\varOmega },\qquad \vert \partial _{1}\beta _{3} \vert \leq Ch^{2.5}_{z} \Vert u \Vert _{4,\infty ,\varOmega }. $$
(2.48)

Hence,

$$ \biggl\vert \int _{\varOmega }a_{31}\partial _{3}r_{3} \partial _{1}v\,dx\,dy\,dz \biggr\vert \leq Ch^{4}_{z} \Vert u \Vert _{4,\infty ,\varOmega } \sum_{e} \int _{e}\bigl( \vert \partial _{3}\partial _{3}v \vert + \vert \partial _{3}v \vert + \vert v \vert \bigr)\,dx\,dy\,dz. $$
(2.49)

By the Poincaré inequality in (2.49), we get

$$ \biggl\vert \int _{\varOmega }a_{31}\partial _{3}r_{3} \partial _{1}v\,dx\,dy\,dz \biggr\vert \leq Ch^{4}_{z} \Vert u \Vert _{4,\infty ,\varOmega } \vert v \vert ^{h}_{2,1,\varOmega }. $$
(2.50)

Similarly, in the case of \(j=2\), we also have

$$ \biggl\vert \int _{\varOmega }a_{32}\partial _{3}r_{3} \partial _{2}v\,dx\,dy\,dz \biggr\vert \leq Ch^{4}_{z} \Vert u \Vert _{4,\infty ,\varOmega } \vert v \vert ^{h}_{2,1,\varOmega }. $$
(2.51)

For the right term from (2.39), integration by parts yields

$$\begin{aligned} \int _{\varOmega }a_{3}\partial _{3}r_{3}v\,dx\,dy\,dz =& \sum_{e} \int _{e} a_{3} \beta _{3}(x,y)l_{2}(z)v\,dx\,dy\,dz \\ =&\sum_{e} \int _{e}\tilde{D}^{-2}l_{2}( \partial _{3} \partial _{3}va_{3}\beta _{3} +2\partial _{3}v\partial _{3}a_{3} \beta _{3} +v\partial _{3}\partial _{3}a_{3} \beta _{3})\,dx\,dy\,dz. \end{aligned}$$

Using (2.48) and the Poincaré inequality, we obtain

$$ \biggl\vert \int _{\varOmega }a_{3}\partial _{3}r_{3}v\,dx\,dy\,dz \biggr\vert \leq Ch^{4} _{z} \Vert u \Vert _{3,\infty ,\varOmega } \vert v \vert ^{h}_{2,1,\varOmega }. $$
(2.52)

Combining (2.50)–(2.52) yields

$$ \vert J_{2} \vert \leq Ch^{4}_{z} \Vert u \Vert _{4,\infty ,\varOmega } \vert v \vert ^{h}_{2,1,\varOmega }. $$
(2.53)

As for \(J_{3}\), we first consider the case of \(i=1\). Clearly, integration by parts yields

$$\begin{aligned}& \int _{\varOmega }a_{13}\partial _{1}r_{3} \partial _{3}v\,dx\,dy\,dz\\& \quad =\sum_{e} \int _{e} a_{13}\partial _{1}\beta _{3}(x,y)\omega _{3}(z)\partial _{3}v\,dx\,dy\,dz \\& \quad =-\sum_{e} \int _{e}\partial _{1}\beta _{3}(x,y)\tilde{D} ^{-1}\omega _{3}(z) \partial _{3}(a_{13}\partial _{3}v)\,dx\,dy\,dz. \end{aligned}$$

Furthermore, by (2.48), the Poincaré inequality and \(\tilde{D}^{-1}\omega _{3}=\mathcal{O}(h^{1.5}_{z})\), we have

$$ \biggl\vert \int _{\varOmega }a_{13}\partial _{1}r_{3} \partial _{3}v\,dx\,dy\,dz \biggr\vert \leq Ch^{4}_{z} \Vert u \Vert _{4,\infty ,\varOmega } \vert v \vert ^{h}_{2,1,\varOmega }. $$

Similarly, when \(i=2\), we also get

$$ \biggl\vert \int _{\varOmega }a_{23}\partial _{2}r_{3} \partial _{3}v\,dx\,dy\,dz \biggr\vert \leq Ch^{4}_{z} \Vert u \Vert _{4,\infty ,\varOmega } \vert v \vert ^{h}_{2,1,\varOmega }. $$

Thus, we have

$$ \vert J_{3} \vert \leq Ch^{4}_{z} \Vert u \Vert _{4,\infty ,\varOmega } \vert v \vert ^{h}_{2,1,\varOmega }. $$
(2.54)

Finally, for \(J_{4}\), integration by parts yields

$$\begin{aligned}& \int _{\varOmega }a_{33}\partial _{3}r_{3} \partial _{3}v\,dx\,dy\,dz\\& \quad =\sum_{e} \int _{e} a_{33}\beta _{3}(x,y)l_{2}(z) \partial _{3}v\,dx\,dy\,dz \\& \quad =\sum_{e} \int _{e}\beta _{3}\tilde{D}^{-2}l_{2}(2 \partial _{3}\partial _{3}v\partial _{3}a_{33}+ \partial _{3}v\partial _{3}\partial _{3}a_{33})\,dx\,dy\,dz. \end{aligned}$$

Thus,

$$ \biggl\vert \int _{\varOmega }a_{33}\partial _{3}r_{3} \partial _{3}v\,dx\,dy\,dz \biggr\vert \leq Ch^{4}_{z} \Vert u \Vert _{3,\infty ,\varOmega } \vert v \vert ^{h}_{2,1,\varOmega }. $$

Hence,

$$ \vert J_{4} \vert \leq Ch^{4}_{z} \Vert u \Vert _{3,\infty ,\varOmega } \vert v \vert ^{h}_{2,1,\varOmega }. $$
(2.55)

Combining (2.42), (2.46), and (2.53)–(2.55) results in

$$ \bigl\vert a(R_{z},v) \bigr\vert \leq Ch^{4}_{z} \Vert u \Vert _{4,\infty ,\varOmega } \vert v \vert ^{h}_{2,1, \varOmega }. $$
(2.56)

From (2.36) and (2.56), the desired result (2.1) is immediately obtained. The proof of Lemma 2.1 is therefore completed. □

Pointwise superconvergence estimates

To analyze pointwise superconvergence, for each fixed \(Z\in \varOmega \), we may introduce the discrete Green function defined by

$$ a\bigl(v,G^{h}_{Z}\bigr)=v(Z) \quad \forall v\in S^{h}_{0}(\varOmega ). $$
(3.1)

As for \(G^{h}_{Z}\), we have the following result.

Lemma 3.1

For\(G^{h}_{Z}\in S^{h}_{0}(\varOmega )\)the discrete Green function, we have the following estimate:

$$ \bigl\vert G^{h}_{Z} \bigr\vert ^{h}_{2,1,\varOmega }\leq C \vert \ln h \vert ^{\frac{2}{3}}. $$
(3.2)

The proof of Lemma 3.1 can be found in [16].

From (1.4), (2.1), (3.1), and (3.2), we immediately obtain the following theorem.

Theorem 3.1

Let\(\{\mathcal{T}^{h}\}\)be a uniform family of pentahedral partitions ofΩ, and\(u\in W^{5,\infty }(\varOmega )\cap H^{1}_{0}(\varOmega )\). For\(u_{h}\)andΠu, the\(\mathcal{P}_{2}(x,y)\otimes \mathcal{P}_{2}(z)\)pentahedral finite element approximation and the corresponding interpolant tou, respectively, we have the following pointwise superconvergence estimate:

$$ \vert u_{h}-\varPi u \vert _{0,\infty ,\varOmega }\leq C \bigl(h_{xy}^{4}+h_{z} ^{4}\bigr) \vert \ln h \vert ^{\frac{2}{3}} \Vert u \Vert _{5,\infty ,\varOmega }. $$

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Acknowledgements

The authors would like to thank the editors for their help to improve the quality of the article.

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Funding

This work is supported by Hainan Provincial Natural Science Foundation of China (Grant 119MS038), National Natural Science Foundation of China (Grant 11161039), and Natural Science Foundation of Ningbo (Grant 2017A610133).

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The first author proved Lemma 2.1 and Theorem 3.1, and the second author gave the idea of this article. All authors read and approved the final manuscript.

Correspondence to Jinghong Liu.

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Liu, J., Zhu, Q. Superconvergence of the function value for pentahedral finite elements for an elliptic equation with varying coefficients. Bound Value Probl 2020, 7 (2020) doi:10.1186/s13661-019-01318-y

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Keywords

  • Pentahedral finite element
  • Interpolation fundamental estimate
  • Superconvergence
  • Discrete Green’s function