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.
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1 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 [3–25] 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:
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
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:
where
and
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 \(C_{1}\leq \frac{h_{z}}{h_{xy}}\leq C_{2}\).
We introduce an \(\mathcal{P}_{2}(x,y)\otimes \mathcal{P}_{2}(z)\) polynomial space denoted by \(\mathcal{P}\), that is,
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
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,
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:
Thus, the finite element method of (1.2) is
From (1.2) and (1.3), the following Galerkin orthogonal relation holds:
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 \(a(u-\varPi u,v)\).
2 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:
where\(|v|^{h}_{2,1,\varOmega }=\sum_{e\in {\mathcal{T}}^{h}}|v|_{2,1,e}\).
Proof
Clearly, the interpolation remainder is
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
We set
Clearly,
By the two-dimensional interpolation fundamental estimate of triangular quadratic elements [26], we have
As for \(I_{2}\), by Green’s formula, we have
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
As for \(I_{3}\), we first bound the integral
By Green’s formula and \(v=0\) on ∂Ω, we get
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
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
where \(Q_{0}\) is the center of \(S_{j}\). Thus, by (2.11) and (2.12), we have
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
Furthermore, we easily obtain
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 \(S_{ij}\)
Similar to (2.12), we have
where \(Q^{*}_{0}\) is the center of \(S_{ij}\). Hence, by (2.16) and (2.17), we get
For simplicity, we write
Thus,
Clearly,
Taking \(\tilde{p}_{3}\) a three-degree interpolant to u on \(S_{ij}\) in (2.20), we have
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
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
is a seminorm of w on e. Using the rightmost rule from (2.22), we find that
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
Using the left rule from (2.22), we get
Combining (2.23)–(2.25) yields
Summing over all e in \(\mathcal{T}^{h}\) results in
Combining (2.19), (2.21), and (2.26) yields
Similar to the arguments mentioned above, we also get
Thus, by (2.15) and (2.29), we have
Similar to the proof of (2.30), we have
Combining (2.6), (2.30), and (2.31), we get
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,
Similar to the proof of (2.13), we have
Clearly,
Combining (2.33) and (2.34) yields
From (2.8)–(2.10), (2.32), and (2.35),
Now, we can bound the term
Additionally, we set
Clearly,
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:
where
The coefficients \(\beta _{j}(x,y)\) satisfy \(\beta _{0}(x,y)=u(x,y,z_{i-1})\), and for \(j\geq 1\),
Let \(\varPi _{z}^{e}\) be the quadratic interpolation operator of projection type with respect to z defined by
Thus, the interpolation remainder is
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
By integration by parts, the Poincaré inequality, (2.43), and (2.44), we get
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
As for the other terms from (2.38), using arguments similar to the ones mentioned above, we derive their bounds as follows:
Thus, we have
As for \(J_{2}\), we first analyze the case of \(j=1\). By Green’s formula, we get
For the right term from (2.47), integration by parts yields
where \(\frac{d^{2}(\tilde{D}^{-2}l_{2}(z))}{dz^{2}}=l _{2}(z)\). From (2.43) and (2.44),
Hence,
By the Poincaré inequality in (2.49), we get
Similarly, in the case of \(j=2\), we also have
For the right term from (2.39), integration by parts yields
Using (2.48) and the Poincaré inequality, we obtain
Combining (2.50)–(2.52) yields
As for \(J_{3}\), we first consider the case of \(i=1\). Clearly, integration by parts yields
Furthermore, by (2.48), the Poincaré inequality and \(\tilde{D}^{-1}\omega _{3}=\mathcal{O}(h^{1.5}_{z})\), we have
Similarly, when \(i=2\), we also get
Thus, we have
Finally, for \(J_{4}\), integration by parts yields
Thus,
Hence,
Combining (2.42), (2.46), and (2.53)–(2.55) results in
From (2.36) and (2.56), the desired result (2.1) is immediately obtained. The proof of Lemma 2.1 is therefore completed. □
3 Pointwise superconvergence estimates
To analyze pointwise superconvergence, for each fixed \(Z\in \varOmega \), we may introduce the discrete Green function defined by
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:
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:
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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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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). https://doi.org/10.1186/s13661-019-01318-y
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DOI: https://doi.org/10.1186/s13661-019-01318-y