$W^{1, \infty}$-seminorm superconvergence of the block finite element method for the five-dimensional Poisson equation

1.   Introduction

Superconvergence is a crucial topic in the research field of Galerkin finite element methods. With advancements in research technologies, numerous superconvergence results have been obtained and a theoretical framework on superconvergence has been established ^[1,2,3,4]. Notably, superconvergence research in the finite element method is closely associated with the dimensions of the problems. For one- and two-dimensional problems, superconvergence research has been nearly perfect, and significant progress has also been made for the three-dimensional setting ^{[5,6,7,8,9,10,11,12,13,14,15,16,17]}. However, for dimensions four and higher, superconvergence results are relatively scarce. One of the primary reasons is the difficulty in estimating the discrete Green's function (DGF) and discrete derivative Green's function (DDGF), which play critical roles in superconvergence research, particularly in pointwise superconvergence ^{[10,12,13,14,18,19,20]}. Unfortunately, the high-dimensional DGF and DDGF of dimensions four and higher cannot be straightforwardly extrapolated from those of low-dimensional cases. Consequently, establishing bounds for the high-dimensional DGF and DDGF becomes a vital aspect of researching high-dimensional pointwise superconvergence. Recently, we have obtained some results in this regard ^[12,14,19]. In this paper, we will illustrate a superconvergence analytic technique by considering a five-dimensional problem as an example, which may be applied to other high-dimensional superconvergence issues. First, by utilizing a weighted-function analysis technique, we will derive an optimal estimate for the $W^{2, 1}$-seminorm of the five-dimensional DDGF. Second, in conjunction with a weak estimate for the finite element approximation, we can obtain a pointwise superconvergence estimate for the derivatives of the finite element approximation. Finally, we validate the theoretical results through a numerical example. It is important to note that although we provide the $W^{1, 1}$-seminorm estimate for the five-dimensional DDGF, we do not discuss its applications to superconvergence ^[19]. Furthermore, the $W^{2, 1}$-seminorm estimate for the DDGF in this paper is distinct from the $W^{1, 1}$-seminorm estimate in ^[19]. Additionally, as high-dimensional partial differential equations (PDEs) in dimensions four and higher are encountered in various areas such as financial mathematics, particle physics, statistical physics and general relativity, research on high-dimensional finite element superconvergence holds significant importance.

In the following, we adhere to the standard notations for the Sobolev spaces and their norms. The symbol $C$ represents a constant that is independent of the discretization parameter $h$.

Consider the following Poisson equation with the Dirichlet boundary condition:

where $\Omega\subset{\cal R}^5$ is a bounded polytopic domain. The weak formulation of problem (1.1) is finding $u\in H^1_0(\Omega)$ such that

where

Let $\{{\cal T}^h\}$ be a regular family of rectangular partitions of $\bar{\Omega}$. Denote by $S^h(\Omega)$ the space of continuous piecewise tensor-product $m$-order polynomials with respect to this kind of partition and let $S^h_0 = S^h(\Omega)\cap H^1_0(\Omega)$. Discretizing problem (1.2) using $S^h_0$ as approximating space means finding $u_h\in S^h_0$ such that $a(u_h\, , \, v) = (f\, , \, v)$ for all $v\in S^h_0$. Here, we have the following Galerkin orthogonality relation:

For every $Z\in\Omega$, we define the discrete derivative $\delta$ function $\partial_{Z, \ell}\delta^h_Z\in S^h_0$ and the $L^2$-projection $P_hu\in S^h_0$ such that ^[19]

where $\ell\in {\cal R}^5$ and $|\ell| = 1$. $\partial_\ell v(Z)$ stands for the one-sided directional derivative

Remark 1.1. Since $\Delta Z = |\Delta Z|\ell$, that is, $\Delta Z$ is of the same direction as $\ell$, provided that the direction $\ell$ is given for, the above limit exists, no matter what direction is given, $\partial_\ell v(Z)$ is well defined.

Let $\partial_{Z, \ell}G^*_Z\in H^2(\Omega)\cap H^1_0(\Omega)$ be the solution of the elliptic problem ${\cal L}\partial_{Z, \ell}G^*_Z = \partial_{Z, \ell}\delta^h_Z$. We may call $\partial_{Z, \ell}G^*_Z$ the regularized derivative Green's function (RDGF), and denote by $\partial_{Z, \ell}G^h_Z$ (the so-called DDGF) the finite element approximation to $\partial_{Z, \ell}G^*_Z$. Thus,

One of the main tasks of this article is how to obtain the optimal estimate for the $\partial_{Z, \ell}G^h_Z$.

As for $\partial_{Z, \ell}\delta^h_Z$ and $P_h$, we have ^[19]

The rest of the paper is arranged as follows. In section two, we will bound the terms $|\partial_{Z, \ell}G^*_Z|_{2, \, 1}$ and $|\partial_{Z, \ell}G^h_Z|_{2, \, 1}$. Section three is devoted to the weak estimate of the second type for the finite element method, and section four is about $W^{1, \infty}$-seminorm superconvergence of the finite element approximation. A numerical example is given in section five. Finally, we simply summarize the paper in section six.

2.   The $W^{2, 1}$-seminorm estimate for the DDGF

To derive the estimate for the DDGF, we introduce the weight function defined by

where $\bar{X}\in\bar\Omega$ is a fixed point, $\theta = \gamma h$ and $\gamma\in [5, +\infty)$ is a suitable real number.

For every $\alpha\in {\cal R}$, we give the following notations:

where $\beta = (\beta_1, \beta_2, \beta_3, \beta_4, \beta_5)$, $|\beta| = \beta_1+\beta_2+\beta_3+\beta_4+\beta_5$ and $\beta_i\geq0, \, \, i = 1, 2, 3, 4, 5$ are integers. In particular, for the case of $m = 0$, we write

As for the weight function $\phi$, we have the properties ^[19]

In addition, we need to assume that there exists a $q_0$ $(5 < q_0\leq\infty)$ such that

is a homeomorphism ^[4]. It means that for all $v\in W^{2, \, q}(\Omega)\cap W^{1, \, q}_0(\Omega)$, we have the so-called a priori estimate:

where $C(q)$ denotes a positive constant only depending on $q$. Next, we give some lemmas used in the proofs of our main results.

Lemma 2.1. For $\partial_{Z, \ell}G^*_Z$ the RDGF, we have the weighted-norm estimate

Proof. By the triangular inequality, the a priori estimate (2.5) and the definition of $\partial_{Z, \ell}G^*_Z$, we have

namely,

Further, from the inverse estimate, the stability estimate (1.5), the a priori estimate (2.5) and the Sobolev embedding theorem ^[21],

where ${\cal L}w = \phi^{-\frac15}\partial_{Z, \ell}G^*_Z$ in $\Omega$ and $w|_{\partial\Omega} = 0$. Taking $q = |\ln h|$ in (2.8) yields

However, from (2.1), the definition of $\partial_{Z, \ell}G^*_Z$ and the triangular inequality,

In addition, from the H$\ddot{o}$lder inequality,

Since ^[19]

from (2.2), (2.11) and (2.12),

Using the H$\ddot{o}$lder inequality again, we have

From (1.4), (2.2) and (2.14),

Similarly,

From (2.9), (2.10), (2.13), (2.15) and (2.16),

Taking $\epsilon = \frac12$ in (2.17) yields

Combining (1.4), (2.7) and (2.18) immediately yields the result (2.6).

Lemma 2.2. For $\partial_{Z, \ell}G^*_Z$ the RDGF, we have the $W^{2, \, 1}$-seminorm estimate

Proof. Obviously,

By (2.3), (2.6) and (2.20), we immediately obtain the result (2.19).

In order to derive the estimate of the DDGF $\partial_{Z, \ell}G^h_Z$, we need the following result ^[19].

Lemma 2.3. ^[19] For $\partial_{Z, \ell}G^*_Z$ and $\partial_{Z, \ell}G^h_Z$, the RDGF and the DDGF, respectively, we have

Now, we can derive the following important estimate.

Theorem 2.1. For $\partial_{Z, \ell}G^h_Z$ the DDGF, we have the $W^{2, \, 1}$-seminorm estimate

where $|\partial_{Z, \ell}G^h_Z|^h_{2, \, 1} = \sum\limits_{e\in {\cal T}^h}|\partial_{Z, \ell}G^h_Z|_{2, \, 1, \, e}$.

Proof. We denote by $\Pi\partial_{Z, \ell}G^*_Z$ the interpolant of projection type to $\partial_{Z, \ell}G^*_Z$. Thus, by the triangle inequality, the interpolation error estimate and the inverse property, we have

Combining (2.19), (2.21) and (2.23) yields the result (2.22).

3.   Weak estimate of the second type for the finite element

In this section, we first introduce an interpolation operator of projection type, and then derive the weak estimate of the second type for the finite element by using the interpolation operator of projection type.

Let element

and let $\{l_{1, j}(x_1)\}^\infty_{j = 0}$, $\{l_{2, j}(x_2)\}^\infty_{j = 0}$, $\{l_{3, j}(x_3)\}^\infty_{j = 0}$, $\{l_{4, j}(x_4)\}^\infty_{j = 0}$, $\{l_{5, j}(x_5)\}^\infty_{j = 0}$ be the normalized orthogonal Legendre polynomial systems on $L^2(I_1)$, $L^2(I_2)$, $L^2(I_3)$, $L^2(I_4)$, $L^2(I_5)$, respectively. Now, let $\partial_{x_1}\partial_{x_2}\partial_{x_3}\partial_{x_4}\partial_{x_5}u\in L^2(e)$ then we have the following expansion:

where

Set

By the Parseval equality for $X = (x_1, x_2, x_3, x_4, x_5)\in e$,

where

where $i_k\geq1, k = 1, \cdots, 5$. Similarly, the other coefficients can also be given.

We introduce standard tensor-product polynomial spaces of degree $m\geq1$ denoted by $T_m$, i.e.,

where the indexing set $I$ is as follows:

Define the tensor-product interpolation operator of projection type by $\Pi_m^e$: $H^5(e)\rightarrow T_m(e)$ such that

By the definitions of the finite element space $S^h_0(\Omega)$ and $\Pi_m^e$, we have the tensor-product interpolation operator of project type

where $(\Pi_mu)|_e = \Pi_m^eu$.

For simplicity, we write

Thus, from (3.1) and (3.2),

Next, we will derive the weak estimate of the second type for the finite element.

Theorem 3.1. Let $\{{\cal T}^h\}$ be a regular family of rectangular partitions of $\bar{\Omega}$, $u\in W^{m+2, \, \infty}(\Omega)\cap H^1_0(\Omega)$ and $v\in S_0^h(\Omega)$ then the tensor-product $m$-degree interpolation operator of projection type $\Pi_m$ satisfies the weak estimate of the second type

where $|v|^h_{2, \, 1, \, \Omega} = \sum\limits_{e\in {\cal T}^h}|v|_{2, \, 1, \, e}$.

Proof. By the properties of $\omega_{k, i}(x_k)$ as well as the orthogonality of the Legendre polynomial system, we have

where

Clearly, $r$ only contains finite terms.

Among the indices $i_k, \, k = 1, \cdots, 5$, when some $i_k = m+1$ or $m+2$ and the others are zero, we have by the orthogonality of the Legendre polynomial system

For $m\geq2$, without loss of generality, we assume $i_k\neq0, \, k = 1, \cdots, j$ and $i_{j+1} = \cdots = i_5 = 0$.

We assume $i_1\geq m+1$; thus, $i_1\geq m+1\geq 3$. By the orthogonality of the Legendre polynomial system,

In addition,

and it is easy to prove

In fact,

Combining (3.8)–(3.10) yields

Similarly, we have

From (3.6), (3.7), (3.11) and (3.12),

When each $i_k\neq0, \, k = 1, \cdots, 5$, similar to the above arguments, we easily get

From (3.4), (3.5), (3.13) and (3.14),

Summing over all elements proves the result (3.3).

4.   $W^{1, \infty}$-seminorm superconvergence of the block finite element approximation

In this section, we will give the superconvergent estimate for the $m$-degree block finite element approximation by using the weak estimate of the second type and the $W^{2, 1}$-seminorm estimate for the DDGF.

Let $u_h$ be the $m$-degree block finite element approximation to $u$, the solution of problem (1.2) and $\Pi_m u$ the corresponding interpolant of projection type to $u$. Thus, we have the following theorem.

Theorem 4.1. Let $\{{\cal T}^h\}$ be a regular family of rectangular partitions of $\bar{\Omega}$ and $u\in W^{m+2, \, \infty}(\Omega)\cap H^1_0(\Omega)$ then we have the superconvergent estimate

Proof. For every $Z\in\Omega$, applying the definition of $\partial_{Z, \ell}G^h_Z$ and the Galerkin orthogonality relation (1.3), we derive

From (2.22), (3.3) and (4.2), we immediately obtain the result (4.1).

5.   A numerical example

Example 5.1. Consider the following Poisson equation:

where

The true solution is

Let $u_h$ be the tensor-product two-degree finite element approximation to $u$ on uniform rectangular meshes and $\Pi u$ the corresponding interpolant of the projection type. Set $X^* = (0.5, 0.5, 0.5, 0.5, 0.5)$, $Y^* = (0.25, 0.25, 0.25, 0.25, 0.25)$ and $Z^* = (0.125, 0.125, 0.125, 0.125, 0.125)$. We solve Example 5.1 and obtain the following numerical results (see Table 1):

Here, the operator $\bar{\nabla} = (\bar{\partial}_{x_1}, \bar{\partial}_{x_2}, \bar{\partial}_{x_3}, \bar{\partial}_{x_4}, \bar{\partial}_{x_5})$ and $\bar{\partial}_{x_i}v = \frac{1}{2}(|\partial^{+}_{x_i}v|+|\partial^{-}_{x_i}v|), \, i = 1, \cdots, 5$. The numerical results demonstrate our theoretical results.

6.   Conclusions

In this paper, we proposed two important analytic tools: The DDGF and the weak estimate of the second type. By combining these tools, we obtained a pointwise superconvergence estimate for the finite element approximation in the $W^{1, \infty}$-seminorm, which is a challenging issue in the field of superconvergence for the finite element method. The main difficulty of the paper lies in deriving the optimal order estimate for the DDGF in the $W^{2, 1}$-seminorm. Notably, the estimates for the DDGF vary for different dimensions of elliptic equations, necessitating different approaches for them. Although the methods presented in the paper are specifically developed for the five-dimensional second-order elliptic equation, they can also be applied to other high-dimensional second-order elliptic equations.

Use of AI tools declaration

The author declares he has not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work was supported by the Special Projects in Key Fields of Ordinary Colleges and Universities in Guangdong Province (2022ZDZX3016) and the Projects of Talents Recruitment of Guangdong University of Petrochemical Technology (2021rc003).

Conflict of interest

The author declares no conflict of interest.