Superconvergence patch recovery for the gradient of the tensor-product linear triangular prism element

Abstract

In this article, we study superconvergence of the finite element approximation to the solution of a general second-order elliptic boundary value problem in three dimensions over a fully uniform mesh of piecewise tensor-product linear triangular prism elements. First, we give the superclose property of the gradient between the finite element solution u_h and the interpolant Πu. Second, we introduce a superconvergence recovery scheme for the gradient of the finite element solution. Finally, superconvergence of the recovered gradient is derived.

1 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 [1–6]). Recently, we studied the superconvergence patch recovery (SPR) technique introduced by Zienkiewicz and Zhu [7–9] 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 L^2-norm. In addition, Chen [11] and Goodsell [12] derived superconvergence estimates of the recovered gradient by the L^2-projection technique and the average technique, respectively. This article will use the SPR technique to obtain a superconvergence estimate for the gradient of the tensor-product linear triangular prism element. In this article, we shall use the letter C to denote a generic constant which may not be the same in each occurrence and also use the standard notations for the Sobolev spaces and their norms.

2 General elliptic boundary value problem and finite element discretization

We consider the model problem

\mathcal{L}u\equiv -\sum _{i,j=1}^3{\partial }_j(a_{ij}{\partial }_{i}u)+\sum _{i=1}^3{a}_i{\partial }_{i}u+a_{0}u=f\text{in }\mathrm{\Omega },u=0\text{on }\partial \mathrm{\Omega }.

(2.1)

Here \mathrm{\Omega }={[0,1]}^2\times [0,1]={\mathrm{\Omega }}_{xy}\times {\mathrm{\Omega }}_z\subset {\mathcal{R}}^3 is a rectangular block with boundary ∂ Ω consisting of faces parallel to the x-, y-, and z-axes. We also assume that the given functions a_{ij},a_i\in W^{1,\mathrm{\infty }}(\mathrm{\Omega }), a_0\in L^{\mathrm{\infty }}(\mathrm{\Omega }), and f\in L^2(\mathrm{\Omega }). In addition, we 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}, which are usual partial derivatives.

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 \{{\mathcal{T}}^h\} these uniform partitions as above. Thus \overline{\mathrm{\Omega }}={\bigcup }_{e\in {\mathcal{T}}^h}\overline{e}. Obviously, we can write e=D\times L (see Figure 1), where D and L represent a triangle parallel to the xy-plane and a one-dimensional interval parallel to the z-axes, respectively.

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 tensor-product linear polynomial space denoted by , that is,

q(x,y,z)=\sum _{(i,j,k)\in \mathcal{I}}{a}_{ijk}{x}^i{y}^j{z}^k,a_{ijk}\in \mathcal{R},q\in \mathcal{P},

where \mathcal{P}={\mathcal{P}}_{xy}\otimes {\mathcal{P}}_z, {\mathcal{P}}_{xy} stands for the linear polynomial space with respect to (x,y), and {\mathcal{P}}_z is the linear polynomial space with respect to z. The indexing set ℐ satisfies \mathcal{I}=\{(i,j,k)|i,j,k\ge 0,i+j\le 1,k\le 1\}. Let {\mathrm{\Pi }}_{xy}^e be the linear interpolation operator with respect to (x,y)\in D, and let {\mathrm{\Pi }}_z^e be the linear interpolation operator with respect to z\in L. Thus we may define the tensor-product linear interpolation operator by {\mathrm{\Pi }}^e:H_0^1(e)\to \mathcal{P}(e). Obviously, {\mathrm{\Pi }}^e={\mathrm{\Pi }}_{xy}^e\otimes {\mathrm{\Pi }}_z^e={\mathrm{\Pi }}_z^e\otimes {\mathrm{\Pi }}_{xy}^e. In addition, the weak form of problem (2.1) is

a(u,v)=(f,v)\mathrm{\forall }v\in H_0^1(\mathrm{\Omega }),

(2.2)

where

a(u,v)\equiv {\int }_{\mathrm{\Omega }}(\sum _{i,j=1}^3{a}_{ij}{\partial }_{i}u{\partial }_{j}v+\sum _{i=1}^3{a}_i{\partial }_{i}uv+a_{0}uv)dxdydz,

and

(f,v)={\int }_{\mathrm{\Omega }}fvdxdydz.

Define the tensor-product linear triangular prism finite element space by

S_0^h(\mathrm{\Omega })=\{v\in H_0^1(\mathrm{\Omega }):v{|}_e\in \mathcal{P}(e)\mathrm{\forall }e\in {\mathcal{T}}^h\}.

Thus, the finite element method of problem (2.2) is to find u_h\in S_0^h(\mathrm{\Omega }) such that

a(u_h,v)=(f,v)\mathrm{\forall }v\in S_0^h(\mathrm{\Omega }).

Moreover, from the definitions of {\mathrm{\Pi }}^e and S_0^h(\mathrm{\Omega }), we may define a global tensor-product linear interpolation operator \mathrm{\Pi }:H_0^1(\mathrm{\Omega })\to S_0^h(\mathrm{\Omega }). Here (\mathrm{\Pi }u){|}_e={\mathrm{\Pi }}^{e}u. As for this interpolation operator, the following Lemma 2.1 holds (see [13]).

Lemma 2.1 For Πu and u_h, the tensor-product linear interpolant and the tensor-product linear triangular prism finite element approximation to u, respectively, we have the supercloseness estimate

|u_h-\mathrm{\Pi }u{|}_{1,\mathrm{\infty },\mathrm{\Omega }}\le C{h}^2{|lnh|}^{\frac{4}{3}}{\parallel u\parallel }_{3,\mathrm{\infty },\mathrm{\Omega }}.

(2.3)

3 Gradient recovery and superconvergence

For v\in S_0^h(\mathrm{\Omega }), we consider a SPR scheme of ∇v. We denote by R_h the SPR-recovery operator for ∇v and begin by defining the point values of R_{h}v 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 R_{h}v. Obviously, R_{h}v\in {(S_0^h(\mathrm{\Omega }))}^3.

Let us first assume that N is an interior node of the partition {\mathcal{T}}^h, and denote by ω the element patch around N containing 12 triangular prisms. Under the local coordinate system centered N, we let Q_i(x_i,y_i,z_i) be the barycenter of a triangular prism e_i\subset \omega, i=1,2,\dots ,12. Obviously, {\sum }_{i=1}^{12}(x_i,y_i,z_i)=(0,0,0). SPR uses the discrete least-squares fitting to seek linear vector function \mathbf{p}\in {(P_1(\omega ))}^3 such that

\sum _{i=1}^{12}[\mathbf{p}(Q_i)-\mathrm{\nabla }v(Q_i)]q(Q_i)=(0,0,0)\mathrm{\forall }q\in P_1(\omega ),

(3.1)

where v\in S_0^h(\mathrm{\Omega }). We define R_{h}v(N)=\mathbf{p}(0,0,0). If N is a boundary node, we calculate R_{h}v(N) by linear extrapolation from the values of R_{h}v already obtained at two neighboring interior nodes, N_1 and N_2 (with diagonal directions being used for edge nodes and corner nodes) (see Figure 2). Namely,

R_{h}v(N)=2R_{h}v(N_1)-R_{h}v(N_2).

Figure 2

N , a boundary node. N_1 and N_2 are interior nodes. We can calculate R_{h}v(N) by linear extrapolation from the values of R_{h}v already obtained at two neighboring interior nodes, N_1 and N_2.

Lemma 3.1 Let ω be the element patch around an interior node N, and u\in W^{3,\mathrm{\infty }}(\omega ). For \mathrm{\Pi }u\in S_0^h(\mathrm{\Omega }) the interpolant to u, we have

|\mathrm{\nabla }u(N)-R_h\mathrm{\Pi }u(N)|\le C{h}^2{\parallel u\parallel }_{3,\mathrm{\infty },\omega }.

(3.2)

Proof Choose v=\mathrm{\Pi }u and set q=1 in (3.1) to obtain {\sum }_{i=1}^{12}\mathbf{p}(Q_i)={\sum }_{i=1}^{12}\mathrm{\nabla }\mathrm{\Pi }u(Q_i). Therefore,

\begin{array}{rl}{R}_h\mathrm{\Pi }u(N)-\frac{1}{12}\sum _{i=1}^{12}\mathrm{\nabla }\mathrm{\Pi }u(Q_i)& =\mathbf{p}(0,0,0)-\frac{1}{12}\sum _{i=1}^{12}\mathbf{p}(x_i,y_i,z_i)\\ =-\frac{1}{12}\sum _{i=1}^{12}[{\partial }_1\mathbf{p}(0,0,0)x_i+{\partial }_2\mathbf{p}(0,0,0)y_i+{\partial }_3\mathbf{p}(0,0,0)z_i]\\ =(0,0,0).\end{array}

That is,

R_h\mathrm{\Pi }u(N)=\frac{1}{12}\sum _{i=1}^{12}\mathrm{\nabla }\mathrm{\Pi }u(Q_i).

(3.3)

Further,

\begin{array}{r}\frac{1}{12}\sum _{i=1}^{12}\mathrm{\nabla }\mathrm{\Pi }u(Q_i)-\mathrm{\nabla }u(N)\\ =\frac{1}{12}\sum _{i=1}^{12}\mathrm{\nabla }(\mathrm{\Pi }u-u)(Q_i)+\frac{1}{12}\sum _{i=1}^{12}[\mathrm{\nabla }u(Q_i)-\mathrm{\nabla }u(N)]\\ =\frac{1}{12}\sum _{i=1}^{12}\mathrm{\nabla }(\mathrm{\Pi }u-u)(Q_i)\\ +\frac{1}{12}\sum _{i=1}^{12}[{\partial }_1\mathrm{\nabla }u(N)x_i+{\partial }_2\mathrm{\nabla }u(N)y_i+{\partial }_3\mathrm{\nabla }u(N)z_i]+\mathbf{r}(u),\end{array}

(3.4)

where \frac{1}{12}{\sum }_{i=1}^{12}[{\partial }_1\mathrm{\nabla }u(N)x_i+{\partial }_2\mathrm{\nabla }u(N)y_i+{\partial }_3\mathrm{\nabla }u(N)z_i]=(0,0,0), and the high-order term \mathbf{r}(u) satisfies |\mathbf{r}(u)|\le C{h}^2{|u|}_{3,\mathrm{\infty },\omega }.

In (3.4), we write \mathbf{f}(u)=\frac{1}{12}{\sum }_{i=1}^{12}\mathrm{\nabla }(\mathrm{\Pi }u-u)(Q_i). For every u\in P_2(\omega ), it is not difficult to verify \mathbf{f}(u)=(0,0,0). Thus, by the Bramble-Hilbert lemma [14], we have |\mathbf{f}(u)|\le C{h}^2{|u|}_{3,\mathrm{\infty },\omega }. Therefore,

|\frac{1}{12}\sum _{i=1}^{12}\mathrm{\nabla }\mathrm{\Pi }u(Q_i)-\mathrm{\nabla }u(N)|\le C{h}^2{|u|}_{3,\mathrm{\infty },\omega }.

(3.5)

Combining (3.3) and (3.5), we obtain the result (3.2). □

Lemma 3.2 For \mathrm{\Pi }u\in S_0^h(\mathrm{\Omega }) the tensor-product linear interpolant to u, the solution of (2.2), and R_h the SPR recovery operator, we have the superconvergent estimate

|\mathrm{\nabla }u-R_h\mathrm{\Pi }u{|}_{0,\mathrm{\infty },\mathrm{\Omega }}\le C{h}^2{\parallel u\parallel }_{3,\mathrm{\infty },\mathrm{\Omega }}.

(3.6)

Proof Denote by F:\hat{e}\to e an affine transformation. Obviously, there exists an element e\in {\mathcal{T}}^h, using the triangle inequality and the Sobolev embedding theorem [15], and (3.3), such that

\begin{array}{rcl}|\mathrm{\nabla }u-R_h\mathrm{\Pi }u{|}_{0,\mathrm{\infty },\mathrm{\Omega }}& =& |\mathrm{\nabla }u-R_h\mathrm{\Pi }u{|}_{0,\mathrm{\infty },e}\\ \le & C{h}^{-1}|\mathrm{\nabla }\hat{u}-R_h\widehat{\mathrm{\Pi }u}{|}_{0,\mathrm{\infty },\hat{e}}\\ \le & C{h}^{-1}[|\mathrm{\nabla }\hat{u}{|}_{0,\mathrm{\infty },\hat{e}}+|R_h\widehat{\mathrm{\Pi }u}{|}_{0,\mathrm{\infty },\hat{e}}]\\ \le & C{h}^{-1}[|\mathrm{\nabla }\hat{u}{|}_{0,\mathrm{\infty },\hat{\chi }}+|\widehat{\mathrm{\Pi }u}{|}_{1,\mathrm{\infty },\hat{\chi }}]\\ \le & C{h}^{-1}{\parallel \hat{u}\parallel }_{3,\mathrm{\infty },\hat{\chi }},\end{array}

where \hat{\chi } is a small patch of elements surrounding the triangular prism, \hat{e}. Due to the fact that for \hat{u} quadratic over \hat{\chi },

\mathrm{\nabla }\hat{u}-R_h\widehat{\mathrm{\Pi }u}=(0,0,0)\text{in }\hat{\chi },

so, from the Bramble-Hilbert lemma [14],

|\mathrm{\nabla }u-R_h\mathrm{\Pi }u{|}_{0,\mathrm{\infty },\mathrm{\Omega }}\le C{h}^{-1}|\hat{u}{|}_{3,\mathrm{\infty },\hat{\chi }}\le C{h}^2|u{|}_{3,\mathrm{\infty },\mathrm{\Omega }},

which completes the proof of the result (3.6). Finally, we give the main result in this article. □

Theorem 3.1 For u_h\in S_0^h(\mathrm{\Omega }) the tensor-product linear triangular prism finite element approximation to u, the solution of (2.2), and R_h the SPR recovery operator, we have the superconvergence estimate

|\mathrm{\nabla }u-R_h{u}_h{|}_{0,\mathrm{\infty },\mathrm{\Omega }}\le C{h}^2{|lnh|}^{\frac{4}{3}}{\parallel u\parallel }_{3,\mathrm{\infty },\mathrm{\Omega }}.

(3.7)

Proof Using the triangle inequality, we have

\begin{array}{rcl}|\mathrm{\nabla }u-R_h{u}_h{|}_{0,\mathrm{\infty },\mathrm{\Omega }}& \le & |R_h(u_h-\mathrm{\Pi }u){|}_{0,\mathrm{\infty },\mathrm{\Omega }}\\ +|\mathrm{\nabla }u-R_h\mathrm{\Pi }u{|}_{0,\mathrm{\infty },\mathrm{\Omega }}\\ \le & |u_h-\mathrm{\Pi }u{|}_{1,\mathrm{\infty },\mathrm{\Omega }}\\ +|\mathrm{\nabla }u-R_h\mathrm{\Pi }u{|}_{0,\mathrm{\infty },\mathrm{\Omega }},\end{array}

which combined with (2.3) and (3.6) completes the proof of the result (3.7). □