Supercloseness of the Divergence-Free Finite Element Solutions on Rectangular Grids

Abstract

By the standard theory, the stable Q _k+1,k −Q _k,k+1/$Q_{k}^{dc'}$ divergence-free element converges with the optimal order of approximation for the Stokes equations, but only order k for the velocity in H ¹-norm and the pressure in L ²-norm. This is due to one polynomial degree less in y direction for the first component of velocity, which is a Q _k+1,k polynomial of x and y. In this manuscript, we will show by supercloseness of the divergence free element that the order of convergence is truly k+1, for both velocity and pressure. For special solutions (if the interpolation is also divergence-free), a two-order supercloseness is shown to exist. Numerical tests are provided confirming the accuracy of the theory.

1 Introduction

The divergence-free finite element method is mainly for solving incompressible flow problems, such as Stokes or Navier–Stokes equations, where the finite element space for the pressure is exactly the divergence of the finite element space for the velocity. In such a method, the finite element velocity is divergence-free pointwise, i.e., the incompressibility condition is enforced strongly. Traditional finite elements enforce the incompressibility weakly, cf. [10, 21]. That is, in order to satisfy the inf-sup stability condition, the incompressibility condition is weakened by either enlarging the velocity space or decreasing the pressure space. This often leads to some sub-optimal methods, or a waste of computation, due to the imperfect matching of two spaces. It may also lead to an inaccurate mass conservation, which is critical in certain computational problems. For example, for a high pressure flow problem, the Taylor–Hood finite element method produces a solution of large error of order O(Re) where Re is the Reynolds number [11].

A fundamental study on the divergence-free element method was done by Scott and Vogelius [22, 23], which shows that the P _k+1/$P_{k}^{dc}$ method is stable (except on grids with nearly-singular vertexes) and consequently of the optimal order on 2D triangular grids, for k≥3. Here the finite element space for velocity is the space of continuous piecewise-polynomials of degree (k+1) or less; The space approximating pressure is the space of discontinuous piecewise-polynomials of degree k or less, or the divergence of the discrete velocity space, to be precise. There are several other such divergence-free finite elements, cf. [2, 14, 16, 17, 19, 20, 29–31, 33].

Starting from the most popular element, the Q ₁/P ₀ element [6, 7], there is a series of works on Q _k mixed finite elements on rectangular grids in 2D and 3D. Brezzi and Falk showed that the Q _k+1/$Q_{k}^{dc}$ element is unstable in [9], for any k≥0. Here $Q_{k}^{dc}$ denotes the space of discontinuous piecewise-polynomials. In [27], Stenberg and Suri showed the stability, but a sub-optimal order of approximation, for the Q _k+1/$Q_{k-1}^{dc}$ element for all k≥1 in 2D. Bernardi and Maday proved the stability and the optimal order of convergence for the Q _k+1/$P_{k}^{dc}$ element, cf. [4]. Ainsworth and Coggins established [1] the stability and the optimal order of convergence for the Taylor–Hood Q _k+1/Q _k element, where the pressure space is continuous too. The Bernardi–Raugel element [5] optimizes the Q _k+1/$Q_{k-1}^{dc}$ element, when k=1, by reducing the velocity space to Q _1,2−Q _2,1 polynomials. Here the first component of velocity in the Bernardi–Raugel element is a polynomial of degree 1 in x direction, but of degree 2 in y direction. To be precise, the Bernardi–Raugel element enriches the Q ₁ velocity space by face-bubble functions. Similar to the Bernardi–Raugel element, a divergence-free finite element, Q _k+1,k −Q _k,k+1/$Q_{k}^{dc'}$ (k≥2), was proposed in [31], which further optimizes the Bernardi–Raugel element by increasing the polynomial degree of pressure from (k−1) to k. The nodal degrees of freedom of this divergence-free element and the Bernardi–Raugel element are plotted in Fig. 1. This divergence-free element was extended to its lowest-order form, k=1, i.e., Q _2,1−Q _1,2/$Q_{1}^{dc'}$, in [17]. Here the space $Q_{k}^{dc'}$ for the pressure is the space of discontinuous Q _k polynomials with all spurious modes removed, i.e., eliminating one degree of freedom at each vertex, cf. (2.7). In the construction, the pressure space is exactly the divergence of the velocity. Thus, the resulting finite element is divergence-free pointwise. In such a case, the discrete pressure space can be omitted in the computation. By an iterated penalty method, we obtain the pressure solution as a byproduct, cf. [30] and Sect. 4 below. However, by the standard finite element theory developed in [17, 31], this divergence-free element converges at order k only, due to a degree k polynomial in y for the first component of u _h . This cannot be improved by the standard theory, where the optimal order of convergences is derived from the inf-sup stability.

Fig. 1

Nodes of u _h /p _h for divergence-free (top) and Bernardi–Raugel elements

In this manuscript, we further study this Q _k+1,k −Q _k,k+1 divergence-free element and show its supercloseness that the element does converge at order k+1. Further the velocity solution of the Q _k+1,k −Q _k,k+1 divergence-free element may be ultraclose, i.e., two orders higher than the standard approximation, provided the interpolation is divergence-free. The extension of this divergence-free element to 3D is straightforward, so is its supercloseness property. By the supercloseness of the finite element solution, traditionally we interpolate the finite element solution by either a (Q _k+2)² or a (Q _2k )² polynomial piecewise on two by two sub-grids, to obtain a superconvergent solution. It may not be meaningful to do so here as such an interpolation is no longer divergence free. We do get a higher-order solution though, but we lose its mass conservation property. To keep its divergence-free property, we need to post-process this higher-order interpolation. It may cost even more to post-process this interpolated high-order polynomial solution to get a divergence-free solution than to compute directly a higher order divergence-free solution from the family of finite elements.

The divergence-free element is connected to the C ¹−Q _k element [18, 32]. A mathematical interest of this work is its application to the superconvergence analysis of the whole family of C ¹−Q _k elements. We intend to do so in a forthcoming work. Only the superconvergence of the degree-three C ¹−Q _k element, the Bogner–Fox–Schmit element, is established at the moment [18, 26]. We note that such a Q _k -type divergence-free element exists only on rectangular grids, not on general quadrilateral grids. Correspondingly, a construction of C ¹−Q _k elements is not possible yet, on quadrilateral grids.

The rest of the paper is organized as follows. In Sect. 2, we define the finite element for the stationary Stokes equations. In Sect. 3, we establish supercloseness for the divergence-free element. In Sect. 4, we provide some test results confirming the analysis. In particular, we show the order of convergence of the divergence-free element is one higher than that of the rotated Bernardi–Raugel element.

2 The Q _k+1,k −Q _k,k+1 Divergence-Free Element

In this section, we shall define the divergence-free finite element for the stationary Stokes equations on rectangular grids. The resulting finite element solutions for the velocity are divergence-free point wise.

We consider a model stationary Stokes problem: Find the velocity u and the pressure p on a 2D polygonal domain Ω, which can be subdivided into rectangles, such that

$$ \begin{aligned} - \Delta \mathbf{u} + \nabla p &=\mathbf{f} \quad \mbox{in}\ \varOmega, \\ \operatorname{div} \mathbf{u} &= 0 \quad \mbox{in}\ \varOmega, \\ \mathbf{u} &= \mathbf{0}\quad \mbox{on}\ \partial\varOmega.\end{aligned} $$

(2.1)

The weak form for (2.1) is: Find $\mathbf{u}\in H_{0}^{1}(\varOmega)^{2}$ and $p\in L_{0}^{2}(\varOmega):=L^{2}(\varOmega)/C=\{ p\in L^{2} \mid \int_{\varOmega}p = 0 \}$ such that

$$ \begin{aligned} a(\mathbf{u},\mathbf{v})+b(\mathbf{v},p) &=(\mathbf{f},\mathbf{v}) \quad \forall \mathbf{v} \in H^1_0(\varOmega)^2 , \\ b(\mathbf{u},q) &=0 \quad \forall q \in L^2_0(\varOmega). \end{aligned} $$

(2.2)

Here $H^{1}_{0}(\varOmega)^{2}$ is the subspace of the Sobolev space H ¹(Ω)² (cf. [13]) with zero boundary trace, and

The finite element grids are defined by, cf. Fig. 2,

We further assume, only for the lowest-order element k=1 in (2.3), that the rectangles in grid ${\mathcal{T}}_{h}$ can be combined into groups of four to form a macro-element grid:

See the fourth diagram in Fig. 2. The polynomial spaces are defined by

The Q _k+1,k −Q _k,k+1 (k≥1) element spaces are

(2.3)

(2.4)

Since $\int_{\varOmega}p_{h} = \int_{\varOmega} \operatorname{div} \mathbf{u}_{h} =\int_{\partial\varOmega} \mathbf{u}_{h} =0$ for any p _h ∈P _h , we conclude that

i.e., the mixed-finite element pair is conforming. The resulting system of finite element equations for (2.2) is: Find u _h ∈V _h and p _h ∈P _h such that

$$ \begin{aligned} a(\mathbf{u}_h,\mathbf{v})+b(\mathbf{v},p_h) &=(\mathbf{f},\mathbf{v}) \quad \forall \mathbf{v} \in \mathbf{V}_h, \\ b(\mathbf{u}_h,q) &=0\quad \forall q \in P_h. \end{aligned} $$

(2.5)

Fig. 2

Three levels of grids, and a macro-element grid (for k=1 only)

Traditional mixed-finite elements require the inf-sup condition to guarantee the existence of discrete solutions. As (2.4) provides a compatibility between the discrete velocity and the discrete pressure spaces, the linear system of equations (2.5) always has a unique solution, cf. [30]. Furthermore, such a solution u _h is divergence-free: by the second equation in (2.5) and the definition of P _h in (2.4),

$$ b(\mathbf{u}_h, q) = b(\mathbf{u}_h, -\operatorname{div} \mathbf{u}_h) = \| \operatorname{div} \mathbf{u}_h \|_{L^2(\varOmega)^2}^2 =0. $$

(2.6)

In this case, i.e., $\mathbf{V}_{h}\subset \mathbf{Z}:=\{\operatorname{div} \mathbf{v} \mid \mathbf{v} \in H^{1}_{0}(\varOmega)^{2} \}$, we call the mixed finite element a divergence-free element. It is apparent that the discrete velocity solution is divergence-free if and only if the discrete pressure finite element space is the divergence of the discrete velocity finite element space, i.e., (2.4).

As singular vertices are present (see [17, 22, 23, 31]), by the definition (2.4), P _h is a subspace of the discontinuous, piecewise Q _k polynomials:

(2.7)

where K _i are four squares numbered counterclockwise around a vertex x in the grid ${\mathcal{T}}_{h}$. It is possible, but very difficult to find a local basis for P _h . But on the other side, it is the special interest of the divergence-free finite element method that the space P _h can be omitted in computation and the discrete solutions approximating the pressure function in the Stokes equations can be obtained as byproducts, if an iterated penalty method is adopted to solve the system (2.5), cf. [8, 10, 15, 25, 30] for more information.

3 Supercloseness

As usual, the supercloseness is obtained by the method of integration by parts, cf. [12, 28]. But we have a long series of lemmas dealing with each term in the bilinear forms a(⋅,⋅) and b(⋅,⋅).

For a convenience in referring components of the vector velocity, we define the two inhomogeneous polynomial spaces:

(3.1)

(3.2)

k≥1. That is,

The interpolation operator I _h is defined for the two components of u:

(3.3)

We define u _I by its values at the Lagrange nodes. For the nodes at vertexes, we could use the Scott–Zhang [24] interpolation, i.e., the nodal value of u _I is an average on an edge, against a dual basis function at the vertex. When u is a P _k polynomial locally, $u_{I}(a_{i}^{K})=u(a_{i}^{K})$. But for convenience, also because the function u to be interpolated is very smooth in the analysis, we use the nodal value interpolation at vertexes. The nodal values inside an edge, and inside a square, are defined by proper L ²-projections, i.e., by solving the following equations sequentially (see Fig. 3):

(3.4)

(3.5)

(3.6)

(3.7)

where p _k ∈P _k , the space of 1D polynomials of degree k or less, and q _k,l ∈Q _k,l . By rotating x and y, v _I is defined similarly/symmetrically to u _I .

Fig. 3

Three types of interpolation nodes, k=3

Lemma 3.1

(two-order supercloseness)

For any Q _k+1,k function ψ∈V _h,1, defined in (3.1), for any $u\in H^{k+3}(\varOmega)\cap H^{1}_{0}(\varOmega)$, and for all k>1,

$$ \biggl|\int_\varOmega (u-u_I)_x \psi_x \,d\mathbf{x}\biggr| \le Ch^{k+2}\|u\|_{H^{k+3}} \|\psi \|_{H^1}. $$

(3.8)

Proof

We first consider the estimation on the reference element $\hat{K}=[-1,1]^{2}$. Since ψ∈Q _k+1,k , we have an exact Taylor expansion:

(3.9)

where ψ _x (x,0) and all $\psi_{xy^{j}}(x,0)$ are P _k polynomials in x only. We will perform the integration by parts repeatedly. First, for the lower order terms in (3.9), we notice that, by the definition of u _I in (3.5) and (3.7),

(3.10)

Please be aware that $\psi_{x^{2}y^{j}}(x,0)\in P_{k-1}(x)$ above. Hence, we need to deal with only the last two terms in (3.9).

For the last two terms in (3.9), in order to do integration by parts, we express the polynomials y ^k−1 and y ^k by derivatives of another polynomial:

(3.11)

(3.12)

(3.13)

Here $\tilde{p}_{2k}(y)$ and p _k−2(y) denote a polynomial of degree 2k and (k−2), respectively. We note that, as in (3.10), the integral of (u−u _I ) _x against p _k−2(y) is zero. Thus, by (3.10), dropping the first k−1 terms, we have

(3.14)

Let us consider the first boundary integral in (3.14), on the top edge of the square $\hat{K}$. By (3.4) and (3.5),

(3.15)

noting again that $\psi_{x^{2}y^{k-1}}(x,0)$ is a P _k−1 polynomial in x only. The other boundary integral in (3.14) is also 0 as $\psi_{x^{2}y^{k}}(x,0)\in P_{k-1}$ too:

(3.16)

That is, the boundary integrals in (3.14) are all zero. We repeat the integration by parts in this direction, while the boundary terms would be zero by (3.12) and (3.5). Note that in the last case, in (3.15) and (3.16), the boundary terms vanish because of (3.4): (u−u _I )(±1,±1)=0, while they vanish below because of (3.12): $s_{k-1}^{(j)}(\pm 1)=0$ and $s_{k}^{(j+1)}(\pm 1)=0$ for j<k. By k times more integration by parts, (3.14) would be

(3.17)

We will perform the integration by parts one last time. But this time, we will treat the two terms in the last integral differently:

For the second integral, the boundary term disappears by the condition (3.12). For the first integral, the boundary integrals will be canceled due to the opposite line integrals (one is from the top limit x=1 and one from the bottom limit x=−1 of the integral on the neighboring square) on two sides of a vertical edge x=x _i or due to the boundary condition on ψ:

We also note that the (k+1)st and (k+2)nd partial derivatives on u _I above are all zero. Hence, we get (3.8) by summing over the estimation on all rectangles $K\in {\mathcal{T}}_{h}$, plus a scaling and the Schwarz inequality,

We note that the semi H ¹-norm is needed above to bound $\psi_{y^{k-1}}$, when there is no boundary condition on one element K. But higher order norms are equivalent to semi-H ¹ norm on one element, $\|\psi_{x y^{k}}\|_{L^{2}(\hat{K})} \le C_{k} |\psi |_{H^{1}(\hat{K})}$. Thus k>1 is required. □

In the proof, we can see that the decrease of one degree polynomial in y does not change the super-approximation of Q _k+1,k in x direction. After (3.17), if we skip the last step of integration by parts, we would get the following corollary. That is, we avoid $\|\psi_{y^{k-1}}\|_{L^{2}}$ when k=1 which cannot be bounded by $|\psi|_{H^{1}}$.

Corollary 3.2

(one-order supercloseness)

For any Q _k+1,k function ψ∈V _h,1, defined in (3.1), for any u∈H ^k+2(Ω), and for all k≥1,

$$ \biggl|\int_\varOmega (u-u_I)_x \psi_x \,d\mathbf{x}\biggr| \le Ch^{k+1}\|u\|_{H^{k+2}} \|\psi \|_{H^1}. $$

(3.18)

Symmetrically, switching x and y in Lemma 3.1, we prove the following lemma.

Lemma 3.3

(two-order supercloseness)

For any Q _k,k+1 function ψ∈V _h,2, defined in (3.2), and for any v∈H ^k+3(Ω), if k>1,

$$ \biggl|\int_\varOmega (v- v_I)_y \psi_y \,d\mathbf{x}\biggr| \le Ch^{k+2}\|v\|_{H^{k+3}} \|\psi \|_{H^1}. $$

(3.19)

For the same reasons in Corollary 3.2, we get the following corollary from Lemma 3.3.

Corollary 3.4

(one-order supercloseness)

For any Q _k,k+1 function ψ∈V _h,2, defined in (3.2), for any $v\in H^{k+2}(\varOmega)\cap H^{1}_{0}(\varOmega)$, and for all k≥1,

$$ \biggl|\int_\varOmega (v-v_I)_y \psi_y \,d\mathbf{x}\biggr| \le Ch^{k+1}\|v\|_{H^{k+2}} \|\psi \|_{H^1}. $$

(3.20)

Though the interpolation order is (k+2) in the above two lemmas, only the (k+1) order in two corollaries can be achieved in computation, due to the coupling of terms in mixed formulation. We prove the approximation properties in the lower polynomial direction next, i.e., the y-derivative convergence of the first component of velocity. Now, even for k=1, we have a two-order supercloseness.

Lemma 3.5

(two-order supercloseness)

For any Q _k+1,k function ψ∈V _h,1, defined in (3.1), for any $u\in H^{k+3}(\varOmega)\cap H^{1}_{0}(\varOmega)$, and for all k≥1,

$$ \biggl|\int_\varOmega (u-u_I)_y \psi_y \,d\mathbf{x}\biggr| \le Ch^{k+2} \|u\|_{H^{k+3} } \|\psi \|_{H^1}. $$

(3.21)

Proof

Again, we first consider the estimation on the reference element $\hat{K}=[-1,1]^{2}$. Since the polynomial degree in y is too low, we do Taylor expansion in x direction, different from the last lemma:

Again, similar to (3.9), the integral of (u−u _I ) _y against x ^j terms are zero if j≤k−1,

noting that $x^{j} \psi_{x^{j}y^{2}}(0,y)\in Q_{k-1,k-2}$. Using the polynomial function s _k (x) defined in (3.11) we have, cf. (3.14),

Here, for the first time integration by parts, the boundary integral disappeared by (3.4), (u−u _I )(±1,±1)=0. In the next (k+1) times of integration by parts, the boundary integrals on x=±1 would be zero, directly by the boundary condition (3.12) of s _k (x):

Thus,

The rest of the proof repeats that of Lemma 3.1. □

As for above lemmas and corollaries, we can get the following corollary from Lemma 3.5.

Corollary 3.6

For any Q _k+1,k function ψ∈V _h,1, defined in (3.1), for any $u\in H^{k+2}(\varOmega)\cap H^{1}_{0}(\varOmega)$, and for all k≥1,

$$ \biggl|\int_\varOmega (u-u_I)_y \psi_y \,d\mathbf{x}\biggr| \le Ch^{k+1} \|u\|_{H^{k+2} } \|\psi \|_{H^1}. $$

(3.22)

Corollary 3.7

For any Q _k,k+1 function ψ∈V _h,2, defined in (3.2), and for any $u\in H^{k+3}(\varOmega)\cap H^{1}_{0}(\varOmega)$, and for all k≥1,

(3.23)

(3.24)

Now we study the supercloseness in both bilinear forms.

Lemma 3.8

For any (v _h ,q _h )∈V _h ×P _h , defined in (2.3) and (2.4), and for any $\mathbf{u}\in H^{3}(\varOmega)\cap H^{1}_{0}(\varOmega)$,

(3.25)

(3.26)

(3.27)

where I _h u is the interpolation of u defined by (3.3).

Proof

(3.25) is a combination of (3.8), (3.21), (3.23), and (3.19). (3.26) is a combination of (3.18), (3.22), (3.24), and (3.20).

For (3.27), we will lose one order of convergence. Let $q_{h}=\operatorname{div} \mathbf{w}_{h}$ for some w _h =(ϕ,ψ)∈V _h . We have, denoting u=(u,v),

Here we have two old integrals, ∫ _K (u−u _I ) _x ϕ _x   d x and ∫ _K (v−v _I ) _y ψ _y   d x, and two new integrals, ∫ _K (u−u _I ) _x ψ _y   d x and ∫ _K (v−v _I ) _y ϕ _x   d x. The approximation order can be one order higher for the two old integrals. For the two new integrals, by symmetry, we consider ∫ _K (u−u _I ) _x ψ _y   d x. We use the following Taylor expansion on the reference element $\hat{K}$ in the y direction. We note that the Taylor expansion in x direction would lead to a too high order polynomial in y direction each term in (3.28) below:

(3.28)

Here all $\psi_{y^{j}}(x,0)$ are polynomials of degree k in x. That is, a generic term $y^{j} \psi_{y^{j+1}}(x,0) \in Q_{k,j}$. This is the same as the generic term $y^{j} \psi_{xy^{j}}(x,0)$ in the early Taylor expansion (3.9). Thus repeating the proof of Lemma 3.1, we get

For the second integral, we can do an integration by parts to raise one more order. But we are limited by the first integral above to get only

Similarly, we have the same bound for $\vert \int_{\hat{K}} (u-u_{I})_{y} \psi_{x} \,d\mathbf{x}\vert$. (3.27) follows by the Schwarz inequality and the scaling of referencing mappings. □

Finally, we estimate the approximation to p.

Lemma 3.9

For any function v _h ∈V _h , defined in (2.3), and for any $p\in H^{k+1}(\varOmega)\cap L^{2}_{0}(\varOmega)$,

$$ \biggl|\int_\varOmega \operatorname{div} \mathbf{v}_h (p-p_I) \,d\mathbf{x}\biggr| \le Ch^{k+1} \|\mathbf{v}_h \|_{H^1} \| p \|_{H^{k+1}}, $$

(3.29)

where p _I is a special nodal interpolation of p in P _h , defined in (3.30) below.

Proof

We note that P _h are discontinuous Q _k functions, $P_{h}=\operatorname{div} \mathbf{V}_{h}$. We define an interpolation operator for P _h via that I _h for V _h defined in (3.3). For a $p\in H^{2}(\varOmega)\cap L^{2}_{0}(\varOmega)$, Arnold, Scott and Vogelius have shown in [3] that there is a $\mathbf{w}\in H^{3}(\varOmega)^{2}\cap H^{1}_{0}(\varOmega)^{2}$, such that

For simplicity, we assume the above lifting exists up to order k+1. We define

$$ p_I = \operatorname{div} \mathbf{w}_I, $$

(3.30)

for w _I =I _h w defined by (3.3). In order to use (3.27), we use the notations

Repeating the proof in Lemma 3.8, we get

 □

We derive the main theorem.

Theorem 3.10

The finite element solution (u _h ,p _h ) of (2.5) has the following supercloseness property, one order higher than the optimal order,

(3.31)

where the interpolations (I _h u,p _I ) are defined in (3.3) and (3.30).

Proof

By the inf-sup condition shown in [17, 31], it follows that, cf. [21], for all (w _h ,r _h )∈V _h ×P _h ,

$$ \sup_{(\mathbf{v}_h,q_h)\in \mathbf{V}_h\times P_h} \frac {a(\mathbf{w}_h,\mathbf{v}_h) + b(\mathbf{v}_h, r_h) + b(\mathbf{w}_h, q_h) }{\| \mathbf{v}_h\|_{H^1} + \| q_h\|_{L^2} } \ge C \bigl(\|\mathbf{w}_h\|_{H^1} +\|r_h\|_{L^2}\bigr). $$

(3.32)

By Corollary 3.7 and Lemma 3.9, we have

Note that, due to the pointwise divergence free property, above we have

 □

Here, to be precise, we do not have a supercloseness for p in Theorem 3.10. As P _h are degree-k polynomials, the best order approximation to p in L ²-norm would be (k+1). However, due to the mixed formulation, the convergence of p _h to p is limited to the optimal order convergence of u _h , which is (k−1) in H ¹-norm as u _h has polynomial degree k only in y direction for its first component. In this sense, the supercloseness result (3.31) does lift the order of approximation of p _h by one. For k>1, we may have a two-order supercloseness for the velocity. Such numerical examples are shown in [31] and in next section. That is, for some special functions u, I _h u might be also in the divergence-free subspace of V _h . If so, we have a two-order supercloseness result.

Theorem 3.11

(two-order supercloseness)

For some solution u of (2.1), if

where I _h is defined in (3.3), and if k>1, then

(3.33)

Proof

By (3.25), limited to the divergence-free subspace,

 □

4 Numerical Tests

In this section, we report some results of numerical experiments on the Q _k+1,k −Q _k,k+1 element for the stationary Stokes equations (2.1) on the unit square Ω=[0,1]². The grids ${\mathcal{T}}_{h}$ are depicted in Fig. 2, i.e., each squares are refined into four sub-squares each level. The initial grid, level one grid, is simply the unit square.

We choose an exact solution for the Stokes equations (2.1):

$$ \mathbf{u}= \operatorname{\textbf{curl}}g, \quad p=\Delta g. $$

(4.1)

Here

So we can compute the right hand side function f for (2.1) as

$$ \begin{aligned} \mathbf{f} & = -\Delta \operatorname{\textbf{curl}}g + \nabla \Delta g. \end{aligned} $$

(4.2)

We note that, unlike [17, 31], we intentionally choose a non-symmetric solution so that no ultraconvergence would happen, which does not exist in general. The solution p is plotted in Fig. 4.

Fig. 4

The solution p (the errors are shown in Fig. 5)

We compute the Stokes solution on refined grids, cf. Fig. 2, by the divergence Q _k+1,k −Q _k,k+1 element (2.3) and by the rotated Bernardi–Raugel element [5, 10, 21]:

(4.3)

Following the analysis in [17], the stability of the rotated Bernardi–Raugel element would be proved. For the rotated Bernardi–Raugel element, the system of finite element equations is solved by the Uzawa iterative method, cf. [10, 15, 21]. The stop criterion is the difference $|p_{h}^{(n)} -p_{h}^{(n-1)}|\le 10^{-6}$. We list the number of Uzawa iterations in the data tables by #Uz. Here the interpolation operators are standard Lagrange nodal interpolations [13].

For the Q _k+1,k −Q _k,k+1 divergence-free element, the pressure does not enter into computation, but is obtained as a byproduct, because $P_{h}=\operatorname{div} \mathbf{V}_{h}$. The resulting linear system of Q _k+1,k −Q _k,k+1 divergence-free element equations can be formulated as symmetric positive definite. Then the iterated penalty method [15, 31] can be applied to obtain the divergence-free finite element solution for the velocity, and a byproduct $p_{h}=\operatorname{div} \mathbf{w}_{h}$ for the pressure. In our computation, the iterated penalty parameter is 2000. The stop criterion is the divergence $\|\operatorname{div} \mathbf{u}_{h}^{(n)}\|_{0} \le 10^{-9}$. The number of iterated penalty iterations is also listed as #it in the data tables.

In Table 1, we list the errors in various norms for the Q _k+1,k −Q _k,k+1 divergence-free element and for the rotated Bernardi–Raugel element, for k=1. It is clear that the order of convergence is 2, one order higher than that of latter. We note that the convergence order is only 2 for Q _2,1−Q _1,2 divergence-free elements in L ²-norm, i.e., there is no L ²-supercloseness. But we do see L ²-supercloseness for k>1 next.

Table 1 The errors e _h =I _h u−u _h and ϵ _h =p _I −p _h (one-order supercloseness) for (4.1) with k=1

In Table 2, we list the computation results for k=2 elements. Again, the divergence-free element is one order higher than the rotated Bernardi–Raugel element. To show the difference in the two elements, we plot the errors by two elements on level 4 grid in Fig. 5. One can see the advantage of the divergence-free element, which fully utilizes the approximation power of u _h by lifting the pressure polynomial degree. Of course, another advantage is the divergence-free solution after such a lift. We next report the results for k=3 in Table 3. All numerical results confirm the theory, and also show the accuracy of the supercloseness analysis.

Fig. 5

The errors of p _h for the divergence-free (top) and BR elements

Table 2 The errors e _h =I _h u−u _h and ϵ _h =p _I −p _h (one-order supercloseness) for (4.1) with k=2

Table 3 The errors e _h =I _h u−u _h and ϵ _h =p _I −p _h (one-order supercloseness) for (4.1) with k=3

Finally, we test the two-order supercloseness in Theorem 3.11. We choose a symmetric function as the exact solution of the Stokes equations (2.1):

(4.4)

Comparing to the data in Table 3, we can see, in Table 4, that the velocity does converge with another order higher than the optimal order. This is predicted in (3.33). Here the order of convergence for the pressure is the same as that in Table 3. It indicates that the analysis in Theorem 3.10 is sharp. Here we have a two-order supercloseness in L ²-norm too, for the velocity. But the supercloseness in L ²-norm is not studied in this manuscript.

Table 4 The errors e _h =I _h u−u _h and ϵ _h =p _I −p _h (two-order supercloseness) for (4.4) with k=3