On the Divergence-Free Finite Element Method for the Stokes Equations

On the divergence-free ﬁnite element method for the Stokes equations and the P1 Powell-Sabin divergence-free element

Shangyou Zhang∗

Abstract

A general framework for the conforming k- k−1 mixed element method is set up.

Because such a method would always produce point wise divergence-free solutions for the

velocity and the mixed element can always be reduced to a conforming k element for the vector Laplace equations, we name it the divergence-free ﬁnite element method. It is

shown that the method always has a unique solution. Further, the method is guaranteed

to provide optimal-order solutions if some subspace of C1- k+1 on the same triangular or

hexahedral grid has the optimal-order approximation property. The framework is applied

to the 1 divergence-free element on Powell-Sabin triangulations. Numerical tests are provided.

Keywords. Powell Sabin triangles, mixed ﬁnite elements, Stokes, divergence-free element. AMS subject classiﬁcations (2000). 65M60, 65N30, 76D07.

1 Introduction

A natural ﬁnite element method for the Stokes equations would be the k- k−1 mixed element

which approximates the velocity by continuous k piecewise-polynomials and approximates the pressure by discontinuous k−1 piecewise-polynomials. One advantage of the element is to preserve the incompressibility condition of incompressible ﬂuids ([4, 6, 27, 38, 39, 48]). Another advantage is its simplicity in computation that the mixed element can be reduced to the standard C0 ﬁnite element solving a vector Laplace equation, see Section 5 in this paper. A fundamental study on the method was done by Scott and Vogelius in 1983 ([38, 39]) that the method is stable and consequently of the optimal order on 2D triangular grids for any k ≥ 4, provided the grids have no singular vertex (i.e., edges meeting at a vertex do not form two cross lines.) For k ≤ 3, Scott and Vogelius showed the method would not be stable for general triangular grids, in [38, 39]. What is this magic number k in 3D? Scott and Vogelius posted this question explicitly after discovering that k = 4 in 2D. The problem remains open after 20

years.

Even in the 2D case, there are many open problems for low order (k < 4) k- k−1 elements. It is known that for k < 4, there are spurious pressure modes in the discrete pressure spaces and the classic inf-sup condition does not hold. Nevertheless, ﬁltering out such modes could be trivial in this case (see Section 5 in the paper), diﬀerent from other mixed elements where

either the pressure space is continuous, or the grid for the pressure diﬀers from that for the

velocity, or the space for the velocity is not continuous or is not k, e.g. ¡ k. By ﬁltering out

the spurious pressure modes, all types of the k- k−1 element would have a unique solution

∗Department of Mathematical Sciences, University of Delaware, DE 19716. [email protected].

for the Stokes equations (see Proposition 2.1 in the paper). However, the approximation of ¢ solutions by low-order ¢ k- k−1 elements is not guaranteed, known as locking phenomenon (see the numerical tests in Section 6 of the paper.) The approximation is shown for the Hsieh- Clough-Tocher triangles with k ≥ 2 ([33]), for the quadrilateral-triangulations with k = 2 ([4]), for the uniform criss-cross grids with k = 1 ([34]) and for the 3D Hsieh-Clough-Tocher

tetrahedra with k ≥ 3 ([49]). ¢

In this paper, we set up a framework for the divergence-free element, i.e., the ¢ k- k−1 ¢

element for all k and space dimensions. To be precise, the ¢ k- k−1 mixed element is replaced ¢ by the ¢ k-div k element, after ﬁltering out the spurious pressure modes. In addition to showing the existence of the ﬁnite element solution, we show that the convergence and the optimal-order approximation of both the velocity and the pressure are guaranteed by the

optimal order approximation of any one subspace of C1- ¢ k+1 scalar ﬁnite element space on the same grid. Using the frame work, one would eagerly try to answer the Scott-Vogelius question (in a little diﬀerent setting) that the magic number k in 3D is 8 or less because a subspace

of the C1- ¢ 9 ﬁnite element on tetrahedral grids was constructed in [47]. However, due to the

complexity of 3D singular or nearly-singular edges and/or vertices, we leave the study of 3D ¢

¢ k- k−1 elements somewhere else. Instead, in this paper, we will apply the framework only ¢ to the 2D ¢ 1 element on Powell-Sabin triangulations (cf. [31, 32]), i.e., a 1 divergence-free

element. The stability of this ¢ 1 element has not been fully studied previously, but the element was used in computation, cf. [13]. We remark that the inf-sup condition is no longer used in the analysis of this framework for

the divergence-free element. Since the discrete pressure space is div ¢ k, the inf-sup condition always holds, except the inf-sup constant might be arbitrarily close to 0. For example, when a nearly-singular vertex approaches to singular (cf. [38, 39]), the inf-sup constant approaches to zero. Nevertheless, when the vertex become a singular one, the inf-sup constant would jump back from zero to a regular one, because the extra spurious pressure mode is ﬁltered out automatically in the divergence-free element method. On the other side, even if the inf- sup condition holds uniformly on some quasiuniform grids (independent of the grid size), the convergence of the divergence-free element method may still fail. This is because the traditional discrete pressure space of ¢ k−1 piecewise polynomials (modulo the global constant) is replaced

by a subspace, div ¢ k, while the latter may lack local support and may not have enough, or the optimal order, approximation property.

In this paper, we will also show how to ﬁnd the divergence-free element solutions by the ¢ classic iterative penalty method. We will show how the mixed ¢ k- k−1 element can always

be reduced to a conforming ¢ k element for the vector Laplace equations. Therefore, in this case, we do not really have a mixed element. We name the method as the divergence-free finite element method. We note that the name of divergence-free mixed element is used often for non-conforming ([9, 30, 45]) or discontinuous Galerkin methods ([12, 15, 21]), or even for weakly divergence-free methods ([46]). But here, the discrete solutions for the velocity is truly divergence-free, including the points on the inter-element boundary. The rest of the paper is divided into 5 sections. In Section 2, we define the divergence-free finite element method and show that the method always has a unique solution. In Section 3, we set a general framework for the convergence of the divergence-free finite element method.

In Section 4, we apply the general theory to the ¢ 1 Powell-Sabin element. In Section 5, we will

¢ ¢ show how to reduce the ¢ k- k−1 mixed element to the C0- k element, and how to apply the classic iterative penalty method to solve the discrete, but positive deﬁnite, linear systems of

equations. In Section 6, we provide a numerical test using the Powell-Sabin ¢ 1 divergence-free element.

2 2 The divergence-free ﬁnite element

In this section, we shall define a class of finite elements for the stationary Stokes equations, where the finite element solution for the velocity is divergence-free point wise. The resulting linear systems of equations, by such elements, are shown to have a unique solution. We consider the stationary Stokes problem: Find functions u (the fluid velocity) and p (the pressure) on a 2D or 3D polygonal domain Ω ⊂ Rd such that

−∆u + ∇p = f in Ω, div u = 0 in Ω, (2.1) u = 0 on ∂Ω, where f is the body force. Via the integration by parts, we get a variational problem for the d Stokes equations: Find u ∈ H1,0(Ω) and p ∈ L2,0(Ω) := L2(Ω)/C = {p ∈ L2 | Ω p = 0} such that R d a(u, v) + b(v, p) = (f, v), ∀v ∈ H1,0(Ω) , (2.2) b(u, q) = 0, ∀q ∈ L2,0(Ω). d d Here H1,0(Ω) is the subspace of the Sobolev space H1(Ω) (cf. [14]) with zero boundary trace, and

a(u, v) = ∇u · ∇v dx, ZΩ b(v, p) = − div u p dx, ZΩ (f, v) = f v dx. ZΩ Let Ω be quasi-uniformly triangulated ([14]):

Ωh = {T | T is a triangle or a tetrahedron, |T | ≤ h} .

Let Vh be the space of of continuous piecewise polynomial of degree k on Ω with the homoge- neous boundary condition:

d Vh = uh ∈ C(Ω) | uh|T ∈ Pk(T ) ∀T ∈ Ωh, and uh|∂Ω = 0 . (2.3) n o Then we deﬁne Ph = {div uh | uh ∈ Vh} . (2.4)

Since Ω ph = Ω div uh = ∂Ω uh · n = 0 for any ph ∈ Ph, we conclude that R R R d Vh ⊂ H1,0(Ω) , Ph ⊂ L2,0(Ω), i.e., the mixed-ﬁnite element pair (Vh, Ph) is conforming. Then the ﬁnite element discretization of problem (2.2) reads: Find uh ∈ Vh and ph ∈ Ph such that a(uh, v) + b(v, ph) = (f, v), ∀v ∈ Vh, (2.5) b(uh, q) = 0, ∀q ∈ Ph.

We note that by (2.4), Ph is a subspace of discontinuous, piecewise polynomials of degree (k − 1) or less. Except a few cases such as the high-order tetrahedral elements on Hsieh- Clough-Tocher grids (cf. [48]), Ph does not include all mean-value zero piecewise polynomials

3 of degree (k − 1). So it is not possible to find a local basis for Ph in general, and it is not possible to derive a linear system of equations for (2.5), different from all other mixed finite methods. It seems that the definition of Ph (2.4) is not practical to implement the mixed element. But on the other side, it is the special interest of the new formulation where the space Ph is not implemented at all, and the discrete solutions approximating the pressure function in the Stokes equations will be obtained as byproducts, as we shall see in Section 5. So there will be only one finite element space for the velocity, and the mixed element becomes a single function element. It does not only greatly simplify the coding work, but also it avoids the difficulty of solving non-positive definite systems of linear equations encountered in the classic mixed element method. Let us introduce a key finite element space, the divergence-free finite element space,

Zh = {uh ∈ Vh | div uh = 0} . (2.6)

Because of the special deﬁnition (2.4) of Ph, the solution uh of (2.5) is always divergence-free point wise, i.e., in Zh, shown below.

Proposition 2.1 Let Vh be deﬁned by (2.3) on any 2D or 3D triangulation, with or without singular or nearly singular vertices, of quasi-uniform or graded grids, of regular or degenerating grids, and for any degree of polynomial k ≥ 1. Let Ph be deﬁned by (2.4). The discrete linear system (2.5) has a unique solution:

uh ∈ Zh, ph ∈ Ph.

Proof. We show the uniqueness ﬁrst. Let uh be a solution of (2.5). We have, by the second equation,

b(uh, q) = 0 ⇒ div uh div uh = 0 ⇒ uh ∈ Zh, ZΩ where we have chosen q = div uh ∈ Ph. Now, if u˜h is another solution of (2.5), by the ﬁrst equation, because div u˜h = 0 as shown above, we get

a(uh − u˜h, uh − u˜h) = 0, where we have chosen v = uh − u˜h so that the term involving two pressure solutions drops as div v = 0. Therefore we have shown uh is unique (if it does exist.) Next, let (uh, ph) and (uh, p˜h) be two solutions of (2.5). It follows that

2 b(v, ph − p˜h) = 0 ⇒ (ph − p˜h) = 0, ZΩ

where we have chosen one v so that div v = ph − p˜h, by the deﬁnition (2.4). So we conclude the solution ph is unique too. Finally, because (2.5) is a linear system of ﬁnitely many equations, the uniqueness implies the existence.

We remark that an inf-sup condition is required to ensure the existence of solution for general mixed ﬁnite elements. Therefore, for divergence-free ﬁnite elements, the inf-sup condition always holds, except that the inf-sup constant in the inequality may depend on the grid, i.e., it may be very close to zero sometimes.

4 To conclude this section, we show a lemma about the equivalence, in fact, the identity, of the 1 semi H -inner product and the div + curl product, for C0-Pk polynomials. It is well known. But we could not ﬁnd a rigorous proof anywhere, as people usually assume the continuous diﬀerentiability.

Lemma 2.1 For all u, v ∈ Vh, in both 2D and 3D, it holds that

a(u, v) = (div u, div v) + (curl u, curl v). (2.7)

Proof. We note that curl u is a scalar function in 2D, but a vector function in 3D, curl u. We will show (2.7) in 2D ﬁrst. Let u = hu1, u2i, v = hv1, v2i, and x = hx, yi.

a(u, v) = (u1xv1x + u1yv1y + u2xv2x + u2yv2y)dx ZT TX∈Ωh

(div u, div v) = (u1xv1x + u1xv2y + u2yv1x + u2yv2y)dx ZT TX∈Ωh

(curl u, curl v) = (u2xv2x − u2xv1y − u1yv2x + u1yv1y)dx ZT TX∈Ωh

We need to show the cancellation of 4 terms in the middle of (div u, div v) and (curl u, curl v). This is done by the integration by parts.

u1xv2ydydx = (u1v2y)xdydx − u1v2yxdydx ZT ZT ZT

= u1v2ydy − ( (u1v2x)ydydx − u1yv2xdydx) Z∂T ZT ZT

= u1v2ydy − u1v2xdx + u1yv2xdydx Z∂T Z∂T ZT ∂v2 = u1 ds + u1yv2xdydx. Z∂T ∂t ZT

∂v2 We note that the normal derivative ∂n is not continuous across two elements, but the tangential ∂v2 derivative ∂t is. When summing above integrals on all triangles, as u1 is continuous on internal edges, the two integrals on the two sides of each internal edge are canceled. In addition, u1 is zero at the boundary of Ω. So it follows that

u1xv2ydx − u1yv2xdx = 0. ZΩ ZΩ Similarly we can do such integration by parts for the other pair of terms. Hence (2.7) holds in 2D. We now study the 3D case. The terms inside the integral over a tetrahedron T for a(u, v), for (div u, div v), and for (curl u, curl v), are, respectively,

u1xv1x + u1yv1y + u1zv1z + u2xv2x + u2yv2y + u2zv2z + u3xv3x + u3yv3y + u3zv3z,

u1xv1x + u1xv2y + u1xv3z + u2yv1x + u2yv2y + u2yv3z + u3zv1x + u3zv2y + u3zv3z,

u2xv2x − u2xv1y − u1yv2x + u1yv1y + u1zv1z − u1zv3x − u3xv1z + u3xv3x

+u3yv3y − u3yv2z − u2zv3y + u2zv2z.

5 Thus what we need to show is that

(u1xv2y + u1xv3z + u2yv1x + u2yv3z + u3zv1x + u3zv2y ZΩ −u1yv2x − u1zv3x − u2xv1y − u2zv3y − u3xv1z − u3yv2z) dx = 0. (2.8)

We study one pair of such terms on one tetrahedron T ∈ Ωh, by the divergence theorem, as follows.

(u1xv2y − u1yv2x)dx = (u1v2ynx − u1v2xny + 0)dS ZT Z∂T

= u1(n × ∇v2)3dS. Z∂T

Here we denote the normal vector on the triangular faces of T by n = hnx, ny, nzi. We note that all three components of “tangential derivatives”

n × ∇v2 = h(n × ∇v2)1, (n × ∇v2)2, (n × ∇v2)3i

are continuous at the inter element boundaries and are zero at the domain boundary. Therefore (2.7) holds in 3D and the lemma is proven.

3 Convergence

The existence of solution always holds for the divergence-free elements. However, the approximation property of such solutions is not guaranteed at all. Comparing to the traditional mixed elements, the discrete pressure space Ph = div Vh is now a subspace of the discontinuous Pk−1 polynomials. One could suspect the pressure approximation might not be of optimal order. Further, the approximation of velocity by the divergence-free elements might be problematic too. The reason is apparent that, enforcing div uh = 0, we may have a too small subspace Zh ⊂ Vh for the optimal-order velocity approximation. We will study the approximation property of the divergence-free element under no condition ﬁrst.

Theorem 3.1 Let (u, p) and (uh, ph) be the solutions of (2.2) and (2.5) respectively. The following inequalities hold:

|u − uh|H1 = inf |u − zh|H1 , (3.1) zh∈Zh

kp − phkL2 = inf kp − ph − qhkL2 , (3.2) qh∈{div wh|curl wh=0}⊂Ph

kp − phkL2 ≤ inf kp − qhkL2 + inf | curl(vh − zh)|L2 qh=div vh∈Ph zh∈Zh

+ inf |u − zh|H1 . (3.3) zh∈Zh

Proof. (3.1) is well known and we can ﬁnd it in most books, for example, [10], [35] and [11]. Let z ∈ Zh in (2.2) and (2.5).

a(u − uh, z) = 0.

6 Therefore (3.1) follows immediately that

2 |u − uh|H1 = a(u − uh, u − z) ≤ |u − uh|H1 |u − z|H1 .

Let w1 ∈ {wh | curl wh = 0} ⊂ Vh and qh = div w1. This time, by (2.2) and (2.5), we have, instead, by Lemma 2.1,

a(u − uh, v) + b(v, p − ph) = 0 ∀v ∈ Vh. (3.4)

Then (3.2) follows the following standard argument,

2 2 2 kp − phkL2 = (p − ph, p − ph − qh) ≤ kp − phkL kp − ph − qhkL ,

because a(u−uh, w1)+b(w1, p−ph) = b(w1, p−ph) = 0. For a general v ∈ Vh and qh = div v, by (3.4), we can get that

(qh, p − ph) = a(u − uh, v − z) ∀z ∈ Zh

and that

2 kp − phkL2 ≤ (p − ph, p − qh) + | curl u − curl uh|L2 | curl(v − z)|L2 .

Here Lemma 2.1 is applied again. Therefore we obtain (3.3) by the Cauchy-Schwarz inequality.

We next deﬁne the potential space, C 1 piecewise polynomials of one degree higher than that of Vh:

1 2d−3 Sh = sh ∈ C (Ω) | sh T ∈ Pk+1(T ) ∀T ∈ Ωh , (3.5) n o ∂sh Sh,0 = sh ∈ Sh | sh ∂ = 0, = 0 . (3.6) Ω ∂n ∂Ω

We note that Sh is a scalar function space if d = 2, but a vector function space if d = 3. Let the biharmonic solutions th ∈ Sh,0 be deﬁned by

(∆th, ∆sh) = −(curl f, sh) ∀sh ∈ Sh,0, (3.7) where f is from (2.1). Here we abuse the notation curl f a little, which would be the scalar curl if d = 2. Corresponding to (3.7) we have the smooth solution t ∈ H2,0(Ω) (the closure of C∞,0 under H2 norm) for the biharmonic problem too:

(∆t, ∆s) = −(curl f, s) ∀s ∈ H2,0. (3.8)

It is shown in [35] that

curl H2,0 = Z := {div v = 0} ∩ H1,0, curl t = u. (3.9)

(3.9) should hold for the discrete spaces too. For example, it holds for a uniform type of 2D grids shown in [34] and [49]. In general, it is very diﬃcult to ﬁnd even the dimension of Zh and Sh,0, even in 2D, which is stated as Strang’s conjecture, cf. [7, 26, 25, 42, 43, 49]. Therefore, in the paper, we would not prove curl Sh,0 = Zh. Instead, we will use it as a condition.

7 Lemma 3.1 Let uh and th be deﬁned in (2.5) and (3.7) respectively. If curl Sh,0 = Zh, then

curl th = uh.

Proof. We note that curl sh ∈ Zh for any sh ∈ Sh,0. The assumption here indicates that the other direction inclusion holds too. Let v = curl sh and uh = curl th in (2.5).

a(curl th, curl sh) = (f, curl sh) ∀sh ∈ Sh,0.

(3.7) follows by an integration by parts.

In computation, it is diﬃcult to construct a basis for the whole space Sh,0. As stated above, it is not even known mostly the dimension of the vector space Sh,0, except for some special cases such as the ones in [25], [1], [26] and [49]. Often a basis may be constructed for a subspace of Sh,0, the so-called super-splines, where some extra smoothness is used at vertices

or at some interface points, cf. for example, [16, 36, 2]. For example, the Argyris £ 5 triangles

([10, 14]) is C2 at each vertex, i.e., it is a proper subspace of the C1- £ 5 space. Nevertheless, if the C1 solution in any subspace to the biharmonic equation is of optimal order, so is the C0 solution to the Stokes equations, to be shown in next theorem. Computationally we do not really compute C1 elements for the Stokes equations. What we need here is to obtain a proof of convergence for C0 divergence-free elements via that for C1 elements.

Theorem 3.2 Let the smooth solution t in (3.8) have the required elliptic regularity. Let G˜ ⊂ Sh,0 and the solution t˜h below be of optimal order:

(∆t˜h, ∆sh) = −(curl f, sh) ∀sh ∈ G,˜

i.e., we assume that

min{k,r−2} min{k,r−2} kt − t˜hkH2 ≤ Ch ktkHr ≤ Ch kfkHr−3 .

Then the solution uh of (2.5) is of optimal order,

min{k,r−2} ku − uhkH1 ≤ ku − curl t˜hkH1 ≤ Ch kfkHr−3 .

Proof. The proof is done simply by noting that curl t˜h is the (curl ·, curl ·) projection of u in a smaller subspace curl S˜h ⊂ Zh. Here uh is the projection in Zh.

We study next the convergence for the pressure. We note that the estimates in Theorem 3.1 for the pressure function are sharp. However, we do not know how to use the bounds to derive the optimal order of approximation. We will next show some less sharp error estimates and show the optimal order of convergence for the pressure. But ﬁrst, we need the following result on the regular inversion (or the maximal right inverse) of the divergence operator:

kUkHr+1 ≤ CkF kHr (3.10)

d d for some r ≥ 0, where C is independent of r and U ∈ Hr+1 ∩ H1,0 such that div U = F . For smooth domains, (3.10) is trivial. So is it on a 2D convex polygonal domain when r = 0. In both cases, letting U = ∇t, the resulting Poisson equation for t would provide an H2 solution. We refer to [19], [5], [38] and [44] for more discussion. For d = 2 (2 space dimension), (3.10) is shown in [5] for any r > 0 (except a few 1/2 indexes) and for any Lipschitz domain.

8 Lemma 3.2 Let (u, p) and (uh, ph) be the solutions of (2.2) and (2.5) respectively. Assume (3.10) holds for r = 0. The following estimate holds:

kp − phkL2 ≤ |u − uh|H1 + inf |u − vh|H1 + inf kp − qhkL2 . (3.11) vh∈Vh qh∈Ph

Proof. By (2.2) and (2.5), we have

d (p, div v) = a(u, v) − (f, v) ∀v ∈ H1,0, (3.12)

(ph, div vh) = a(uh, vh) − (f, vh) ∀v ∈ Vh. (3.13)

We will separate the error in ph in two parts. One is due to the L2 ﬁnite-dimensional projection. The other is caused by the perturbation of uh to u. So we introduce the L2-orthogonal projection of p in p¯h ∈ Ph by

(p¯h, qh) = (p, qh), ∀qh ∈ Ph. (3.14)

It follows by (3.12) and the deﬁnition of Ph in (2.4), letting qh = div vh, that

(p¯h, div vh) = (p, div vh) = a(u, vh) − (f, vh) ∀vh ∈ Vh. (3.15)

By (3.13) and (3.15),

(ph − p¯h, div vh) = a(uh − u, vh) ∀vh ∈ Vh. (3.16)

We let w¯ h be the solution of

a(w¯ h, vh) = a(uh − u, vh) ∀vh ∈ Vh.

Then, uh − w¯ h is the a(·, ·)-orthogonal projection of u in Vh, because

a(u − (uh − w¯ h), vh) = 0 ∀vh ∈ Vh.

By the Cauchy-Schwarz inequality, we have the property of best apprxoimation for projections:

|u − (uh − w¯ h)|H1 ≤ inf |u − vh|H1 . vh∈Vh

By the triangle inequality, we get

|w¯ h|H1 ≤ |u − uh|H1 + |u − (uh − w¯ h)|H1

≤ |u − uh|H1 + inf |u − vh|H1 . (3.17) vh∈Vh

We go back to (3.16). Because (ph − p¯h) ∈ Ph, there is a wh ∈ Vh such that

div wh = ph − p¯h.

Letting vh = wh in (3.16),

2 kph − p¯hkL2 = a(uh − u, wh) = a(w¯ h, wh)

≤ |w¯ h|H1 |wh|H1 = |w¯ h|H1 kph − p¯hkL2 .

9 Therefore, by (3.17),

kph − p¯hkL2 ≤ |w¯ h|H1 ≤ |u − uh|H1 + inf |u − vh|H1 . (3.18) vh∈Vh

Using the deﬁnition (3.14) of p¯h, as it is the L2-orthogonal projection of p, we have

kp − p¯hkL2 ≤ inf |p − qh|L2 . (3.19) qh∈Ph

Combining (3.18) and (3.19), by the triangle inequality, we get the lemma as

kp − phkL2 ≤ kp − p¯hkL2 + kp¯h − phkL2

≤ |u − uh|H1 + inf |u − vh|H1 + inf |p − qh|L2 . vh∈Vh qh∈Ph

We conclude the section by the optimal order of convergence for the pressure.

Theorem 3.3 Let Theorem 3.2 and Lemma 3.2 hold. Then the solution ph of (2.5) is of optimal order, i.e.,

min{k,r−2} kp − phkL2 ≤ Ch kfkHr−3 .

Proof. Let F = p in (3.10) and div U = p. We simply let Uh be the nodal-value interpolation of U in Vh if U is smooth enough, or the boundary-averaging interpolation of U deﬁned in [40]. Let qh = div Uh.

min{k,r−2} kp − qhkL2 = k div(U − Uh)kL2 ≤ kU − UhkH1 ≤ Ch kUkHr−1 min{k,r−2} min{k,r−2} ≤ Ch kpkHr−2 ≤ Ch kfkHr−3 .

4 The Powell-Sabin P1 divergence-free element

We will deﬁne a Powell-Sabin triangulation and the ¤ 1 divergence-free element. The general

theory developed in the last section will be applied to the Powell-Sabin ¤ 1 element. We note that the general theory can be applied to any divergence-free element where the approximation

of C1 elements on the same grid is known, for example, the Powell-Sabin-12 split ¤ 1 element [37]. Let Ω be a 2D (d = 2) polygonal domain triangulated into a quasiuniform grid Ω˜ h (see ˜ ˜ [10]). For each triangle T ∈ Ωh, let cT˜ be the center of the inscribed circle of T , see Figure ˜ 1. To deﬁne the Powell-Sabin triangulation Ωh based on Ωh, we connect each cT˜ to both the three vertices of T˜ (this is the Hsieh-Clough-Tocher triangulation, cf. [14, 33, 23, 28, 49]) and ˜ ˜ ˜ the three cT˜i of the three neighboring triangles Ti of Ωh (see Figure 1). If T is a boundary triangle, we simply connect cT˜ to the mid-point of the boundary edge(s). In this fashion, each triangle of Ω˜ h splits into 6 subtriangles, see Figure 2. We call the resulting triangulation a Powell-Sabin triangulation, Ωh. It can be shown (see the references in [22]) that the family of Powell-Sabin triangulations derived from a quasi-uniform family triangulations is also quasiuniform. We note also that the interior point on each triangle T˜ can be chosen diﬀerently, in

c T˜3

c T˜H HH H jH

cT˜ cT˜ 1 2

Figure 1: A Powell-Sabin split of the central triangle T˜.

general. However, we can see that by choosing the point cT˜i to be the center of the inscribed ˜ circle of Ti, it is guaranteed that the intersection of the edge cT˜1 cT˜2 and the common edge between triangles T˜1 and T˜2 is an interior point on the common edge. In fact, the intersection point is between the two projection points of cT˜1 and cT˜2 on the common edge. We remark that there is no restriction on Ω˜ h here. However, if we would like to have a nested multigrid- reﬁnement of Ω˜ h and a nested family of Powell-Sabin ﬁnite element spaces, Ω˜ h would be no longer an arbitrary triangulation (see [29]). In the numerical test in Section 6, we use the

center of mass of each triangle as cT˜, for simplicity. We note that the center of the inscribed circle is used in Figure 1, but the center of mass is used in Figure 2.

(Ω˜ h) (Ωh) @¨¡ ¨¡@¡¨ ¨¡@¡¨ ¨¡@¡¨ ¨¡ ¨¡ @¨¡ ¨¡ @¨¡ ¨¡ @¨¡ ¨¡ @¨¡ ¨ ¡ ¨¡@¨ ¡ ¨¡@¨ ¡ ¨¡@¨¡ ¨¡@ @¡¨ ¨¡@¡¨ ¨¡@¡¨ ¨¡@¡¨ ¨¡ ¨¡ @¨¡ ¨¡ @¨¡ ¨¡ @¨¡ ¨¡ @¨¡ ¨ ¨ ¨ ¨ ⇒ ¨ ¡ ¡@¨ ¡ ¡@¨ ¡ ¡@¨¡ ¡@ @¡¨ ¨¡@¡¨ ¨¡@¡¨ ¨¡@¡¨ ¨¡ ¨¡ @¨¡ ¨¡ @¨¡ ¨¡ @¨¡ ¨¡ @¨¡ ¨ ¨ ¨ ¨ ¡¨ ¡¨@¨ ¡ ¡¨@¨ ¡ ¡¨@¨¡ ¡¨@ ¨@¡¨ ¡¨@¡¨ ¡¨@¡¨ ¡¨@¡¨ ¡ ¡¨ @¨¡ ¡¨ @¨¡ ¡¨ @¨¡ ¡ ¨@¨¡ ¨ ¡ ¡@¨ ¡ ¡@¨ ¡ ¡@¨¡ ¡@

Figure 2: A uniform triangulation Ω˜ h splits into a Powell-Sabin grid Ωh. ¥ The Powell-Sabin C0- ¥ 1 space and the Powell-Sabin C1- 2 space on the grid Ωh are deﬁned as, respectively,

P S Vh = {v | v|T ∈ ¥ 1 ∀T ∈ Ωh} ∩ H1,0(Ω), P S Sh,0 = Span {φi,j | j = 0, 1, 2, xi ∈ ¦ h} ⊂ H2,0(Ω), ˜ where ¦ h is the set of all internal vertices in the original triangulation Ωh, and φi,j ∈ C1(Ωh) are the three Powell-Sabin nodal basis functions at vertex i, which are piecewise quadratic

11 polynomials on Ωh:

∂φ0,j φi,0(xk) = δi,k, (xk) = 0 ∀k, l = 0, 1, 2, ∂xl ∂φi,j φi,j(xk) = 0 ∀i & k, j > 0, (xk) = δi,kδj,l. ∂xl

In [31] it is shown that φi,j is unique and supported on all the Ω˜ h triangles sharing xi as a vertex. It is known that the span of all such nodal basis functions {φi,j} does generate all

C1- § 2 polynomials on the grid. But in our framework, we do not require the approximation § property of the full C1- § 2 space, but a subspace instead. In fact, some subspaces of C1- 2 do have the full order of approximation too, studied in [3]. With the Powell-Sabin nodal basis {φi,j}, the standard nodal interpolation operator would

preserve § 2 polynomials, and be stable in the H2-norm (see the standard finite element theory, e.g. [14, 10]). Consequently the interpolation produces the optimal order of approximation, which guaranteed the finite element project would be of the optimal order of approximation for the biharmonic equation. On the other side, we assume the 2D polygonal domain provides the full H4 elliptic regularity, i.e., for any g ∈ L2(Ω), the unique weak solution s ∈ H2,0(Ω) of the biharmonic equation satisfies

kskH4(Ω) ≤ CkgkL2(Ω). (4.1)

For example, Blum and Rannacher showed the H4 elliptic regularity in [8] for convex polygons with inner angles smaller than 126.28...o. We can ﬁnd more results on the elliptic regularity in [18]. In addition, Arnold, Scott and Vogelius showed the H1 (and higher) bound (3.10) for the regular inversion of the divergence operator in [5]. With these assumptions, Theorems 3.2

and 3.3 hold for the Powell-Sabin § 1 divergence-free element too. We state it as the following theorem.

Theorem 4.1 Let (3.10) and (4.1) hold. Then the Powell-Sabin § 1 divergence-free element solution for the Stokes equations (2.5) is of optimal order in approximation:

ku − uhkH1 + kp − phkL2 ≤ ChkfkL2 .

5 An iterative method for the divergence-free element

In this section, we will introduce the classic iterative penalty method ([17, 10, 11, 41, 20]) and we will show how to reduce the mix-element space to a single divergence-free element space

in the method. The method is an iterative method solving a conforming § k element for the vector Laplace equations, in essence. Therefore the divergence-free element method is not a mixed-element method. This can be done obviously as the discrete space for the pressure is a derivative:

d Ph = div Vh; Vh ⊂ H1,0(Ω), Ph ⊂ L2,0(Ω).

12 Definition 5.1 (The iterative penalty method for the divergence-free element.) Let the initial 0 n iterate uh = 0 for the finite element Stokes equation (2.5). The rest iterates uh are defined iteratively to be the unique solution of

n−1 n n j a(uh, vh) + α(div uh, div vh) = (f, vh) + (div uh, div vh) ∀vh ∈ Vh, Xj=0

n = 1, 2, · · · . Here α is positive constant between 1 and 10. At the end of iteration, we let

n n j ph = div uh. Xj=0

Remark 5.1 In the iterative penalty method of Deﬁnition 5.1, we need only to do computer

coding for the conforming C0- ¨ k element for the vector Laplacian like equations. This avoids a heavy work for coding the discontinuous ﬁnite elements for the pressure.

n Remark 5.2 By Deﬁnition 5.1, at the end of iteration, div uh = 0 and we obtain the solution uh of (2.5):

n−1 n j a(uh, vh) = (f, vh) + (div uh, div vh) ∀vh ∈ Vh, Xj=0 n a(uh, vh) = (f, vh) ∀vh ∈ Zh.

n n j Consequently, the unique solution ph of (2.5) is then ph = div j=0 uh. P Remark 5.3 The iterative speed for convergence of the iterative penalty method in Deﬁnition 5.1 is shown to be constant, independent of the grid size h, in [10] under a slightly stronger condition, the inf-sup condition. We would predict that the convergence speed for the divergence- free element, i.e., without the inf-sup condition or with an unbounded inf-sup constant, remains constant. A further study will be done somewhere else.

Remark 5.4 The iterative penalty method of Deﬁnition 5.1 is a nested-iteration method. As usual, the solution for the inner equation would be solved by an iterative method, up to certain accuracy, not too accurate. In other words, the inner iteration and the outer iteration would be combined into one, for example, the multigrid iteration ([41]).

6 Numerical tests

In this section, we report some results of numerical experiments of the ¨ 1 divergence-free element on the uniform cross grids and on the Powell-Sabin grids, shown in the left graph and the right graph of Figure 2. The computational domain Ω for (2.5) is the unit square. The unit square is ﬁrst partitioned into n × n small squares, then 2(n × n) triangles. The partition is denoted by Ω˜ h, shown in the left graph of Figure 2. Each triangle of Ω˜ h is further subdivided into 6 subtriangles by connecting the center of mass with the three vertices and the three mid-edge points, shown by

13 Figure 3: The ﬁrst component of uh on the 4th level grid.

Ωh in the right graph of Figure 2. This is a simpliﬁed version of Powell-Sabin grids, described in Section 4. It works because the base triangulation is regular. The right hand side function f for (2.5) is

−gyxx − gyyy − gxxx f = −∆ curl g − ∇gxx = . (6.1) gxxx + gxyy − gyxx where

g = 28(x − x2)2(y − y2)2.

The continuous solution for the Stokes equations (2.1) is

u = curl g, p = −gxx.

We depict the ﬁrst component of the numerical solution uh in Figure 3 and ph in Figure 4 with

In Table 1, we list the errors of the Powell-Sabin © 1 divergence-free element on several levels of grids. The order of convergence for the velocity in the H1-norm and the order for the pressure in the L2-norm are apparently both 1, verifying the theory presented in Theorem 4.1. Also in Table 1, we can see that the nodal error for the velocity converges at order 2, while that for the pressure at order 1, as expected, though not proved in this paper.

In Table 2, we listed the errors of the © 1 divergence-free element on the uniform grids, shown in the left graph of Figure 2. It is well-known that such a conforming © 1 method suﬀers the locking problem, i.e., the solutions are forced to approach 0, instead of the exact solution, on ﬁner grids. The numerical data in Table 2 show that there is no approximation at all for

the © 1 divergence-free element on the uniform grids. It shows that the discrete solutions for the velocity go to zero and that the discrete solutions for the pressure diverge.

Acknowledgments. This work was done while the author visited the University of Sci- ence and Technology of China (USTC) during his sabbatical leave year, 2005–2006, and was presented in the International Conference on Partial Diﬀerential Equations and Numerical

14 Figure 4: The pressure ph on the 4th level grid.

Table 1: The Powell-Sabin divergence-free 1 element.

Level Grid # triangles |u − uhkH1 kp − phkL2 |u − uh|l∞ kp − phkl∞ 2 2 × 2 48 2.42444 0.42188 5.65923 9.54726 3 4 × 4 192 1.31865 2.91791 0.15469 5.51233 4 8 × 8 768 0.67491 1.44462 0.05398 3.23210 5 16 × 16 3072 0.33514 0.71194 0.01544 1.72471 6 32 × 32 12288 0.16663 0.35458 0.00413 0.90545 7 64 × 64 49152 0.08306 0.17711 0.00107 0.46507

Analysis, held at Changsha, China, June 22–26, 2006. The author wishes to thank Chairman Falai Chen of the Department of Mathematics, USTC, and Professor Zhimin Zhang of Wayne State University, for their invitation and hospitality.

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