LOCAL AND POINTWISE ERROR ESTIMATES OF THE LOCAL DISCONTINUOUS GALERKIN METHOD APPLIED TO STOKES PROBLEM

DISCONTINUOUS GALERKIN METHOD APPLIED TO STOKES PROBLEM

J. GUZM ´ AN

Abstract. We prove local and pointwise error estimates for the local discon- tinuous Galerkin method applied to Stokes problem in two and three dimen- sions. By using techniques originally developed by A. Schatz [Math. Comp., 67 (1998), 877-899] to prove pointwise estimates for the Laplace equation, we prove optimal weighted pointwise estimates for both the velocity and the pressure for domains with smooth boundaries.

1. Introduction

In this paper, we study the local and pointwise behavior of the Local Discontin- uous Galerkin (LDG) method for the following problem

−4~ u + ∇p = ~ f in Ω,

∇ · ~ u = g in Ω,

~

u = 0 on ∂Ω, (1.1)

where Ω ⊂ R ^N (N = 2, 3) is bounded and has a smooth boundary. Here ~ u = (u ₁ , · · · , u _N ) represents the velocity of the fluid, p ∈ L ² ₀ (Ω) the pressure, ~ f = (f ₁ , · · · , f _N ) is a smooth external force and g ∈ L ² ₀ (Ω) is a smooth function (for Stokes problem we take g ≡ 0). The space L ² ₀ (Ω) consist of functions in L ² (Ω) with mean zero.

The LDG method for Stokes problem was introduced by Cockburn et al. [10];

see the review [8]. The LDG finite dimensional spaces for the both the velocity and pressure are discontinuous across interelement boundaries. Therefore, the LDG method allows meshes with hanging nodes and allows flexibility when choosing the local finite element spaces. Cockburn et al. [6] generalized this method to Oseen equations. Finally, in [7] the LDG method was extended to the stationary incompressible Navier-Stokes equation; see also the follow up note [9]. Although the LDG method considered in [10] satisfies the incompressibility condition only weakly, it is shown in [7] that one can enforce exact incompressibility by a simple element by element post-processing technique.

Date: September 26, 2006.

1991 Mathematics Subject Classification. 65N30,65N15.

Key words and phrases. Finite Elements, Discontinuous Galerkin, Stokes Problem.

The author was supported by a National Science Foundation Mathematical Science Postdoc- toral Research Fellowship (DMS-0503050).

1

Global L ² error analysis was performed in [10] for the LDG method applied to (1.1). In this paper we prove local L ² error estimates along with pointwise error estimates. Roughly speaking, the local L ² analysis shows that the error for both the pressure and the gradient of the velocity measured by the L ² (D ₀ ) − norm for a subdomain D ₀ ⊂ Ω is bounded by the best approximation error in the L ² (D ₁ ) − norm for a slightly larger subdomain D ₁ plus the error in a weaker norm. These estimates are very similar to the local error estimates obtained by Arnold and Liu [2] for conforming mixed methods applied to (1.1). However, the results in [2] are for interior subdomains D 0 whereas in this paper we allow D 0 to touch

∂Ω. Many of the techniques to prove local error estimates presented in this paper and in [2] are borrowed from the techniques developed by Nitsche and Schatz [18]

for proving local estimates of conforming finite element methods for the Laplace equation. However, the pressure term and the incompressibility equation adds extra difficulties when analyzing the Stokes problem. Moreover, the fact that the LDG spaces are discontinuous and that the primal formulation of the LDG method does not satisfy the Galerkin orthogonality property adds even more challenges when analyzing the LDG method for (1.1). Local error estimates for the LDG method applied to Laplace’s equation were carried out by Chen [5]. Later Guzm´ an [17]

proved similar results for three DG methods, including the LDG method, in primal form.

We use the local L ² error estimates to prove weighted pointwise estimates. These pointwise estimates are optimal and describe how the error at a point x depends on the behavior of the exact solution in regions away from x. Recently, Chen [3]

used the local estimates derived in [2] to prove pointwise estimates of conforming mixed methods for (1.1) on a domain Ω with a smooth boundary. Chen makes use of techniques originally developed by Schatz [19] to prove pointwise estimates for the Laplace equation. In this paper we also use the techniques found in [19] and our results are very similar to the results contained in [3]. However, in order to prove pointwise estimates Chen assumed local error estimates for subdomains that touch

∂Ω which are not contained in [2]. As mentioned above, in this paper we prove local estimates for subdomains that touch ∂Ω for the LDG method. Furthermore, Chen assumed that functions in the finite element subspace for the velocity are zero on ∂Ω, but such spaces are difficult to construct for curved edges. Since we are analyzing the LDG method there is no need to choose subspaces that agree with the boundary data.

To further put our work in perspective, we describe previous work concerning

pointwise error estimates for Stokes problem. Pointwise error estimates for con-

forming mixed methods applied to Stokes problem was first carried out by Dur´ an

et al. [12]. For a stabilized Petrov-Galerkin mixed method the analysis was carried

out in [14]. The drawback of these articles is that the analysis is two dimensional

and the estimates are sub-optimal by a logarithmic factor for higher order ele-

ments. Recently, Girault et al. [16] removed the logarithmic factor and extended

the results to three dimensions. In this paper and in [3] the logarithmic factor is

also not present for higher order elements. The proof in [16] uses techniques for

maximum-norm estimates for finite element approximations of the Laplace equa-

tion [23] whereas in this paper and in [3] techniques from [19] were used. This

allows us to establish a more local dependence of the error on the exact solution

as compared to the results in [16]. However, our results are restricted to domains

with smooth boundaries whereas the results in [16] hold for polygonal/polyhedral domains. We use an integral representation of solutions to (1.1) and sharp bounds for the kernels whereas in [16] an integral representation for the the inverse of the divergence operator and sharp bounds for that kernel are used; see [15].

Instead of discretizing the viscosity term −4~ u with the LDG method one can discretize this term using methods in [1] to come up with different DG methods for (1.1); see [21]. If we use the methods in [1] that are consistent, adjoint consistent and have bilinear forms that our coercive to discretize the viscosity term of (1.1), then we can easily prove similar results for the resulting methods for (1.1).

The rest of the paper is organized as follows: In the next section we define the LDG method and present our main results. Section 3 contains the proofs of the theorems.

2. The Main Results

2.1. The LDG Method. We assume we have a family of triangulations T h which fit the boundary of Ω exactly, where Ω = ∪ T ∈T

_h

T . We allow hanging nodes, but we assume our family of meshes are quasi-uniform and that the elements are shape- regular. The collection of edges/faces will be denoted by E _h = E _h ^I ∪ E _h ^B where E _h ^I is the set of interior edges/faces and E _h ^B is the set of boundary edges/faces.

The LDG approximations belong to the following spaces:

V ~ _h ^k = {~ v ∈ [L ² (Ω)] ^N : ~ v| _T ∈ [P _k (T )] ^N , ∀T ∈ T _h }, Σ ^k _h = {σ ∈ [L ² (Ω)] ^{N ×N} : σ| _T ∈ [P _k (T )] ^{N ×N} , ∀T ∈ T _h }, Q ^k _h = {q ∈ L ² ₀ (Ω) : q| T ∈ P k−1 (T ) ∀T ∈ T h },

Q ˜ ^k _h = {q ∈ L ² (Ω) : q| K ∈ P k−1 (T ) ∀T ∈ T h }.

Here P _l (T ) are the set of polynomials of degree less than or equal to l defined on T . An arrow above a function means that the function is vector-valued and a line under the function means that the function is matrix-valued.

To write a compact form of the method we will need to define the jump and average operators. The jump operator is given by

[[(φ  ~ n)]] =

( (φ  ~ n) on boundary edges in E _h ^B

(φ ⁺  ~ n K

⁺

) + (φ ⁻  ~ n K

⁻

) on interior edges in E _h ^I , where φ ^± denote traces of φ on the edge e = ∂K ⁺ ∩ ∂K ⁻ taken from within the interior of K ^± . The vector ~ n K is the outward unit vector normal to K. The symbol  denotes a multiplication operator. The average operator is defined as

{{φ}} =

( φ on boundary edges in E _h ^B

1

2 (φ ⁺ + φ ⁻ ) on interior edges in E _h ^I .

We can now define the LDG approximation. To simplify notations we take the

stabilization parameters to be 1 (i.e. c 11 = d 11 = 1 in (2.21) [10]). Since we are

working with quasi-uniform meshes we use h everywhere instead of the local mesh

size.

Find (~ u h , p h ) ∈ ~ V _h ^k × Q ^k _h such that A _h (~ u _h , ~ v) + B _h (~ v, p _h ) =

Z

Ω

f · ~ ~ v dx

−B _h (~ u _h , q) + D _h (p _h , q) = Z

Ω

gqdx ∀(v, q) ∈ ~ V _h ^k × Q ^k _h , (2.1)

where A h (~ u, ~ v) =

Z

Ω

(∇ h ~ u − L(~ u)) : (∇ h ~ v − L(~ v)) dx + h ⁻¹ X

e∈E

h

Z

e

[[~ u ⊗ ~ n]] : [[~ v ⊗ ~ n]] ds,

B h (~ v, q) = − Z

Ω

q∇ h · ~v dx + X

e∈E

^I_h

Z

e

{{q}}[[~v · ~ n]] ds + X

e∈E

^B_h

Z

e

q~ v · ~ nds,

D h (p, q) = h X

e∈E

_h^I

Z

e

[[p~ n]] · [[q~ n]] ds.

For ~ u ∈ [H _h ¹ (Ω)] ^N the lifting operator L(~ u) ∈ Σ ^k _h is defined by Z

Ω

L(~ u) : σ dx = X

e∈E

_h

Z

e

[[~ u ⊗ ~ n]] : {{σ}} ds ∀σ ∈ Σ ^k _h .

We used the standard notation (∇~ v) ij = ∂ j v i and (∇ · σ) i = P N

i=1 ∂ j σ ij . We also have ~ v · ~ n = P N

i=1 v _i n _i , (~ v ⊗ ~ n) _ij = v _i n _j and σ : τ = P N

i,j=1 σ _ij τ _ij . Here ∇ _h ~ u is the piecewise defined function such that ∇ h ~ u = ∇u on each element T ∈ T h .

By using the lifting operator L we eliminated the unknown σ h appearing the in the original LDG method [10]. As a result, the Galerkin orthogonality property is not satisfied. That is, if (~ u, p) solves (1.1), then we have

A h (~ u, ~ v) + B h (~ v, p) = Z

Ω

f · ~ ~ v dx + R(~ u, ~ v)

−B h (~ u, q) + D h (p, q) = Z

Ω

g q dx ∀(v, q) ∈ H _h ¹ (Ω) × L ² ₀ (Ω).

(2.2)

The residual term R(~ u, ~ v) is given by R(~ u, ~ v) = X

e∈E

h

Z

e

{{Π(∇~ u) − ∇~ u}} : [[v ⊗ n]]ds.

2.2. Sobolev Norms. In order to describe the main results we need to introduce some norms. If Ω ₀ ⊂ Ω, we define our discontinuous Sobolev space as in [3]:

W _h ^r,p (Ω 0 ) = {v : v ∈ W ^r,p (T ∩ Ω 0 ), ∀ T ∈ T h }.

Let Ω ₀ ⊂ Ω then we define the broken norm for r = 1 and 1 ≤ p < ∞

||~v|| ^p

W

_h^1,p

(Ω

₀

) = X

T ∈T

h

||∇~v|| ^p _L

p

(T ∩Ω

₀

) + h ^1−p X

e∈E

h

||[[~v ⊗ ~ n]]|| ^p _L

p

(e∩Ω

₀

) .

If p = ∞, we define

||~v|| _W

1,∞

h

(Ω

0

) = sup

T ∈T

_h

||∇~v|| _L

p

(T ∩Ω

0

) + h ⁻¹ sup

e∈E

_h

||[[~v ⊗ ~ n]]|| _L

^∞

_(e∩Ω

₀

₎ .

For the pressure we use the following norm for 1 ≤ p < ∞

||q|| ^p _L

p

h

(Ω

₀

) = ||q|| ^p _L

p

(Ω

₀

)

+h X

e∈E

_h^I

||[[q~ n]]|| ^p _L

p

(e∩Ω

0

) + h X

e∈E

h

||{{q~ n}}|| ^p _L

p

(e∩Ω

0

) .

For r > 1 and 1 ≤ p ≤ ∞, we define

||~v|| ^p _W

r,p

h

(Ω

₀

) = X

T ∈T

_h

||~v|| ^p _W

r,p

(T ∩Ω

0

)

The case p = ∞ can be defined similarly. We write H _h ^r = W _h ^r,2 for any r ≥ 1. We will also need to define negative-order Sobolev norms. Let D ⊂ Ω and q ∈ L ² (D) then we define the H ⁻¹ (D) norm as follows

||q|| _H

⁻¹

_(D) = sup

r∈C∞c (D)

||r||H1 (D)=1

Z

D

qrdx.

We present a function space, as in [25], that will let us define a slightly different negative-order norm. If S ⊂ D ⊂ Ω, let ∂ _< (S, D) = dist(∂S \ ∂Ω, ∂D \ ∂Ω). The space is defined as follows:

C _< ^∞ (D) = {v ∈ C ^∞ : ∂ _< (supp(v), D) > 0}.

The H _< ⁻¹ (D) norm is defined by

||q|| _H

⁻¹

<

(D) = sup

r∈C∞<(D)

||r||

H1 (D)=1

Z

D

qrdx.

Notice that H _< ⁻¹ (D) and H ⁻¹ (D) norms coincide if D ⊂⊂ Ω.

2.3. Local Estimates. For the rest of this paper Π will denote the L ² projection into ˜ Q ^k _h , ~ Π will denote the L ² projection into ~ V _h ^k and Π the L ² projection into Σ ^k _h Theorem 2.1. Suppose that (~ u h , p h ) ∈ ~ V _h ^k × Q ^k _h and (~ u, p) ∈ [H _h ¹ (Ω)] ^N × L ² ₀ (Ω) satisfy

A h (~ u − ~ u h , ~ v) + B h (~ v, p − p h ) = R(~ u, ~ v)

−B _h (~ u − ~ u _h , q) + D _h (p − p _h , q) = 0 ∀(~v, q) ∈ ~ V _h ^k × Q ^k _h , (2.3)

then for D ₀ ⊂ D d ⊂ Ω with ∂ < (D ₀ , D _d ) = d ≥ 2 h

||~ u − ~ u _h || _H

¹

h

(D

0

) + ||p − p _h || L

²

(D

₀

) + (h X

e∈E

_h^I

||[[(p − p h )~ n]]|| ² _L

2

(e∩D

0

) ) ^1/2

≤ C(||~ u − ~ Π~ u|| _H

1

h

(D

d

) + h||~ u − ~ Π(~ u)|| _H

2

h

(D

d

) + ||p − Πp|| _L

2 h

(D

d

) ) +Cd ⁻¹ (||~ u − ~ Π~ u|| _L

2

(D

d

) + ||p − Πp|| _H

⁻¹

<

(D

d

) ) +Cd ⁻¹ (||~ u − ~ u h || _L

2

(D

d

) + ||p − p h || _H

−1

<

(D

d

) ).

In these estimates we have the norm H _< ⁻¹ norm of the pressure appearing in the

right hand side instead of the H ⁻¹ norm. The H ⁻¹ norm appears in the estimates

found in [2] since in their analysis only interior subdomains were considered.

2.4. Pointwise Estimates. We need to define weighted norms in order to describe the results of this section. We will use the weight used in [19], σ x (y) = _h+|x−y| ^h . The weighted norms for 1 ≤ p < ∞ are given by

||~v|| ^p

W

_h^1,p

(Ω

₀

),x,s = X

T ∈T

_h

||σ _x ^s ∇~v|| ^p _L

_p

_{(T ∩Ω}

0

) + h ^1−p X

e∈E

_h

||σ _x ^s [[~ v ⊗ ~ n]]|| ^p _L

_p

_(e∩Ω

0

) , and for p = ∞

||~v|| _W

1,∞

h

(Ω

0

),x,s = sup

T ∈T

h

||σ ^s _x ∇~v|| _L

p

(T ∩Ω

0

) + sup

e∈E

h

h ⁻¹ _e ||σ ^s _x [[~ v ⊗ ~ n]]|| _L

^∞

_(e∩Ω

₀

₎ . Also,

||q|| ^p _L

p

h

(Ω

0

),x,s = ||σ ^s _x q|| ^p _L

p

(e∩Ω

₀

)

+h X

e∈E

_h^I

||σ _x ^s [[q~ n]]|| ^p _L

p

(e∩Ω

₀

) + h X

e∈E

_h

||σ ^s _x {{q}}|| ^p _L

p

(e∩Ω

₀

) ,

and for p = ∞

||q|| _L

∞

(Ω),x,s = ||σ ^s _x q|| _L

∞

(Ω

₀

)

Now we can state the pointwise error estimates for the velocity.

Theorem 2.2. Suppose (~ u, p) ∈ [W ^1,∞ (Ω)] ^N × L ^∞ (Ω) ∩ L ² ₀ (Ω) and (~ u h , p h ) ∈ V ~ _h ^k × Q ^k _h satisfy (2.3). Let x ∈ ¯ Ω and s satisfy 0 ≤ s ≤ k − 1. Then, there exists a constant C independent of x, (~ u, p), (~ u _h , p _h ) and h such that

|(~ u − ~ u h )(x)| ≤ Ch log(1/h) ^{s ¯} (||~ u − ~ Π(~ u)|| _W

1,∞

h

(Ω),x,s + ||p − Π(p)|| _L

^∞

h

(Ω),x,s

+||σ ^s _x (∇ _h ~ u − Π(∇ _h ~ u))|| _L

∞

(Ω) ) where ¯ s = 0 if 0 ≤ s < k − 1, and ¯ s = 1 if s = k − 1.

Notice that if k ≥ 2 we can take s = 0 < k − 1 and we get the optimal L ^∞ (Ω) found in [16] for the velocity. But, if we take s > 0 then the error at x depends much more on the behavior of the exact solution in regions close to x rather than the behavior of the exact solution in regions far from x. In fact, one can prove error expansion inequalities; see Theorem 4.1 in [19] for the corresponding result for the Laplace equation. If k = 1 then we are forced to take s = 0 in Theorem 2.2. In this case, the logarithmic factor does appear. Also, we see that the estimates is no longer local. That is, the error of the velocity at the point x depends on the exact solution equally on all of Ω; see [11] for the sharpness of this result for the Laplace equation.

The pointwise estimate for the pressure is given in the next theorem.

Theorem 2.3. Suppose (~ u, p) ∈ [W ^1,∞ (Ω)] ^N × L ^∞ (Ω) ∩ L ² ₀ (Ω) and (~ u h , p h ) ∈ V ~ _h ^k × Q ^k _h satisfy (2.3). Let x ∈ ¯ Ω and s satisfy 0 ≤ s ≤ k. Then, there exists a constant C independent of x, (~ u, p), (~ u _h , p _h ) and h such that

|(p − p h )(x)| ≤ C log(1/h) ^{¯ ¯ s} (||~ u − ~ Π(~ u)|| _W

1,∞

h

(Ω),x,s + ||p − Π(p)|| _L

^∞

h

(Ω),x,s

+||σ ^s _x (∇~ u − Π(∇~ u))|| L

^∞

(Ω) )

where ¯ s = 0 if 0 ≤ s < k, and ¯ ¯ s = 1 if s = k. ¯

The logarithmic will not appear in the estimates for the pressure as long as we take 0 ≤ s < k which can always be done since k ≥ 1. Since we can always choose 0 < s < k, we see that error of the pressure at the x has more of a dependence on the behavior of the exact solution near x rather than the behavior of the exact solution far from x.

3. Proofs

Before we prove Theorems 2.1, 2.2 and 2.3, we state some preliminaries results.

3.1. Preliminary results.

3.1.1. Continuity Of Bilinear Forms. We can easily prove the following bound for our lifting operator. For any ~ v ∈ [W _h ^1,p (Ω)] ^N we have

||L(~v)|| _L

p

(Ω) ≤ Ch ^1/p−1 ( X

e∈E

_h

||[[~v ⊗ ~ n]]|| ^p _L

p

(e) ) ^1/p

Now it easily follows that

A h (~ u, ~ v) ≤ C||~ u|| _W

1,l

h

(Ω) ||~v|| _W

1,r h

(Ω)

where ¹ _l + ¹ _r = 1. In fact, one has

A h (~ u, ~ v) ≤ C||~ u|| _W

1,l

h

(Ω),x,−s ||~v|| _W

1,r h

(Ω),x,s . We also have

B(~ v, q) ≤ C||~ u|| _W

1,l

h

(Ω),x,−s ||q|| _L

^r

h

(Ω),x,s .

3.1.2. Regularity and Global Error Estimates. The following result is standard; see [24].

Proposition 3.1. If ~ f ∈ [H ^l (Ω)] ^N and g ∈ H ^l+1 (Ω) ∩ L ² ₀ (Ω) with for l ≥ −1, then there exists a unique solution (~ u, p) ∈ H ^l+2 × H ^l+1 (Ω) of (1.1). Furthermore, the following bound holds

||~ u|| _H

l+2

(Ω) + ||p|| _H

l+1

(Ω) ≤ C(||f || _H

l

(Ω) + ||g|| _H

l+1

(Ω) ).

Global error estimates were obtained in [10]. Here we state the result in a slightly different form.

Proposition 3.2. Let (~ u, p) solve (1.1) and let (~ u _h , p _h ) be the LDG approximation defined by (2.1), then

||∇(~ u − ~ u h )|| _L

2

(Ω) + ||p − p h || _L

2

(Ω)

≤ C(||∇(~ u − ~ Π(~ u))|| L

²

(Ω) + ||p − Π(p)|| L

²

(Ω) )

+C(||∇~ u − Π(∇~ u)|| _L

2

(Ω) + h||∇~ u − Π(∇~ u)|| _H

1

(Ω) ).

3.1.3. Approximation. We start by stating well-known trace inequalities. Let e be an face of T ∈ T h . Then, for 1 ≤ p ≤ ∞, we have

(3.1) ||φ|| _L

p

(e) ≤ C(h ⁻

^p1

||φ|| _L

p

(T ) + h ¹⁻

^1p

|φ| _W

1,p

(T ) ).

If we restrict φ to P k (T ), then

(3.2) ||φ|| _W

l,t

(T ) ≤ Ch ^[

^Ns

⁻

^Nt

^]+l−q ||φ|| _W

q,s

(T ) , (3.3) ||φ|| _L

_p

_(e) ≤ Ch ⁻

^1p

||φ|| _L

_p

_{(T )} ,

where C does not depend on φ, h, e, or T . Here 1 ≤ t ≤ s ≤ ∞ and 0 ≤ l ≤ q The following is a standard elementwise approximation result. Let v ∈

W _h ^j,p (Ω) with 0 ≤ i ≤ j ≤ k − 1. Then,

(3.4) ||v − Π(v)|| _W

i,p

(T ) ≤ Ch ^j−i |v| _W

j,p

(T ) , ∀T ∈ T h , where C does not depend on v, h, or T .

The same result holds for ~ Π and Π, but in these cases 0 ≤ i ≤ j ≤ k. We now define a standard negative-order Sobolev norm. The following inequalities are similar to Lemma 2.2 in [17].

Lemma 3.3. Let χ ∈ P r−1 (T ), for and let ω be a smooth function. Suppose there exist constants C > 0 and d ≥ h, ||D ^l ω|| _L

^∞

_(Ω) ≤ Cd ^−l for l = 0, 1, · · · , r +1. Then, for r ≥ 2

|ω ² χ| H

^r

(T ) ≤ C

h ^r−2 (d ⁻¹ ||ωχ|| H

¹

(T ) + d ⁻² ||χ|| L

²

(T ) ), (3.5a)

|ωχ| _H

r

(T ) ≤ C

h ^r−2 (d ⁻¹ ||χ|| _H

1

(T ) + d ⁻² ||χ|| ² _L

2

(T ) ), (3.5b)

and for r = 1

|ω ² χ| _H

1

(T ) ≤ Cd ⁻¹ ||ωχ|| _L

2

(T ) , (3.5c)

|ωχ| _H

1

(T ) ≤ Cd ⁻¹ ||χ|| _L

2

(T ) . (3.5d)

Here C is independent of ω, χ, T , and h.

Now we state a super-approximation result (see [17]) which easily follows from (3.4) and (3.5a) if we set r − 1 = k.

Lemma 3.4. Let ∂ < (D 0 , D d ) = d > 2h, where ω ∈ C _< ^∞ (D 0 ). Suppose

||D ^l ω|| _L

∞

(S

0

) ≤ Cd ^−l for l = 0, 1, · · · , k + 2. Then, for all ~ v ∈ ~ V _h ^k 1

h ||ω ² ~ v − ~ Π(ω ² ~ v)|| _L

2

(D

0

) + ||ω ² ~ v − ~ Π(ω ² ~ v)|| _H

1 h

(D

₀

)

≤ Ch(d ⁻¹ ||ω~v|| _H

1

h

(D

_d

) + d ⁻² ||v|| _L

₂

_(D

_d

₎ ), where C is independent of ~ v and ω.

We will also need the following superapproximation result.

Lemma 3.5. Let ω be as in Lemma 3.4. Then, for all p ∈ Q h

1

h ||ω ² p − Π(ω ² p)|| _L

2

(D

₀

) + ||∇ _h (ω ² p − Π(ω ² p))|| _L

2

(D

₀

)

≤ Cd ⁻¹ (||ωp|| _L

2

(D

_d

) + h

d ||p|| _L

2

(D

_d

) ).

where C is independent of h, p and ω.

3.2. Proof of Theorem 2.1. With a covering argument, as was used in [20], it is enough to show Theorem 2.1 with D 0 and D d replaced with S d and S 2d , respectively. Here S d = B d ∩ Ω and S 2d = B 2d ∩ Ω and B d ⊂ B 2d are concentric balls with common center in Ω and of radius d and 2d, respectively. We prove this result in several steps.

3.2.1. Step1: Reduce to weighted stability estimates.

Lemma 3.6. Let ω ∈ C _< ^∞ (S _3d/2 ) with ω ≡ 1 on S _d and |D ^l ω| _L

^∞

≤ Cd ^−l for l = 1, 2, . . . , k + 2, then Theorem 2.1 is implied by the following inequality

||ω~ u _h || _H

1

h

(Ω) + ||ωp _h || _L

2

(Ω) + D _h (ωp _h , ωp _h )

≤ C(||~ u|| _H

1

h

(S

2d

) + h||~ u|| _H

2

h

(S

2d

) + ||p|| _L

2 h

(S

2d

) ) +Cd ⁻¹ (||~ u h || _L

2

(S

_2d

) + ||p h || _H

−1

<

(S

_2d

) ).

(3.6)

Proof. Since ~ u − ~ u h = (~ u − ~ Π(~ u)) − (~ u h − ~ Π(~ u)), Theorem 2.1 follows from

||~ u − ~ u h || _H

1

h

(S

_d

) + ||p − p h || _L

2

(S

d

) + (h X

e∈E

_h^I

||[[(p − p h )~ n]]|| _L

2

(e∩S

d

) ) ^1/2

≤ C(||~ u|| _H

¹

h

(S

_2d

) + h||~ u|| _H

²

h

(S

_2d

) + ||p|| _L

²

h

(S

_2d

) ) +Cd ⁻¹ (||~ u|| _L

2

(S

_2d

) + ||p|| _H

−1

<

(S

_2d

) ) +Cd ⁻¹ (||~ u − ~ u h || L

²

(S

_2d

) + ||p − p h || _H

⁻¹

<

(S

_2d

) ).

By the triangle inequality this in turn follows from

||~ u h || _H

1

h

(S

_d

) + ||p h || _L

2

(S

d

) + (h X

e∈E

_h^I

||[[p h ~ n]]|| _L

2

(e∩S

d

) ) ^1/2

≤ C(||~ u|| _H

1

h

(S

_2d

) + h||~ u|| _H

2

h

(S

_2d

) + ||p|| _L

2 h

(S

_2d

) ) +Cd ⁻¹ (||~ u h || L

²

(S

_2d

) + ||p h || _H

⁻¹

<

(S

_2d

) ).

Since ω ≡ 1 on S d we have

||~ u _h || _H

1

h

(S

d

) + ||p _h || _L

2

(S

_d

) + (h X

e∈E

_h^I

||[[p _h ~ n]]|| _L

2

(e∩S

_d

) ) ^1/2

≤ ||ω~ u h || _H

1

h

(S

_d

) + ||ωp h || _L

2

(S

d

) + D h (ωp h , ωp h ).

Lemma 3.6 now follows. 

3.2.2. Step 2: Weighted Stability estimates for the Pressure. We first estimate the term ||ωp h || L

²

(S

_d

) in terms of the other terms in the right-hand side of (3.6).

Lemma 3.7. Let ω be as in Lemma 3.6, then

||ωp h || ² _L

2

(S

_d

) ≤ C||ω~ u h || ² _H

1

h

(S

d

) + CD h (ωp h , ωp h ) +C(||~ u|| ² _H

1

h

(S

_2d

) + h||~ u|| ² _H

2

h

(S

_2d

) + ||p|| ² _L

2 h

(S

_2d

) ) +Cd ⁻² (||~ u _h || ² _L

2

(S

_2d

) + ||p _h || ² _H

−1

<

(S

_2d

) ).

Proof. By the triangle inequality we have

(3.7) ||ωp h || L

²

(S

_2d

) ≤ ||ωp h − avg _S

_2d

(ωp _h )|| _L

2

(S

_2d

) + ||avg _S

2d

(ωp _h )|| _L

2

(S

_2d

)

where avg _S

_2d

(ωp h ) = _|S ¹

2d

|

R

S

_2d

ωp h dx. It is easy to show that

(3.8) ||avg _S

2d

(ωp h )|| _L

2

(S

_2d

) ≤ C

d ||p h || _H

−1

<

(S

_2d

) .

It is well known that one can find a function ~ v ∈ [H ¹ (S _2d )] ^N (see [15]) that satisfies

−∇ · ~v = ωp h − avg _S

_2d

(ωp h ), in S 2d

~ v = 0, on ∂S 2d ,

and

(3.9) ||∇~v|| L

²

(S

_2d

) ≤ C||ωp h − avg _S

_2d

(ωp _h )|| _L

2

(S

_2d

) . We define ~ v on all of Ω by defining it to be zero outside of S _2d .

By the definition of B _h , we have

||ωp h − avg _S

_2d

(ωp h )|| ² _L

2

(S

_2d

) = B h (~ v, ωp h − avg _S

_2d

(ωp h )) Using integration by parts we can rewrite B(·, ·) as

B h (~ v, q) = Z

Ω

~

v · ∇ h q dx + X

e∈E

_h^I

Z

e

[[q~ n]] · {{~ v}} ds.

Hence, B h (~ v, c) = 0 if c is a constant. Therefore,

||ωp h − avg _S

_2d

(ωp h )|| ² _L

2

(S

_2d

) = B h (~ v, ωp h )

= B h (~ v − ~ Π(~ v), ωp h ) + B h (~ Π(~ v), ωp h ).

The first term is

B _h (~ v − ~ Π(~ v), ωp _h ) = Z

Ω

(~ v − ~ Π(~ v)) · ∇ _h (ωp _h )dx

− X

e∈E

_h^I

Z

e

{{(~v − ~ Π(~ v))}} · [[ωp _h ~ n]]ds.

It easily follows using (3.1), (3.4) and (3.9) that

− X

e∈E

_h^I

Z

e

{{~v − ~ Π(~ v)}} · [[ωp h ~ n]]ds ≤ ||ωp h − avg _S

_2d

(ωp h )|| ² _L

2

(S

_2d

)

+ C

 D h (ωp h , ωp h ).

Also, we have using (3.4) and (3.9) Z

Ω

(~ v − ~ Π(~ v)) · ∇ h (ωp h ) = Z

Ω

(~ v − ~ Π(~ v)) · ∇ h (ωp h − Π(ωp h ))dx

≤ ||ωp h − avg _S

_2d

(ωp h )|| ² _L

2

(S

_2d

) + Ch ²

d ² ||p h || ² _L

2

(S

_2d

) , where we also used

||∇ h (ωp h − Π(ωp h ))|| L

²

(S

_2d

) ≤ C

d ||p h || L

²

(S

_2d

)

which follows from (3.5d).

Therefore, after applying inverse estimates we see that

||ωp h − avg _S

2d

(ωp h )|| ² _L

2

(S

_2d

) ≤ 2||ωp h − avg _S

2d

(ωp h )|| ² _L

2

(S

_2d

)

+ C

 D h (ωp h , ωp h ) + C

d ² ||p h || ² _H

−1

(S

_2d

)

+B h (~ Π(~ v), ωp h ).

A simple exercise shows that

B _h (~ Π(~ v), ωp _h ) = B(ω~ Π(~ v), p _h ) + Z

Ω

p _h ∇(ω) · ~ Π(~ v).

Clearly, Z

Ω

p h ∇(ω) · ~ Π(~ v) = Z

Ω

p h ∇(ω) · (~ Π(~ v) − ~ v)dx + Z

Ω

p h ∇(ω) · ~vdx.

Using (3.4), (3.9) and inverse estimates we get that Z

Ω

p h ∇(ω) · (~ Π(~ v) − ~ v)dx ≤ ||ωp h − avg _S

_2d

(ωp h )|| ² _L

2

(S

2d

) + C

d ² ||p h || ² _H

−1

(S

_2d

) . Also, we have

Z

Ω

p h ∇(ω) · ~vdx ≤ ||∇(ω) · ~v|| _H

1

(S

2d

) ||p h || _H

⁻¹

<

(S

2d

)

≤ 1

d ||∇~v|| _L

2

(S

_2d

) ||p h || _H

−1

<

(S

_2d

)

≤ ||ωp h − avg _S

_2d

(ωp h )|| ² _L

2

(S

_2d

) + C

d ² ||p h || ² _H

−1

(S

_2d

) . Here we used Poincare’s inequality ||~ v|| _L

2

(S

_2d

) ≤ Cd||∇~v|| _L

2

(S

_2d

) .

Therefore,

||ωp h − avg _S

_2d

(ωp h )|| ² _L

2

(S

2d

) ≤ 4||ωp h − avg _S

_2d

(ωp h )|| ² _L

2

(S

2d

)

(3.10)

+ C

d ² ||p h || ² _H

−1

<

(S

2d

)

+ C

 D h (ωp h , ωp h ) + B h (ω~ Π(~ v), p h ).

(3.11)

We are left to bound B h (ω~ Π(~ v), p h ). To this end, we have by (2.3) B h (ω~ Π(~ v), p h ) = B h (~ Π(ω~ Π(~ v)), p h ) + B h (ω~ Π(~ v) − ~ Π(ω~ Π(~ v)), p h )

= A _h (~ u, ~ Π(ω~ Π(~ v))) + B _h (~ Π(ω~ Π(~ v)), p) +

−A h (~ u _h , ~ Π(ω~ Π(~ v))) − R(~ u, ~ Π(ω~ Π(~ v))) +B h (ω~ Π(~ v) − ~ Π(ω~ Π(~ v)), p h )

=

6

X

i=1

J _i , (3.12)

where

J ₁ = A _h (~ u, ~ Π(ω~ Π(~ v))), J ₂ = B _h (~ Π(ω~ Π(~ v)), p)

J 3 = −A h (~ u h , ω~ Π(~ v) − ~ Π(ω~ Π(~ v))), J 4 = B h (ω~ Π(~ v) − ~ Π(ω~ Π(~ v)), p h )

J ₅ = A _h (~ u _h , ω~ Π(~ v)), J ₆ = −R(~ u, ~ Π(ω~ Π(~ v))).

By the continuity of A h and B h we have

|J 1 | + |J 2 | ≤ C||~ Π(ω~ Π(~ v))|| _H

1

h

(S

_2d

) (||~ u|| _H

1

h

(S

_2d

) + ||p|| _L

2 h

(S

_2d

) ).

Using the triangle inequality we get

||~ Π(ω~ Π(~ v))|| _H

1

h

(S

2d

) ≤ ||~ Π(ω~ Π(~ v)) − ω~ Π(~ v)|| _H

1 h

(S

2d

)

+||ω~ Π(~ v) − ω~ v|| _H

¹

h

(S

_2d

) + ||ω~ v|| _H

¹

h

(S

_2d

) .

It is not difficult to show using approximation properties of ~ Π, Poincare’s inequality and the fact that ||D ¹ ω|| _L

^∞

_(S

_2d

₎ ≤ Cd ⁻¹ that

||~ Π(ω~ Π(~ v)) − ω~ Π(~ v)|| _H

1

h

(S

2d

) + ||ω~ Π(~ v) − ω~ v|| _H

1

h

(S

2d

) ≤ C||∇~v|| L

²

(S

_2d

) . Moreover, using that the jumps of ~ v are zero and Poincare’s inequality we have

||ω~v|| _H

1

h

(S

2d

) = ||∇(ω~ v)|| _L

2

(S

_2d

) ≤ C||∇~v|| _L

2

(S

_2d

) . Therefore, after using (3.9) we get

||~ Π(ω~ Π(~ v))|| _H

1

h

(S

2d

) ≤ C||ωp _h − avg _S

2d

(ωp _h )|| _L

2

(S

_2d

) . Hence,

|J 1 | + |J 2 | ≤ ||ωp h − avg _S

_2d

(ωp _h )|| ² _L

2

(S

2d

) + C

 (||~ u|| ² _H

1

h

(S

_2d

) + ||p|| ² _L

2 h

(S

_2d

) ).

By the continuity of A h

|J 3 | ≤ ||u h || _H

1

h

(S

_2d

) ||ω~ Π(~ v) − ~ Π(ω~ Π(~ v))|| _H

1 h

(S

_2d

)

Using (3.4) and (3.5b) we have

||ω~ Π(~ v) − ~ Π(ω~ Π(~ v))|| _H

1

h

(S

2d

) ≤ Ch(d ⁻¹ ||∇ _h Π(~ ~ v)|| _L

2

(S

_2d

) + d ⁻² ||~ Π(~ v)|| _L

2

(S

_2d

) )

≤ Ch

d ||∇~v|| L

²

(S

_2d

) , (3.13)

where in the last inequality we used the stability of ~ Π and Poincare’s inequality.

Hence, using inverse estimates and Young’s inequality we have

|J 3 | ≤ ||ωp h − avg _S

2d

(ωp h )|| ² _L

2

(S

_2d

) + C

d ² ||~ u h || ² _L

2

(S

_2d

) .

One can easily show using the Cauchy-Schwarz inequality, (3.1) and (3.13) that

J ₄ = − X

e∈E

_h^I

Z

e

{{ω~ Π(~ v) − ~ Π(ω~ Π(~ v)}} · [[p _h ~ n]]ds

≤ ||ωp _h − avg _S

2d

(ωp _h )|| ² _L

2

(S

_2d

) + C

d ² ||p _h || ² _H

−1

(S

2d

) . To handle the next term we first use integration by parts to get

J 5 = A(ω~ u h , ~ Π(~ v)) + Z

Ω

∇(ω) ⊗ ~ u h : (∇~ Π(~ v) − L(~ Π(~ v))dx +

Z

Ω

∇ h u h : (L(ω~ Π(~ v)) − ωL(~ Π(~ v))dx +

Z

Ω

∇ h Π(~ ~ v) : (L(ω~ u h ) − ωL(~ u h ))dx.

It simple to see using the continuity of A h , properties of ~ Π and (3.9) that A(ω~ u h , ~ Π(~ v)) ≤ ||ωp h − avg _S

2d

(ωp h )|| ² _L

2

(S

_2d

)

+ C

 ||ω~ u h || ² _H

1 h

(S

_2d

) . Similarly, one has

Z

Ω

∇(ω) ⊗ ~ u h : (∇~ Π(~ v) − L(~ Π(~ v))dx ≤ ||ωp h − avg _S

_2d

(ωp h )|| ² _L

2

(S

2d

)

+ C

d ² ||~ u _h || ² _L

2

(Ω) . Using the definition of the Π and L we have

Z

Ω

∇ h u h : (L(ω~ Π(~ v)) − ωL(~ Π(~ v))dx = X

e∈E

_h

Z

e

[[~ Π(~ v) ⊗ ~ n]] : {{ω∇ h u h − Π(ω∇ h u h )}}dx.

Therefore, using Lemma (3.5d) and (3.9) we get Z

Ω

∇ h u h : (L(ω~ Π(~ v)) − ωL(~ Π(~ v))dx

≤ C||ωp h − avg _S

_2d

(ωp h )|| _L

2

(S

2d

)

h d ||u h || _H

1

h

(S

_2d

)

≤ ||ωp h − avg _S

_2d

(ωp _h )|| ² _L

2

(S

2d

) + C

d ² ||u h || ² _L

2

(S

2d

) . In a similar fashion we can show

Z

Ω

∇ h Π(~ ~ v) : (L(ω~ u h ) − ωL(~ u h ))dx

≤ ||ωp h − avg _S

_2d

(ωp h )|| ² _L

2

(S

_2d

) + C

d ² ||u h || ² _L

2

(S

_2d

) . Hence,

|J 5 | ≤ 4||ωp _h − avg _S

_2d

(ωp _h )|| ² _L

2

(S

2d

)

+ C

d ² ||u h || ² _L

2

(S

2d

) + C

 ||ω~ u _h || ² _H

1 h

(S

_2d

) . One can show using (3.1) and approximation properties ~ Π that

|J 6 | ≤ ||ωp h − avg _S

_2d

(ωp h )|| ² _L

2

(S

2d

) + Ch ²

 ||~ u|| ² _H

2 h

(S

_2d

) . Hence, combining the bounds for the J _i ⁰ s and using (3.12) we have that

B h (ω~ Π(~ v), p h ) ≤ 8||ωp h − avg _S

2d

(ωp h )|| ² _L

2

(S

_2d

)

+ C

 (||ω~ u h || ² _H

1

h

(S

2d

) + D h (ωp h , ωp h )) + C

 (||~ u|| ² _H

1

h

(S

_2d

) + ||p|| ² _L

2

h

(S

_2d

) + h ² ||~ u|| ² _H

2 h

(S

_2d

) ) + C

d ² (||u _h || ² _L

2

(S

_2d

) + ||p _h || ² _H

−1

(S

_2d

) ).

(3.14)

Finally, combining (3.14), (3.11), (3.8) and (3.7) and taking  small enough

proves Lemma 3.7. 

3.2.3. Step 3: Weighted Stability Estimates For The Velocity.

Lemma 3.8. For every 1 > δ > 0 we have

||ω~ u _h || ² _H

1

h

(S

_d

) + D _h (ωp _h , ωp _h ) ≤ δ||ωp _h || ² _L

2

(S

_d

)

+ C δ (||~ u|| ² _H

1

h

(S

2d

) + h||~ u|| ² _H

2

h

(S

2d

) + ||p|| ² _L

2 h

(S

2d

) ) + C

δd ² (||~ u _h || ² _L

2

(S

2d

) + ||p _h || ² _H

−1

(S

2d

) ).

Proof. In order to prove the result we use a stability result concerning only A _h bilinear form. The proof is almost identical to the proof of a similar inequality in [17] for the LDG method applied to Laplace’s equation; see (3.3) in [17].

Proposition 3.9. There exists a fixed number C 1 > 1 such that

||ω~ u h || ² _H

1

h

(S

_2d

) ≤ C 1 A h (~ u h , ω ² ~ u h ) + C 1 d ⁻² ||~ u h || ² _L

2

(S

_2d

) . (3.15)

Therefore, we need only to find a bound for A _h (~ u _h , ω ² ~ u _h ). To this end, we use (2.3) to write

A _h (~ u _h , ω ² ~ u _h )

= A h (~ u h , ω ² ~ u h − ~ Π(ω ² ~ u h )) + A h (u h , ~ Π(ω ² ~ u h ))

= A h (~ u h , ω ² ~ u h − ~ Π(ω ² ~ u h )) + A h (~ u, ~ Π(ω ² ~ u h )) + B h (~ Π(ω ² ~ u h ), p)

−B h (~ Π(ω ² ~ u h ), p h ) − R(~ u, ~ Π(ω ² ~ u h ))

=

6

X

j=1

I j

(3.16) where

I 1 = A h (~ u h , ω ² ~ u h − ~ Π(ω ² ~ u h )), I 2 = A h (~ u, ~ Π(ω ² ~ u h )),

I 3 = B h (~ Π(ω ² ~ u h ), p), I 4 = −B h (~ Π(ω ² ~ u h ) − ω ² ~ u h , p h ), I 5 = −B h (ω ² ~ u h , p h ), I 6 = −R(~ u, ~ Π(ω ² ~ u h )).

Here we used that (2.3) holds if Q ^k _h is replaced with ˜ Q ^k _h since B h (~ v, c) = 0 for constant c.

By the continuity of A _h and Lemma 3.4 we see that I 1 ≤ Ch||~ u h || _H

1

h

(S

_3d/2

) ( 1

d ||ω~ u h || _H

1

h

(S

_2d

) + 1

d ² ||~ u h || _L

2

(S

2d

) )

≤ δ 1 ||ω~ u h || ² _H

1

h

(S

2d

) + 1

δ 1 d ² ||~ u h || ² _L

2

(S

_2d

) . In the last step we used the inverse estimate h||~ u _h || _H

1

h

(S

_3/2d

) ≤ ||~ u _h || _L

2

(S

_2d

) . Here δ ₁ is a small positive number that will be chosen later.

By the continuity of A h , the stability of ~ Π and Young’s inequality we have I 2 ≤ δ 1 ||ω~ u h || ² _H

1

h

(S

2d

) + C δ ₁ ||~ u|| ² _H

1

h

(S

2d

) + C

δ ₁ d ² ||~ u h || ² _L

2

(S

_2d

) .

By applying H¨ older’s inequality, the stability of ~ Π and Lemma 3.4 we can easily show that

I 3 ≤ δ 1 ||ω~ u h || ² _H

1

h

(S

2d

) + C δ 1

||p|| ² _L

2

h

(S

2d

) + C

δ 1 d ² ||~ u h || ² _L

2

(S

_2d

) . The following can be written as

I 4 = − X

e∈E

_h^I

Z

e

{{~ Π(ω ² ~ u h ) − ω ² ~ u h }} · [[p h ~ n]].

By applying the Cauchy-Schwarz inequality, (3.1) and (3.2) I ₄ ≤ C

h ||p h || L

²

(S

_2d

) (||~ Π(ω ² ~ u _h ) − ω ² ~ u _h || L

²

(S

_2d

) + h||∇ _h (~ Π(ω ² ~ u _h ) − ω ² ~ u _h )|| _L

2

(S

_2d

) ).

By applying Lemma 3.4, inverse estimates and Young’s inequality we get I ₄ ≤ δ ₁ ||ω~ u _h || ² _H

1

(S

_2d

) + Cd ⁻² ||~ u _h || ² _L

2

(S

_2d

)

+C(1 + 1 δ 1

)d ⁻² ||p h || ² _H

−1

(S

_2d

) . Using the product rule we get

I 5 = −B h (~ u h , ω ² p h ) − 2 Z

Ω

ωp h ~ u h · ∇(ω) dx

= −B h (~ u h , ω ² p h − Π(ω ² p h )) − B h (~ u h , Π(ω ² p h ))

−2 Z

Ω

ωp _h ~ u _h · ∇(ω) dx.

By using Lemma 3.5 and inverse estimates we get

−B _h (~ u _h , ω ² p _h − Π(ω ² p _h )) ≤ δ ₁ ||ωp _h || ² _L

2

(S

_2d

) + C

d ² ||p _h || ² _H

−1

(S

2d

)

+ C d ² (1 + 1

δ 1

)||~ u h || ² _L

2

(S

_2d

) . It easily follows that

Z

Ω

ωp h ~ u h · ∇(ω) dx ≤ δ 1 ||ωp h || ² _L

2

(S

_2d

) + C

δ 1 d ² ||~ u h || ² _L

2

(S

_2d

) . Using (2.3) we have

−B h (~ u h , Π(ω ² p h )) = B h (~ u, Π(ω ² p h )) − D h (p h − p, Π(ω ² p h ))

= −B h (~ u, Π(ω ² p h )) − D h (p h − p, Π(ω ² p h ))

= −B _h (~ u, Π(ω ² p _h )) − D _h (p _h , ω ² p _h )

−D h (p h , Π(ω ² p h ) − ω ² p h ) + D h (p, Π(ω ² p h )).

By using inverse estimates and stability of the L ² projection Π, we have

−B h (~ u, Π(ω ² p _h )) ≤ δ ₁ ||ωp h || ² _L

2

(S

2d

) + C δ 1

||~ u _h || ² _H

1 h

(S

_2d

) , and

D h (p, Π(ω ² p h )) ≤ δ 1 ||ωp h || ² _L

2

(S

2d

) + C h δ 1

X

e∈E

_h^I

||[[p~ n]]|| ² _L

2

(e∩S

2d

) .

If we use Lemma 3.5 and inverse estimates we get

−D h (p h , Π(ω ² p h ) − ω ² p h ) ≤ δ 1 ||ωp h || ² _L

2

(S

2d

) + C

d ² (1 + δ ₁ ⁻¹ )||p h || _H

⁻¹

_(S

_2d

₎ . Hence,

I 5 ≤ −D h (ωp h , ωp h ) + 4δ 1 ||ωp h || ² _L

2

(S

_2d

)

+ C δ 1

(||p|| ² _L

2

h

(S

2d

) + ||~ u|| ² _H

1 h

(S

2d

) ) + C

δ 1 d ² (||p h || ² _H

−1

(S

_2d

) + ||~ u h || ² _L

2

(S

2d

) ), where we used that δ 1 < 1.

Finally, by applying Lemma 3.4 we can show I ₆ ≤ δ ₁ ||ω~ u _h || ² _H

1

h

(Ω)

+ C

d ² ||~ u h || ² _L

2

(S

_2d

) + Ch ² (1 + δ ₁ ⁻¹ )||~ u|| ² _H

2 h

(S

_2d

) .

Therefore, by combining (3.15), (3.16), the bounds for I j , j = 1, · · · , 6 and choosing δ 1 so that δ 1 C 1 ≤ 1/2 proves Lemma 3.8 where we let δ = 20 δ 1 C 1 .  3.2.4. Step 4: Completion of the proof. By combining Lemmas 3.8 and 3.7 and taking δ sufficiently small we get

||ω~ u h || ² _H

1

h

(S

d

) + D h (ωp h , ωp h )

≤ C(||~ u|| ² _H

1

h

(S

2d

) + h||~ u|| ² _H

2

h

(S

2d

) + ||p|| ² _L

2 h

(S

2d

) ) + C

d ² (||~ u h || ² _L

2

(S

2d

) + ||p h || ² _H

−1

<

(S

2d

) ).

Finally, by combining this inequality with Lemma 3.7 gives us (3.6) and hence completes the proof of Theorem 2.1.

3.3. Proof of Theorem 2.2.

3.3.1. Step 1: Reduce To Error Estimates For Approximate Greens Function.

Lemma 3.10. Let x ∈ T _x where T _x ∈ T h and let ~ ρ ∈ [C _c ^∞ (T _x )] ^N with ||~ ρ|| _L

2

(T

_x

) = h ^−N/2 . Then, Theorem 2.2 follows from

||~g − ~g h || _W

^1,1

h

(Ω),x,−s + ||λ − λ h || L

¹

(Ω),x,−s ≤ log(1/h) ^{¯ s} h, where (~ g, λ) (with R

Ω λ dx = 0) solve

−4~g + ∇λ = ~ ρ

∇ · ~g = 0

~ g = 0 (3.17)

and (~ g _h , λ _h ) ∈ ~ V _h ^k × Q ^k _h satisfy

A h (~ g h , ~ v) + B h (~ v, λ h ) = Z

Ω

~ ρ · ~ vdx

−B h (~ g h , q) + D h (λ h , q) = 0 ∀(~v, q) ∈ ~ V _h ^k × Q ^k _h .

(3.18)

Proof. By the triangle inequality and inverse estimates

|(~ u − ~ u h )(x)| ≤ |(~ u − ~ Π(~ u))(x)| + Ch ^−N/2 ||~ Π(~ u) − ~ u h || _L

2

(T

x

) . By the triangle inequality and H¨ older’s inequality, we have

|(~ u − ~ u _h )(x)| ≤ C||~ u − ~ Π(~ u)|| _L

∞

(T

_x

) + Ch ^−N/2 ||~ u − ~ u _h || _L

2

(T

_x

) . Using the fact that 1/2 ≤ σ _x (y) for any y ∈ T _x , we have

|(~ u − ~ u h )(x)| ≤ Ch||~ u|| _W

1,∞

h

(Ω),x,s + Ch ^−N/2 ||u − u h || _L

2

(T

x

) . Since ~ u − ~ u h = ~ u − ~ Π(~ u) − (~ u h − ~ Π(~ u)), we easily see that

|(~ u − ~ u h )(x)| ≤ Ch||(~ u − ~ Π(~ u))|| _W

^1,∞

h

(Ω),x,s + Ch ^−N/2 ||~ u − ~ u h || L

₂

(T

_x

) . We will use that

h ^−N/2 ||~ u − ~ u _h || _L

2

(T

_x

) = sup

ρ∈C∞c (Tx )

||~ρ||

L2 (Tx)=h−N/2

Z

Ω

(~ u − ~ u _h ) · ~ ρ dx.

For a fix ~ ρ let (~ g, λ) and (~ g h , λ h ) be the solutions of (3.17) and (3.18), respectively.

By using the consistency result for the LDG method and (2.3) we have Z

Ω

~

ρ · (~ u − ~ u h )dx = A h (~ u − ~ u h , g) + B h (~ u − ~ u h , λ) − R(~ g, ~ u − ~ u h )

= A _h (~ u − ~ u _h , ~ g − ~ g _h ) − B(~ g _h , p − p _h ) + R(~ u, ~ g _h )

+B h (~ u − ~ u h , λ − λ h ) + D h (λ h , p − p h ) − R(~ g, ~ u − ~ u h )

= A _h (~ u − ~ u _h , ~ g − ~ g _h ) + B _h (~ u − ~ u _h , λ − λ _h ) − R(~ g, ~ u − ~ u _h ) +B(~ g − ~ g h , p − p h ) − D h (λ − λ h , p − p h ) + R(~ u, ~ g h )

= A h (~ u − ~ Π(~ u), ~ g − ~ g h ) + B h (~ u − ~ Π(~ u), λ − λ h )

−R(~g, ~ u − ~ Π(~ u)) + B(~ g − ~ g h , p − Π(p))

−D h (λ − λ _h , p − Π(p)) + R(~ u, ~ g _h ).

Hence, by the continuity of our bilinear forms and the definition of R(·, ·) we have

Z

Ω

~

ρ · (~ u − ~ u h )dx

≤ C(||~ u − ~ Π(~ u)|| _W

1,∞

h

(Ω),x,s + ||σ ^s _x (∇ h ~ u − Π(∇ h ~ u))|| _L

^∞

_(Ω) +||p − Π(p)|| _L

∞

(Ω),x,s ) ×

(||~ g − ~ g _h || _W

1,1

h

(Ω),x,−s + h X

e∈E

_h

||σ ^−s _x {{∇~g − Π(∇~g)}}|| _L

1

(e) + ||λ − λ _h || _L

1

(Ω),x,−s ).

Here we also used that R(~ u, ~ g _h ) = R(~ u, ~ g _h − ~g) since the jumps of ~g are zero.

Theorem 2.2 will follow if we can show

||~g − ~g h || _W

^1,1

h

(Ω),x,−s + h X

e∈E

_h

||σ _x ^−s {{∇~g − Π(∇~g)}}|| L

¹

(e) + ||λ − λ _h || L

¹

(Ω),x,−s

≤ C log(1/h) ^{s ¯} h.

By using the approximation properties of Π, global regularity bounds and Propo- sition 3.14 one can show

(3.19) h X

e∈E

_h

||σ _x ^−s {{∇~g − Π(∇~g)}}|| _L

1

(e) ≤ C log(1/h) ^{s ¯} h.

We leave the details to the reader. This completes the proof of Lemma 3.10.  3.3.2. Step 2: Dyadic Decomposition And Error Estimates For Approximate Greens Functions.

Lemma 3.11. Let (~ g, λ) and (~ g _h , λ _h ) be as in Lemma 3.10

||~g − ~g h || _W

^1,1

h

(Ω),x,−s + ||λ − λ h || L

¹

(Ω),x,−s ≤ log(1/h) ^{¯ s} h, Proof. Let

d j = 2 ^−j for j = 0, 1, 2, . . . and set

Ω j = {y ∈ Ω : d j+1 < |y − x| < d j }, Ω ⁽¹⁾ _j = {y ∈ Ω : d j+2 < |y − x| < d j−1 }, Ω ⁽²⁾ _j = {y ∈ Ω : d j+3 < |y − x| < d j−2 }, Ω ⁽³⁾ _j = {y ∈ Ω : d j+4 < |y − x| < d j−3 }, Ω ⁽⁴⁾ _j = {y ∈ Ω : d j+5 < |y − x| < d _j−4 }.

We now state two important lemmas that we need. The proofs can be found in the next subsection.

Lemma 3.12. If d _j > 8h, then

||~g − ~g h || _H

1

h

(Ω

_j

) + ||λ − λ h || _L

2

(Ω

j

)

≤ Ch ^k d ^1−k−N/2 _j + d ⁻¹ _j (||~ g − ~ g _h || _L

₂

_(Ω

(1)

j

) + C||λ − λ _h || _H

−1

<

(Ω

⁽¹⁾_j

) ).

Lemma 3.13. If d _j > 8h, then

||~g − ~g h || _L

₂

_(Ω

(1)

j

) + ||λ − λ h || _H

−1

<

(Ω

⁽¹⁾_j

)

≤ Ch ^k d ^1−k−N/2 _j (||~ g − ~ g _h || _W

1,1

h

(Ω) + ||λ − λ _h || _L

1 h

(Ω) ) +Ch(||~ g − ~ g _h || _H

₁

h

(Ω

⁽⁴⁾_j

) + ||λ − λ _h || _L

₂

h

(Ω

⁽⁴⁾_j

) ) + C log( 1

h ) ^{¯ s} h ^k+1 d ^1−N/2−k _j . Let M be a real number to be determined later and let J be an integer such that d _J = M h. Set ~ E = ~ g − ~ g _h and r = λ − λ _h . Notice that

|| ~ E|| _W

1,1

h

(Ω),x,−s + ||r|| _L

1

h

(Ω),x,−s ≤ || ~ E|| _W

1,1

h

(S

M h

),x,−s + ||r|| _L

1

h

(S

_{M h}

),x,−s

+

J

X

j=0

(|| ~ E|| _W

1,1

h

(Ω

_j

),x,−s + ||r|| _L

1

h

(Ω

j

),x,−s ).

Without loss of generality we have assumed that diam(Ω) ≤ 1. Since σ ^−s _x (z) ≤ ^Cd

s j

h

^s

for z ∈ Ω _j , using the fact that meas(Ω _j ) ≤ Cd ^N _j and applying H¨ older’s inequality

we can show

|| ~ E|| _W

1,1

h

(Ω

_j

),x,−s + ||r|| _L

1

h

(Ω

_j

),x,−s ≤ Cd ^N/2+s _j h ^−s (||E|| _H

1

h

(Ω

_j

) + ||r|| _L

2 h

(Ω

_j

) ).

One also has

|| ~ E|| _W

1,1

h

(S

_{M h}

),x,−s + ||r|| _L

1

h

(S

M h

) ≤ CM ^N/2+s h ^N/2 (|| ~ E|| _H

1

h

(S

M h

) + ||r|| _L

2 h

(S

M h

) )

≤ CM ^N/2+s h ^N/2+1 (||~ g|| H

²

(Ω) + ||λ|| H

¹

(Ω) )

≤ CM ^N/2+s hh ^N/2 ||~ ρ|| _L

2

(Ω) ≤ ChM ^N/2+s . Here we used global error estimates and regularity results.

Therefore, we have (3.20) || ~ E|| _W

1,1

h

(Ω),x,−s + ||r|| _L

1

h

(Ω),x,−s ≤ CM ^N/2+s h + Cη, where

η =

J

X

j=0

d ^N/2+s _j h ^−s (|| ~ E|| _H

¹

h

(Ω

_j

) + ||r|| _L

²

h

(Ω) ).

If we apply Lemma 3.12, we get η ≤ ChΘ(k − 1 − s) + C

J

X

j=0

d ^N/2−1+s _j h ^−s (|| ~ E|| _L

₂

_(Ω

(1)

j

) + ||r|| _H

−1

<

(Ω

⁽¹⁾_j

) ), where

Θ(α) =

J

X

j=0

(h/d j ) ^α . Now applying Lemma 3.13, we have

J

X

j=0

d ^N/2−1+s _j h ^−s (|| ~ E|| _L

2

(Ω

⁽¹⁾_j

) + ||r|| _H

−1

<

(Ω

⁽¹⁾_j

) )

≤ CΘ(k − s)(|| ~ E|| _W

1,1

h

(Ω) + ||r|| _L

1 h

(Ω) ) +C

J

X

j=0

d ^N/2−1+s _j h ^1−s (|| ~ E|| _H

1

h

(Ω

⁽⁴⁾_j

) + ||r|| _L

2 h

(Ω

⁽⁴⁾_j

) )

≤ CΘ(k − s)(|| ~ E|| _W

1,1

h

(Ω) + ||r|| _L

1 h

(Ω) ) +Cd ^N/2−1+s _J h ^1−s (|| ~ E|| _H

1

h

(S

_{M h}

) + ||r|| _L

2

h

(S

_{M h}

) ) + C M η

≤ CΘ(k − s)(|| ~ E|| _W

1,1

h

(Ω) + ||r|| _L

1 h

(Ω) ) +ChM ^N/2−1+s + C

M η.

Therefore,

η ≤ ChΘ(k − 1 − s) + CΘ(k − s)(|| ~ E|| _W

1,1

h

(Ω) + ||r|| _L

1 h

(Ω) ) +M ^N/2−1+s h + C

M η.

By Choosing M sufficiently large we have η ≤ ChΘ(k − 1 − s) + CΘ(k − s)(|| ~ E|| _W

1,1

h

(Ω) + ||r|| _L

1

h

(Ω) ) + M ^N/2−1+s h.

Substituting this bound into (3.20), we have

|| ~ E|| _W

1,1

h

(Ω),x,−s + ||r|| _L

1

h

(Ω),x,−s ≤ CM ^N/2+s h + ChΘ(k − 1 − s) +CΘ(k − s)(|| ~ E|| _W

1,1

h

(Ω) + ||r|| _L

1 h

(Ω) ).

(3.21)

In particular, we see that

|| ~ E|| _W

1,1

h

(Ω) + ||r|| _L

1

h

(Ω) ≤ CM ^N/2+s h + ChΘ(k − 1) +CΘ(k − 1)(||E||| _W

1,1

h

(Ω) + ||r|| _L

1 h

(Ω) ).

Since J ≤ log( ¹ _h ), we obtain

Θ(α) ≤ C (log( ¹ _h ) if α = 0 M ^{−α 1−(1/2)} _1−(1/2)

^{α log( 1}α^h)

if α > 0.

Hence, by choosing M large enough so that CΘ(k) ≤ 1/2, we get

|| ~ E|| _W

1,1

h

(Ω) + ||r|| _L

1

h

(Ω) ≤ CM ^N/2+s h + ChΘ(k − 1).

By this inequality and (3.21), we have

|| ~ E|| _W

1,1

h

(Ω),x,−s + ||r|| _L

1 h

(Ω),x,−s

≤ CM ^N/2+s h + ChΘ(k − 1 − s) + ChΘ(k − s)Θ(k − 1)

≤ Ch log( 1 h ) ^{s ¯} .

which proves Lemma 3.11 

3.3.3. Step 3: Proof of Lemmas 3.12 and 3.13. In order to complete the proof of Theorem 2.2 it remains to prove Lemmas 3.12 and 3.13.

We first prove Lemma 3.12

Proof. From Theorem 2.1 and approximation properties we get

||~g − ~g _h || _H

1

h

(Ω

j

) + ||λ − λ _h || _L

2

(Ω

_j

)

≤ Ch ^k (|~ g| _H

_k+1

_(Ω

(1)

j

) + |λ| _H

_k

_(Ω

(1) j

) ) +Cd ⁻¹ _j (||~ g − ~ g _h || _L

₂

_(Ω

¹

j

) + ||λ − λ _h || _H

⁻¹

<

(Ω

¹_j

) ).

We need only to approximate |~ g| _H

_k+1

_(Ω

(2)

j

) + |λ| _H

_k

_(Ω

(2)

j

) . In order to do so we will use a Greens’s function representation of ~ g and λ. The result is contained in ([22], Theorem 1.1).

Proposition 3.14. Let (~ v, q) (with R

Ω q dx = 0)solve

−4~v + ∇q = ~ m in Ω,

∇ · ~v = r in Ω,

~ v = 0 on ∂Ω.

with ~ m ∈ [L ² (Ω)] ^N and r ∈ H ¹ (Ω) with R

Ω rdx = 0. Then, the pair (~ v, q) have the following representations

~ v(x) = Z

Ω

(G(x, y) ~ m(y) + ~ Ψ(x, y)r(y) + Γ(x, y)∇r(y))dy