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The Weak Galerkin (WG) finite element method for the unsteady Stokes equations in the primary velocity-pressure formulation is introduced in this paper. Optimal-order error estimates are established for the corresponding numerical approximation in an
H^{1}
norm for the velocity, and
L^{2}
norm for both the velocity and the pressure by us
e of
the Stokes projection.

The finite element method for the unsteady Stokes equations developed over the last several decades is based on the weak formulation by constructing a pair of finite element spaces satisfying the inf-sup condition of Babuska [

In this paper, we study the initial-boundary value problems of the Stokes.

{ u t − Δ u + ∇ p = f ( x , t ) , t ∈ ( 0, T ) ∇ ⋅ u = 0, ( x , t ) ∈ Ω × ( 0, T ) u = 0, ( x , t ) ∈ ∂ Ω × ( 0, T ) u ( x ,0 ) = ψ ( x ) x ∈ Ω , t = 0 (1)

where u = ( u 1 , u 2 ) T is fluid velocity, p is pressure, f = ( f 1 , f 2 ) T is volumetric power density.

The solution of the Stokes equations forms an important aspect of both theoretical and computational fluid dynamics. A limited number of solutions of these non-linear partial differential equations mostly involving spatially one-dimensional problems are given in the literature. Solutions of practical interest have been obtained for cases where, with suitable approximations, the equations are reduced to linear partial differential equations.

Let W be a bounded domain in R^{2}. We introduce function spaces

X = [ H 0 1 ( Ω ) ] 2 , V = { u ∈ X , d i v u = 0 } , M = { q ∈ L 2 ( Ω ) ; ∫ Ω q d x d y = 0 } , then the unsteady Stokes problem would take the following form: seek ( u , p ) ∈ X × M satisfying

{ ( u t , v ) + ( ∇ u , ∇ v ) − ( p , ∇ ⋅ v ) = ( f , v ) , ∀ v ∈ X ( q , ∇ ⋅ u ) = 0 , ∀ q ∈ W u ( x , 0 ) = ψ ( x ) (2)

We use ‖ ⋅ ‖ s , D and | ⋅ | s , D to be denote the norm and Semi-norm in the Sobolev space H s ( D ) for any s ≥ 0 , respectively. The inner product in H s ( D ) is denoted by ( ⋅ , ⋅ ) s , D . For example, for each s ≥ 0 , the Semi-norm | ⋅ | s , D is given by

| φ | s , D = ( ∑ | γ | = s ∫ D | ∂ γ φ | 2 d D ) 1 2

and ‖ ⋅ ‖ is said to be the norm of L 2 .

For w is [ 0, T ] to H s ( D ) , the definition is given by

‖ w ‖ L q ( 0 , T ; H s ( D ) ) = ( ∫ 0 T ‖ w ( ⋅ , t ) ‖ s , D q d t ) 1 q

for 1 ≤ q ≤ ∞ , we have

‖ w ‖ L ∞ ( 0 , T ; H s ( D ) ) = s u p 0 ≤ t ≤ T ‖ w ( ⋅ , t ) ‖ s , D

The space H s ( D ) and the norm defined in the H s ( D ) defined as

H ( d i v , Ω ) = { q : q ∈ [ L 2 ( Ω ) ] , ∇ ⋅ q ∈ L 2 ( Ω ) }

‖ q ‖ H ( d i v , Ω ) = ( ‖ q ‖ 2 + ‖ ∇ ⋅ q ‖ 2 ) 1 2

Let K be any polygonal or polyhedral domain with boundary ∂ K . A weak vector-valued function on the region K refers to a vector-valued function v = { v 0 , v b } such that v 0 ∈ [ L 2 ( K ) ] d and v b ∈ [ H 1 2 ( ∂ K ) ] d . The first component v 0 can be understood as the value of v in K, and the second component v b represents v on the boundary of K. Note that v b may not necessarily be related to the trace of v 0 on ∂ K should a trace be well-defined. Denote by ν ( K ) the space of weak functions on K;

ν ( K ) = { v = { v 0 , v b } : v 0 ∈ [ L 2 ( K ) ] d , v b ∈ [ H 1 2 ( ∂ K ) ] d } (3)

Definition 1. For any v ∈ ν ( K ) , the weak gradient of v is defined as a linear functional ∇ w v in the dual space of [ H ( d i v , K ) ] d , whose action on each q ∈ [ H ( d i v , K ) ] d is given by

( ∇ w v , q ) K = − ( v 0 , ∇ ⋅ q ) K + 〈 v b , q ⋅ n 〉 ∂ K (4)

where n is the outward normal direction to ∂ K , ( v 0 , ∇ ⋅ q ) K = ∫ K v 0 ( ∇ ⋅ q ) d K is the action of v 0 on ∇ ⋅ q , and 〈 v b , q ⋅ n 〉 ∂ K = ∫ ∂ K v b q ⋅ n d s is the action of q ⋅ n on v b ∈ [ H 1 2 ( ∂ K ) ] d .

The Sobolev space [ H 1 ( K ) ] d can be embedded into the space ν ( K ) by an inclusion map i ν : [ H 1 ( K ) ] d → ν ( K ) defined as follows

i ν ( ϕ ) = { ϕ | K , ϕ | ∂ K } , ϕ ∈ [ H 1 ( K ) ] d

With the help of the inclusion map i ν , the Sobolev space [ H 1 ( K ) ] d can be viewed as a subspace of ν ( K ) by identifying each ϕ ∈ [ H 1 ( K ) ] d with i ν ( ϕ ) .

Let P r ( K ) be the set of polynomials on K with degree no more than r.

Definition 2. The discrete weak gradient operator, denoted by ∇ w , r , K , is defined as the unique polynomial ( ∇ w , r , K v ) ∈ [ P r ( K ) ] d × d satisfying the following equation,

( ∇ w , r , K v , q ) K = − ( v 0 , ∇ ⋅ q ) K + 〈 v b , q ⋅ n 〉 ∂ K , (5)

for all q ∈ [ P r ( K ) ] d × d .

In what follows, we give the definition of weak divergence, first of all, we require weak function v = { v 0 , v b } such that v 0 ∈ [ L 2 ( K ) ] d an v b ⋅ n ∈ L 2 ( ∂ K ) Denote by V ( K ) the space of weak vector-valued functions on K;

V ( K ) = { v = { v 0 , v b } : v 0 ∈ [ L 2 ( K ) ] d , v b ⋅ n ∈ L 2 ( ∂ K ) } (6)

Definition 3. For any v ∈ V ( K ) , the weak divergence of v is defined as a linear functional ∇ w ⋅ v in the dual space of H 1 ( K ) whose action on each φ ∈ H 1 ( K ) is given by

( ∇ w ⋅ v , φ ) K = − ( v 0 , ∇ φ ) K + 〈 v b ⋅ n , φ 〉 ∂ K (7)

where n is the outward normal direction to ∂ K , ( v 0 , ∇ φ ) K is the action of v 0

on ∇ φ , and 〈 v b ⋅ n , φ 〉 ∂ K is the action of v b ⋅ n on φ ∈ H 1 2 ( ∂ K ) .

The Sobolev space [ H 1 ( K ) ] d can be embedded into the space V ( K ) by an inclusion map i v : [ H 1 ( K ) ] d → V ( K ) defined as follows

i ν ( ϕ ) = { ϕ | K , ϕ | ∂ K } , ϕ ∈ [ H 1 ( K ) ] d

Definition 4. A discrete weak divergence operator, denoted by ∇ w , r , K , is defined as the unique polynomial ( ∇ w , r , K ⋅ v ) ∈ P r ( K ) that satisfies the following equation.

( ∇ w , r , K ⋅ v , φ ) K = − ( v 0 , ∇ φ ) K + 〈 v b ⋅ n , φ 〉 ∂ K , (8)

for all φ ∈ P r ( K ) .

Let T h be a partition of the domain W with mesh size h that consists of arbitrary polygons/polyhedra. In this paper, we assume that the partition T h is WG shape regular-defined by a set of conditions as detailed in references. Denote by ε h the set of all edges/flat faces in T h , and let ε h 0 = ε h ∂ Ω be the set of all interior edges/faces. For any integer k ≥ 1 , we define a weak Galerkin finite element space for the velocity variable as follows,

V h = { v = { v 0 , v b } : { v 0 , v b } | T ∈ [ P k ( T ) ] d × [ P k − 1 ( e ) ] d , e ⊂ ∂ T }

We would like to emphasize that there is only a single value v b defined on each edge e ∈ ε h . For the pressure variable, we have the following finite element space

W h = { q : q ∈ L 0 2 ( Ω ) , q | T ∈ P k − 1 ( T ) }

Denote by V h 0 the subspace of V h consisting of discrete weak functions with vanishing boundary value;

V h 0 = { v = { v 0 , v b } ∈ V h , v b = 0 on ∂ Ω }

The discrete weak gradient ∇ w , k − 1 and the discrete weak divergence ( ∇ w , k − 1 ) on the finite element space V h can be computed by using (5) and (8) on each element T, respectively. More precisely, they are given by

( ∇ w , k − 1 v ) | T = ∇ w , k − 1 , T ( v | T ) , ∀ v ∈ V h

( ∇ w , k − 1 ⋅ v ) | T = ∇ w , k − 1, T ⋅ ( v | T ) , ∀ v ∈ V h

For simplicity of notation, from now on we shall drop the subscript k − 1 in the notation ∇ w , k − 1 and ( ∇ w , k − 1 ) for the discrete weak gradient and the discrete weak divergence. The usual L 2 inner product can be written locally on each element as follows

( ∇ w v , ∇ w w ) = ∑ T ∈ T h ( ∇ w v , ∇ w w ) T

( ∇ w ⋅ v , q ) = ∑ T ∈ T h ( ∇ w ⋅ v , q ) T

Denote by Q 0 the L^{2} projection operator from [ L 2 ( T ) ] d onto [ P k ( T ) ] d . For each edge/face e ∈ ε h , denote by Q b the L^{2} projection from [ L 2 ( e ) ] d onto [ P k − 1 ( e ) ] d . We shall combine Q 0 with Q b by writing Q h = { Q 0 Q b } .

We are now in a position to describe a weak Galerkin finite element scheme for the Stokes Equations (1). To this end, we first introduce three bilinear forms as follows

s ( v , w ) = ∑ T ∈ T h h T − 1 〈 Q b v 0 − v b , Q b w 0 − w b 〉 ∂ T

a ( v , w ) = ( ∇ w v , ∇ w w ) + s ( v , w )

b ( v , q ) = ( ∇ w ⋅ v , q )

WG Algorithm. Seek ( u h = { u 0 , u b } , p h ) ∈ V h × W h satisfying

{ ( u h , t , v ) + a ( u h , v ) − b ( v , p h ) = ( f , v ) , ∀ v ∈ V h 0 b ( u h , q ) = 0, ∀ q ∈ W h u h ( x ,0 ) = ψ 0 ( x ) (9)

In the following, the proof process of Lemma 1-6 refers to reference [

Lemma 1. For any v ∈ V h 0 , the following equation hold true,

‖ | v | ‖ 2 = ∑ T ∈ T h ‖ ∇ w v ‖ T 2 + ∑ T ∈ T h h T − 1 ‖ Q b v 0 − v b ‖ ∂ T 2

Lemma 2. For any v , w ∈ V h 0 we have

| a ( v , w ) | ≤ ‖ | v | ‖ ‖ | w | ‖

a ( v , v ) = ‖ | v | ‖ 2 .

In addition to the projection Q h = { Q 0 , Q b } defined in the previous section, let Q h and Q h be two local L^{2} projections onto P k − 1 ( T ) and [ P k − 1 ( T ) ] d × d , respectively.

Lemma 3. The projection operators Q h , R h , and S h satisfy the following commutative properties

∇ w ( Q h v ) = R h ( ∇ v ) , ∀ v ∈ [ H 1 ( Ω ) ] d

∇ w ⋅ ( Q h v ) = S h ( ∇ ⋅ v ) , ∀ v ∈ H ( d i v Ω )

Lemma 4. There exists a positive constant b independent of h such that

s u p v ∈ V h 0 b ( v , ρ ) ‖ | v | ‖ ≥ β ‖ ρ ‖

for all ρ ∈ W h .

Lemma 5. Poincare inequality of Weak gradient operator: If v ∈ V h 0 , then exists a constant c satisfying

‖ v ‖ 2 ≤ c ‖ ∇ w v ‖ 2 ≤ c ‖ | v | ‖ 2

First of all, we study the existence and uniqueness of the solution for (9). The space defined as follows

T h = { v h ∈ V h ( j ) ; ( q , ∇ w ⋅ v ) = 0, ∀ q ∈ W h } .

Then we need to seek u h ( x , t ) : ( 0, T ) → T h satisfying

{ ( u h , t , v ) + a ( u h , v ) = ( f , v ) , v ∈ T h u h ( x , 0 ) = ψ 0 ( x ) (10)

Let u h ( x , t ) be the solution of (10) and which is unique, the linear bounded functional l = l ( u h ) on V h defined as follows.

〈 l , v 〉 = ( u h , t , v ) + a ( u h , v ) − ( f , v ) (11)

Then problem (9) is equivalent to seek P h ∈ W h satisfying

( P h , ∇ w ⋅ v ) = 〈 l , v 〉 , ∀ v ∈ V h (12)

Using LBB condition and Lax-Milgram Lemma, we know that the solution P h ∈ W h of (12) is unique.

Combing (11) and (12), it is concluded that if initial approximation u h ( x , 0 ) = ψ 0 ( x ) ∈ T h , the solution ( u h , P h ) ∈ V h × M h of (9) is unique.

In what follows, we introduce Stokes projection, which is the important approximation of projection.

Lemma 6. First of all, we introduce Stokes projection of ( u , p ) ∈ X × W , which is ( Q h 1 u , S h 1 p ) ∈ V h × W h need satisfying

{ a ( u h − Q h 1 u , v ) − b ( v , p h − S h 1 p ) = − φ u , p ( v ) , v ∈ V h b ( u h − Q h 1 u , q ) = 0 , q ∈ W h (13)

If let f ∗ = − Δ u + Δ p , easy to know that ( Q h 1 u , S h 1 p ) ∈ V h × W h satisfying

{ a ( Q h 1 u , v ) − b ( v , S h 1 p ) = ( f ∗ , v 0 ) , v ∈ V h b ( Q h 1 u , q ) = 0, q ∈ W h (14)

Then ( Q h 1 u , S h 1 p ) ∈ V h × W h is the finite element approximation of ( u , p ) ∈ X × W , so we have

{ ‖ u h − Q h 1 ‖ + h ‖ | u h − Q h 1 | ‖ ≤ C h k + 1 ( ‖ u ‖ k + 1 + ‖ p ‖ k ) ‖ p h − S h 1 p ‖ ≤ C h k ( ‖ u ‖ k + 1 + ‖ p ‖ k ) ‖ u h − Q h 1 ‖ + h ‖ | ( u h − Q h 1 ) t | ‖ ≤ C h k + 1 ( ‖ u ‖ k + 1 + ‖ p ‖ k + ‖ u t ‖ k + 1 + ‖ p t ‖ k ) (15)

In what follows, we list Lemma 7 to prove the error estimation of approximate solution for Semi-discrete scheme.

We know that ( u , p ) ∈ X × M and ( u h , p h ) ∈ X h × M h be solution of (1) and Galerkin finite element solution of (9), respectively. The L^{2} projection of u in the finite element space V h is given by Q h u = { Q 0 u , Q b u } . Similarly, the pressure p is projected into W h as S h p . Denote by e h and ε h the corresponding error given by

{ e h = { e 0 , e b } = { Q 0 u − u 0 , Q b u − u b } ε h = S h p − p h (16)

Lemma 7. Let ( w , p ) ∈ [ H 1 ( Ω ) ] d × L 2 ( Ω ) be sufficiently smooth and satisfy the following equation

w t − Δ w + ∇ ρ = η (17)

in the domain W. Let Q h w = { Q 0 w , Q b w } and S h ρ be the L^{2} projection of ( w , p ) into the finite element space V h × W h . Then, the following equation holds true

( Q h w t , v 0 ) + ( ∇ w ( Q h w ) , ∇ w v ) − ( ∇ w ⋅ v , S h ρ ) = ( η , v 0 ) + l w ( v ) − θ ρ ( v ) , (18)

for all v ∈ V h 0 . Where l w and θ ρ are two linear functionals on V h 0 defined by

l w ( v ) = ∑ T ∈ T h 〈 v 0 − v b , ∇ w ⋅ n − R h ( ∇ w ) ⋅ n 〉 ∂ T

θ ρ ( v ) = ∑ T ∈ T h 〈 v 0 − v b , ( ρ − S h ρ ) n 〉 ∂ T

Proof. Together Lemma 3, Equation (5) and integration by parts. we obtain

( ∇ w ( Q h w ) , ∇ w v ) T = ( R h ( ∇ w ) , ∇ w v ) T = − ( v 0 , ∇ ⋅ R h ( ∇ w ) ) T + 〈 v b , ∇ ⋅ R h ( ∇ w ) ⋅ n 〉 ∂ T = ( ∇ v 0 , R h ( ∇ w ) ) T − 〈 v 0 − v b , ∇ ⋅ R h ( ∇ w ) ⋅ n 〉 ∂ T = ( ∇ w , ∇ v ) T − 〈 v 0 − v b , ∇ ⋅ R h ( ∇ w ) ⋅ n 〉 ∂ T (19)

Next, Combing Lemma 3 and Equation (8), the fact that ∑ T ∈ T h 〈 v b , p n 〉 ∂ T = 0 ,

then using integration by parts, we obtain

( ∇ w ⋅ v , S h ρ ) = − ∑ T ∈ T h 〈 v 0 , ∇ ( S h ρ ) 〉 T + ∑ T ∈ T h 〈 v b , ( S h ρ ) n 〉 ∂ T = ∑ T ∈ T h 〈 ∇ ⋅ v 0 , S h ρ 〉 T − ∑ T ∈ T h 〈 v 0 − v b , ( S h ρ ) n 〉 ∂ T = ∑ T ∈ T h 〈 ∇ ⋅ v 0 , ρ 〉 T − ∑ T ∈ T h 〈 v 0 − v b , ( S h ρ ) n 〉 ∂ T

= − ∑ T ∈ T h 〈 v 0 , ∇ ρ 〉 T + ∑ T ∈ T h 〈 v 0 , ρ n 〉 ∂ T − ∑ T ∈ T h 〈 v 0 − v b , ( S h ρ ) n 〉 ∂ T = − ∑ T ∈ T h 〈 v 0 , ∇ ρ 〉 T + ∑ T ∈ T h 〈 v 0 − v b , ρ n 〉 ∂ T − ∑ T ∈ T h 〈 v 0 − v b , ( S h ρ ) n 〉 ∂ T = − ( v 0 , ∇ ρ ) + ∑ T ∈ T h 〈 v 0 − v b , ( ρ − S h ρ ) n 〉 ∂ T

We can imply that

( v 0 , ∇ p ) = − ( ∇ w ⋅ v , S h ρ ) + ∑ T ∈ T h 〈 v 0 − v b , ( ρ − S h ρ ) n 〉 ∂ T (20)

Next, we test (17) by using v 0 in v = { v 0 , v b } ∈ V h 0 to obtain, we can obtain

( w t , v 0 ) − ( Δ w , v 0 ) + ( ∇ ρ , v 0 ) = ( η , v 0 ) (21)

It follows from the usual integration by parts that

− ( Δ w , v 0 ) = ∑ T ∈ T h ( ∇ w , ∇ v 0 ) T − ∑ T ∈ T h 〈 v 0 − v b , ∇ w ⋅ n 〉 ∂ T

Where we have used the fact that ∑ T ∈ T h 〈 v b , ∇ w ⋅ n 〉 ∂ T = 0 . using Equations (19) and (20), we have

− ( Δ w , v 0 ) = ( ∇ w ( Q h w ) , ∇ w v ) − ∑ T ∈ T h 〈 v 0 − v b , ∇ w ⋅ n − R h ( ∇ w ) ⋅ n 〉 ∂ T (22)

Substituting (20), (22) and ( Q h w t , v 0 ) = ( w t , v 0 ) into (21) yields

( Q h w t , v 0 ) + ( ∇ w ( Q h w ) , ∇ w v ) − ( ∇ w ⋅ v , S h ρ ) = ( η , v 0 ) + l w ( v ) − θ ρ (v)

which completes the proof of the lemma.

In what follows, we give the derivation of the error equation of (9).

Lemma 8. Let e h and ε h be the error of the weak Galerkin finite element solution arising from (9), as defined by (16). Then, we have

{ ( e h , t , v ) + a ( e h , v ) − b ( v , ε h ) = φ u , p ( v ) b ( e h , q ) = 0 , (23)

for all v ∈ V h 0 and q ∈ W h , where φ u , p ( v ) = l u ( v ) − θ p ( v ) + s ( Q h u , v ) is a linear functional defined on V h 0 .

Proof. Since ( u , p ) satisfies the Equation (17) with η = f , then from Lemma 6 we have

( Q h u t , v 0 ) + ( ∇ w ( Q h u ) , ∇ w v ) − ( ∇ w ⋅ v , S h p ) = ( f , v 0 ) + l u ( v ) − θ ρ (v)

Adding s ( Q h u , v ) to both side of the above equation give

( Q h u t , v 0 ) + a ( Q h u , v ) − b ( v , S h p ) = ( f , v 0 ) + l u ( v ) − θ ρ ( v ) + s ( Q h u , v ) (24)

The difference of (24) and (9) yields the following equation,

( e h , t , v 0 ) + a ( e h , v ) − b ( v , ε h ) = l u ( v ) − θ ρ ( v ) + s ( Q h u , v )

for all v ∈ V h 0 , where e h = { e 0 , e b } = { Q 0 u − u 0 , Q b u − u b } . This completes the derivation of (23).

As to (24), we test Equation (1) by q ∈ W h and use (9) to obtain

0 = ( ∇ ⋅ u , q ) = ( ∇ w ⋅ Q h u , q ) = b ( Q h u , q ) (25)

The difference of (25) and (9) yields the following equation

b ( e h , q ) = 0

for all q ∈ W h .

Which completes the proof of the lemma.

In the following, the proof process of Lemma 9 refers to reference [

Lemma 9. If ( w , ρ , r ) ∈ [ H r + 1 ( Ω ) ] d × H r ( Ω ) × V h and 1 ≤ r ≤ k , with the

precondition of regular-shape T h , we have the following estimation.

| s ( Q h w , v ) | ≤ C h r ‖ w ‖ r + 1 ‖ | v | ‖

| l w ( v ) | ≤ C h r ‖ w ‖ r + 1 ‖ | v | ‖

| θ ρ ( v ) | ≤ C h r ‖ w ‖ r + 1 ‖ | v | ‖

The following theorem is the main result of this paper.

Theorem 1. Let ( u , p ) ∈ [ H 0 1 ( Ω ) ∩ H k + 1 ( Ω ) ] d × [ L 0 2 ( Ω ) ∩ H k ( Ω ) ] and ( u h , p h ) ∈ V h × W h be the solution of (1) and (9), respectively. the following error estimates is true.

‖ Q h u − u h ‖ ≤ C h k + 1 ( ‖ u ‖ k + 1 + ‖ p ‖ k + ∫ 0 t ( ‖ u ‖ k + 1 + ‖ p ‖ k + ‖ u t ‖ k + 1 + ‖ p t ‖ k ) d τ ) ‖ Q h u − u h ‖ ≤ C h k ( ‖ u ‖ k + 1 + ‖ p ‖ k + ‖ u t ‖ k + 1 + ‖ p t ‖ k ) ‖ S h p − p h ‖ ≤ C h k + 1 ( ‖ u ‖ k + 1 + ‖ p ‖ k + ‖ u t ‖ k + 1 + ‖ p t ‖ k + ∫ 0 t ( ‖ u ‖ k + 1 + ‖ p ‖ k + ‖ u t ‖ k + 1 + ‖ p t ‖ k + ‖ u t t ‖ k + 1 + ‖ p t t ‖ k ) d τ )

Proof. Let

e h = Q h u − u h = Q h u − Q h 1 u + Q h 1 u − u h = θ + η

e h ( ⋅ , 0 ) = θ ( ⋅ , 0 ) = η ( ⋅ , 0 ) = 0

By the error of Equation (23), we have

( θ t , v ) + a ( θ , v ) − b ( v , S h p − S h 1 p ) = φ u , p ( v ) − ( η t , v ) − a ( η , v ) + b ( v , S h 1 p − p h ) (26)

Substituting (13) into (26), we obtain

( θ t , v ) + a ( θ , v ) − b ( v , S h p − S h 1 p ) = − ( η t , v ) (27)

Let v = θ = Q h u − Q h 1 u , combing the Equation (25) and (14), we have

b ( θ , S h p − S h 1 p ) = 0

That is

( θ t , θ ) + a ( θ , θ ) = − ( η t , θ )

By Lemma 2 and Cauchy inequality, we have

1 2 d d t ‖ θ ‖ 2 + ‖ | θ | ‖ 2 ≤ ‖ η t ‖ ‖ θ ‖ ≤ ‖ η t ‖ 2 + ‖ θ ‖ 2 (28)

By Gronwall Lemma, we have

1 2 d d t ‖ θ ‖ 2 + ‖ | θ | ‖ 2 ≤ ( C h k + 1 ( ‖ u ‖ k + 1 + ‖ p ‖ k + ‖ u t ‖ k + 1 + ‖ p t ‖ k ) ) 2 (29)

By Cauchy inequality, we have

‖ | θ | ‖ ≤ C h k + 1 ( ‖ u ‖ k + 1 + ‖ p ‖ k + ‖ u t ‖ k + 1 + ‖ p t ‖ k ) (30)

Then take the integration about t of both side of Equation (28)

‖ θ ( ⋅ , t ) ‖ 2 + 2 ∫ 0 t ‖ | θ | ‖ 2 d τ ≤ ‖ θ ( ⋅ ,0 ) ‖ 2 + C ( ∫ 0 t ‖ η t ‖ d τ ) 2 + 1 4 s u p τ ≤ t ‖ θ ( τ ) ‖ 2

Since ‖ θ ( ⋅ , 0 ) ‖ = 0 , then

‖ θ ‖ ≤ s u p τ ≤ t ‖ θ ( τ ) ‖ ≤ C ∫ 0 t ‖ η t ‖ d τ ≤ C h k + 1 ∫ 0 t ( ‖ u ‖ k + 1 + ‖ p ‖ k + ‖ u t ‖ k + 1 + ‖ p t ‖ k ) d τ (31)

Combing the Equations (15), (29), (30) and triangle inequality, we have

‖ Q h u − u h ‖ ≤ C h k + 1 ( ‖ u ‖ k + 1 + ‖ p ‖ k + ∫ 0 t ( ‖ u ‖ k + 1 + ‖ p ‖ k + ‖ u t ‖ k + 1 + ‖ p t ‖ k ) d τ ) (32)

‖ Q h u − u h ‖ ≤ C h k ( ‖ u ‖ k + 1 + ‖ p ‖ k + ‖ u t ‖ k + 1 + ‖ p t ‖ k ) (33)

Next, we proof the error estimate of pressure approximation ‖ S h p − p h ‖ , by using error Equation (23), we have

b ( v , S h p − p h ) = ( e h , t , v ) + a ( e h , v ) − φ u , p (v)

By using Lemma 2, Lemma 5 and Lemma 9, we obtain

b ( v , S h p − p h ) ≤ ‖ e h , t ‖ ‖ v ‖ + ‖ | e h | ‖ ‖ | v | ‖ + C h k + 1 ( ‖ u ‖ k + 1 + ‖ p ‖ k ) ‖ | v | ‖ ≤ C ‖ e h , t ‖ ‖ v ‖ + ‖ | e h | ‖ ‖ | v | ‖ + C h k + 1 ( ‖ u ‖ k + 1 + ‖ p ‖ k ) ‖ | v | ‖

By Lemma 4, we have

‖ S h p − p h ‖ ≤ C ‖ e h , t ‖ + ‖ | e h | ‖ + C h k + 1 ( ‖ u ‖ k + 1 + ‖ p ‖ k ) (34)

Next we seek error estimate ‖ e h , t ‖ , then take the derivation about t of both sides of Equation (27)

( θ t t , v ) + a ( θ t , v ) − b ( v , S h p t − S h 1 p t ) = − ( η t t , v )

Let v = θ t , take the derivation about t of both side of Equations (14) and (25), we obtain

b ( θ t , S h p t − S h 1 p t ) = 0

That is

( θ t t , θ t ) + a ( θ t , θ t ) = − ( η t t , θ t )

By Lemma 2 and Cauchy inequality, we have

1 2 d d t ‖ θ t ‖ 2 + ‖ | θ t | ‖ 2 ≤ ‖ η t t ‖ ‖ θ t ‖

That is

‖ θ t ( ⋅ , t ) ‖ 2 ≤ ‖ θ t ( ⋅ ,0 ) ‖ 2 + C ( ∫ 0 t ‖ η t t ‖ d τ ) 2 + 1 4 s u p τ ≤ t ‖ θ t ( τ ) ‖ 2 (35)

Since ‖ θ t ( ⋅ , 0 ) ‖ = 0 , that is

‖ θ t ‖ ≤ s u p τ ≤ t ‖ θ t ( ⋅ , t ) ‖ ≤ C h k + 1 ∫ 0 t ( ‖ u ‖ k + 1 + ‖ p ‖ k + ‖ u t ‖ k + 1 + ‖ p t ‖ k + ‖ u t t ‖ k + 1 + ‖ p t t ‖ k ) d τ (36)

Combing the Equations (15) and triangle inequality, we have

‖ e h , t ‖ ≤ C h k + 1 ( ‖ u ‖ k + 1 + ‖ p ‖ k + ‖ u t ‖ k + 1 + ‖ p t ‖ k + ∫ 0 t ( ‖ u ‖ k + 1 + ‖ p ‖ k + ‖ u t ‖ k + 1 + ‖ p t ‖ k + ‖ u t t ‖ k + 1 + ‖ p t t ‖ k ) d τ ) (37)

Substituting (33) and (36) into (34), we have

‖ S h p − p h ‖ ≤ C h k + 1 ( ‖ u ‖ k + 1 + ‖ p ‖ k + ‖ u t ‖ k + 1 + ‖ p t ‖ k + ∫ 0 t ( ‖ u ‖ k + 1 + ‖ p ‖ k + ‖ u t ‖ k + 1 + ‖ p t ‖ k + ‖ u t t ‖ k + 1 + ‖ p t t ‖ k ) d τ )

This completes the proof. Thus, the error estimates of Theorem 1 hold. Optimal-order error estimates are established for the corresponding numerical approximation in an H^{1} norm for the velocity, and L^{2} norm for both the velocity and the pressure by use of the Stokes projection.

Ning, C. and Gu, H.M. (2018) Weak Galerkin Finite Element Method for the Unsteady Navier-Stokes Equation. American Journal of Computational Mathematics, 8, 108-119. https://doi.org/10.4236/ajcm.2018.81009