Analysis of a projection method for the Stokes problem using an $\varepsilon$-Stokes approach
Masato Kimura Kazunori Matsui Adrian Muntean Hirofumi Notsu





ε-Stokes approach        






Masato KimuraFaculty of Mathematics and Physics, Kanazawa University, Kanazawa
920-1192, Japan. 


Kazunori Matsui (Corresponding author)Division of Mathematical and Physical Sciences, 

Graduate School of Natural Science and Technology, Kanazawa University, 

Kanazawa 920-1192, Japan, 

first-lucky@stu.kanazawa-u.ac.jp
Adrian Muntean Department of Mathematics and Computer Science, Karlstad University, 

Universitetsgatan 2, 651 88 Karlstad Sweden. 

Hirofumi Notsu Faculty of Mathematics and Physics, Kanazawa University, Kanazawa
920-1192, Japan, 

Japan Science and Technology Agency, PRESTO, Kawaguchi 332-0012, Japan. 







Analysis of a projection method for the Stokes problem
using an -Stokes approach




    Masato Kimura Kazunori Matsui Adrian MunteanHirofumi Notsu
    December 30, 2023
====================================================================================





We generalize pressure boundary conditions of an -Stokes problem.
Our -Stokes problem connects the classical Stokes problem 
and the corresponding pressure-Poisson equation using one parameter >0.
For the Dirichlet boundary condition, it is proven in K. Matsui and A. Muntean (2018) that
the solution for the -Stokes problem converges to the one for the Stokes problem
as  tends to 0, and to the one for the pressure-Poisson problem
as  tends to ∞. Here, we extend these results 
to the Neumann and mixed boundary conditions.
We also establish error estimates in suitable norms
between the solutions to the -Stokes problem,
the pressure-Poisson problem and the Stokes problem, respectively.
Several numerical examples are provided to show that 
several such error estimates are optimal in .
Our error estimates are improved if one uses the Neumann boundary conditions.
In addition, we show that the solution to the -Stokes problem 
has a nice asymptotic structure.



76D03 35Q35 35B40 65N30 






§ INTRODUCTION





Let Ω be a bounded domain in ^n(n≥ 2,n∈)
with Lipschitz continuous boundary Γ and
let F:Ω→^n be a given applied force field and 
u_b:Γ→^n be a given Dirichlet boundary data satisfying 
∫_Γ u_b·ν=0,
where ν is the unit outward normal vector on Γ.
A strong form of the Stokes problem is given as follows. 
Find u_:Ω→^n and p_:Ω→ such that

    S{[ -Δ u_+∇ p_ = F             Ω,;         u_ = 0             Ω,;         u_=u_b             Γ, ].

where u_ and p_ are the velocity and the pressure of the flow
governed by (S), respectively.
We refer to <cit.> for the details on the Stokes problem
(i.e., physical background and corresponding mathematical analysis).
Taking the divergence of the first equation, we obtain

    F=(-Δ u_+∇ p_)=-Δ ( u_)+Δ p_=Δ p_.

This equation is often called the pressure-Poisson equation and 
is used in numerical schemes such as MAC (marker and cell), SMAC 
(simplified MAC) and the projection methods 
(see, e.g., <cit.>).
Based on the above, we consider a similar problem.
Find u_:Ω→^n and p_:Ω→ satisfying

    PP{[ -Δ u_+∇ p_ = F             Ω,;    -Δ p_ = - F             Ω,;         u_=u_b             Γ,;           +p_.                ].

We call this problem the pressure-Poisson problem.
The idea of using (<ref>) instead of u_=0 is useful
for calculating the pressure numerically in the Navier–Stokes equation.
For example, this idea is used in MAC, SMAC and projection methods.
The Dirichlet boundary condition for the pressure is used in 
an outflow boundary <cit.>. 
See also <cit.>.

We introduce an “interpolation” between problems (S) and (PP).
For >0, find u_:Ω→^n and
p_:Ω→ such that

    ES{[  -Δ u_+∇ p_ = F              Ω,; -Δ p_+ u_ = - F              Ω,;          u_=u_b              Γ,;            +p_.                 ].

This problem is called the -Stokes problem (ES) in <cit.>.
In <cit.>, the authors treat this problem
as an approximation of the Stokes problem to avoid numerical instabilities.
The -Stokes problem approximates the Stokes problem (S)
as →0 and the pressure-Poisson problem (PP)
as →∞ (Fig. <ref>).
It is shown in <cit.> that
if p_∈ then there exists a constant c>0 independent of  such that

    [                 u_-u__^n                        ≤ cγ_0 p_-γ_0 p__H^1/2(Γ),;                 u_-u__^n                        ≤ cγ_0 p_-γ_0 p__H^1/2(Γ), ]

where γ_0∈ B(,H^1/2(Γ)) is the standard trace operator <cit.>. 
From the first inequality, if we have a good prediction value for pressure on Γ,
then u_ is a good approximation of u_.
Moreover, u_ is also a good approximation of u_ from the second inequality. 


Next we specify the boundary conditions for p_ and p_.
We assume that the boundary Γ is a union of two open subsets
Γ_D and Γ_N such that

    |Γ∖(Γ_D∪Γ_N)|=0,  
      |Γ_D|> 0,  Γ_D∩Γ_N=∅,

and number of connected components of Γ_D and Γ_N
with respect to the relative topology of Γ are finite.
We consider a Neumann boundary condition (<ref>) and
a mixed boundary condition (<ref>),

    ∂ p_/∂ν=g_bΓ,  ∂ p_/∂ν=g_bΓ,


    {[      p_=p_b        Γ_D,; ∂ p_/∂ν=g_b        Γ_N, ].  {[      p_=p_b        Γ_D,; ∂ p_/∂ν=g_b        Γ_N, ].

where p_b:Γ_D→ and g_b=Γ→ 
satisfying g_b= F are given boundary data.

In <cit.>, the authors impose Dirichlet boundary conditions
for p_ and p_ 
(i.e., (<ref>) with Γ_D=Γ and Γ_N=∅.) 
For such boundary conditions, they introduce a weak solution 
(u_,p_) to the -Stokes problem (ES) and
prove that (u_,p_) strongly converges in ^n× to
a weak solution to the pressure-Poisson problem (PP)
as →∞ and weakly converges in ^n×(/)
to a weak solution (u_,p_) to the Stokes problem (S)
as →0.
Moreover, if p_∈, then strong convergence of (u_,p_) 
to (u_,p_) as →0 holds.

In this paper, we generalize the Dirichlet boundary condition 
of p_ and p_ to both the Neumann boundary condition (<ref>) 
and the mixed boundary condition (<ref>),
and prove the corresponding convergence result (see Theorem 3.1, 4.2 and 4.3).
Since the mixed boundary condition for pressure often appears 
in engineering problems, this generalization of the boundary conditions
for pressure is important.
In addition, for the Neumann boundary condition,
we estimate the error between the weak solutions to (ES) and (S) provided p_∈.
We also give an asymptotic expansion for the weak solution to (ES).
We furthermore check this convergence result using numerical computations.

The organization of this paper is as follows.
In Section <ref> we introduce the notation used in this work
and the weak form of these problems. We also 
prove the well-posedness of the problems (PP) and (ES)
and show some their properties.
In Section <ref> we study that the solution to (ES) converges
to the solution to (PP) in the strong topology as →∞.
We also explore here the structure of the regular perturbation asymptotics.
Section <ref> is devoted to proving that the solution to (ES)
converges to the solution to (S) in the weak and strong topology
as → 0.
Finally, in Section <ref>,
we show several numerical examples of these problems.
The numerical errors between the problems (ES) and (PP),
and between the problems (ES) and (S)
using the P2/P1 finite element method.
Proofs for several theorems which are similar to ones in <cit.>
are described.





§ WELL-POSEDNESS




In this section, we introduce the notation and the weak form 
of the problems (S), (PP) and (ES), and prove their well-posedness.
We give estimates between these solutions by
using a pressure error on the boundary Γ.





 §.§ Notation



We set

    [               C^∞_0(Ω)^n :={φ∈ C^∞(Ω)^n | (φ)⊂Ω},;                        /            :={ψ∈ | ψ=0},;                        /                  :=∩(/),;                                  :={ψ∈|ψ|_Γ_D=0}.;                          ]

For m∈, ^m:=(^m)^* is equipped with the dual norm

    f_^m:=sup_φ∈ S_m⟨ f,φ⟩f∈^m,

where

    S_m:={φ∈^m | ∇φ_^n× m=1}.


Let Q⊂ be a closed subspace such that 
there exists a constant c>0 for which
q_≤ c∇ q_^n for all q∈ Q.
The dual space Q^* is equipped with the norm

    f_Q^*:=sup_ψ∈ S_Q⟨ f,ψ⟩

for f∈ Q^*,
where 

    S_Q:={ψ∈ Q | ∇ψ_^n=1}.

We define 

    [                         [p]               := p-1/|Ω| p,;                         p_/          :=inf_a∈p-a_=[p]_,;                     ⟨∇ p,φ⟩                     :=- p φ;                      ∇ u:∇φ :=∑^n_i=1∂ u/∂ x_i·∂φ/∂ x_i ]

for all p∈,u,φ∈^n, 
where |Ω| is the volume of Ω.






 §.§ Preliminary results




Let γ_0∈ B(, H^1/2(Γ)) be
the standard trace operator.
The trace operator γ_0 is surjective and satisfies 
(γ_0)= <cit.>.
Let ν be the unit outward normal for Γ.
Since ν is a unit vector, 
^3∋ u↦ u·ν:=(γ_0 u)·ν∈ H^1/2(Γ)
is a linear continuous map.
For all u∈^3 and ω∈, the following Gauss divergence formula holds:

    u·∇ω+( u)ω=∫_Γ(u·ν)ω.


We recall the following four embedding theorems
which plays an important role in the proof of
the existence of pressure solutions to the Stokes problem.
For the proof of Theorem <ref>, 
see <cit.> and
<cit.>.

  There exists a constant c>0 such that
  
    f_≤ c(f_+∇ f_^n)

  for all f∈.


The following result follows from Theorem <ref>.





  <cit.>
  There exists a constant c>0 such that
  
    f_/≤ c∇ f_^n

  for all f∈.


The following two embedding theorems are often called the Poincaré inequality.





  <cit.>
  There exists a constant c>0 such that 
  
    ψ_≤ c∇ψ_^n
 
  for all ψ∈/.






  <cit.>
  There exists a constant c>0 such that 
  
    ψ_≤ c∇ψ_^n
 
  for all ψ∈.






 §.§ Weak formulations of the problems (PP), (S) and (ES)




We assume the following conditions for F,u_b and g_b:

    F∈^n,  
      u_b∈ H^1/2(Γ),  ∫_Γ u_b·ν=0,


    g_b∈ L^2(Γ),   F∈,


    g_b= F,


    p_b∈.

We start by defining the weak solution to (S).
For all φ∈^n, we obtain from the first equation of (S) that

    F·φ
    =-∂ u_/∂ν·φ
    +∇ u_:∇φ+∇ p_·φ
    =∇ u_:∇φ+∇ p_·φ.

Using this expression,
the weak form of the Stokes problem becomes as follows:
Find u_∈^n and p_∈/ such that

    S'{[ ∇ u_:∇φ+⟨∇ p_,φ⟩
    = F·φ                  φ∈^n,;                   u_=0                      ,;                 u_=u_b            H^1/2(Γ)^n. ].


Next, we define the weak formulations of (PP) and (ES)
first for the Neumann boundary condition (<ref>) 
and them for the mixed boundary condition (<ref>).
After that, we define generalized weak formulations for (PP) and (ES) 
which cover both cases.

First, we apply the Neumann boundary condition (<ref>) for (PP) and (ES).
We take a test function ψ∈. From the second equation of (PP), we obtain

    [                        -( F)ψ =-(Δ p_)ψ
    =-∂ p_/∂νψ
    +∇ p_·∇ψ;                                            =- g_bψ+∇ p_·∇ψ. ]

Hence,

    ∇ p_·∇ψ
    = g_bψ-( F)ψ.

We note that g_bψ-( F)ψ= g_b[ψ]-( F)[ψ]
for all ψ∈ by (<ref>).
Therefore, the weak form of the pressure-Poisson problem
with the Neumann boundary condition (<ref>) becomes as follows.
Find u_∈^n and p_∈/ such that

    PP_1{[ ∇ u_:∇φ+∇ p_·φ
    = F·φ                φ∈^n,;     ∇ p_·∇ψ=⟨ G_1,ψ⟩                 ψ∈/,;               u_=u_b          H^1/2(Γ)^n, ].

where G_1∈^* such that
⟨ G_1,ψ⟩= g_bψ-( F)ψ (ψ∈).

The weak form of (ES) with the Neumann boundary condition 
can be defined similarly to that of (PP).
Find u_∈^n and p_∈/ such that

    ES_1{[     ∇ u_:∇φ+∇ p_·φ
    = F·φ                    φ∈^n,; ∇ p_·∇ψ+( u_)ψ
    =⟨ G_1,ψ⟩                     ψ∈/,;                   u_=u_b              H^1/2(Γ)^n. ].


Secondly, we apply the mixed boundary condition (<ref>) for (PP) and (ES).
We take a test function ψ∈. From the second equation of (PP), we obtain

    [                        -( F)ψ =-(Δ p_)ψ
    =-∂ p_/∂νψ
    +∇ p_·∇ψ;                                       =-∫_Γ_N g_bψ+∇ p_·∇ψ. ]

Hence,

    ∇ p_·∇ψ
    =∫_Γ_N g_bψ-( F)ψ.

The weak form of the pressure-Poisson problem
with the mixed boundary condition (<ref>) becomes as follows.
Find u_∈^n and p_∈ such that

    PP_2{[ ∇ u_:∇φ+∇ p_·φ
    = F·φ                φ∈^n,;     ∇ p_·∇ψ=⟨ G_2,ψ⟩                  ψ∈,;               u_=u_b          H^1/2(Γ)^n,;               p_=p_b          H^1/2(Γ_D), ].

where G_2∈^* such that

    ⟨ G_2,ψ⟩=∫_Γ_N g_bψ-( F)ψ

for ψ∈.
The weak form of (ES) with the mixed boundary condition (<ref>) 
can be defined similarly to that of (PP). It reads as follows.
Find u_∈^n and p_∈ such that

    ES_2{[     ∇ u_:∇φ+∇ p_·φ
    = F·φ                    φ∈^n,; ∇ p_·∇ψ+( u_)ψ
    =⟨ G_2,ψ⟩                      ψ∈,;                   u_=u_b              H^1/2(Γ)^n,;                   p_=p_b              H^1/2(Γ_D). ].


Finally, we generalize (PP_1) and (PP_2) to an abstract pressure-Poisson problem.
Let Q⊂ be a closed subspace as defined in Section <ref>.
Find u_∈^n and p_∈ Q such that

    PP'{[ ∇ u_:∇φ +∇ p_·φ= F·φ                φ∈^n,;       ∇ p_·∇ψ=⟨ G,ψ⟩                ψ∈ Q,;               u_=u_b          H^1/2(Γ)^n,;           p_-p_b∈ Q,                      ].

with G∈ Q^*.
Indeed, by Theorem <ref> and <ref>, we obtain (PP_1) from (PP') by putting 
Q:=/ and G:=G_1. Similarly, we obtain (PP_2) from (PP') 
by putting Q:= and G:=G_2.

We generalize (ES_1) and (ES_2) to an abstract -Stokes problem.
Find u_∈^n and p_∈ Q such that

    ES'{[  ∇ u_:∇φ +∇ p_·φ= F·φ                 φ∈^n,; ∇ p_·∇ψ+( u_)ψ=⟨ G,ψ⟩                 ψ∈ Q,;            u_-u_b∈^n,;            p_-p_b∈ Q.                       ].

Indeed, by Theorem <ref>, <ref>, we obtain (ES_1) from (ES') 
by putting Q:=/ and G:=G_1. Similarly, we also obtain  
(ES_2) from (ES') by putting Q:= and G:=G_2. 





 §.§ Well-posedness of (S'), (PP') and (ES')




We show the well-posedness of problems (S'), (PP') and (ES')
in Theorem <ref>, <ref> and <ref>.





Under the condition (<ref>),
there exists a unique solution 

(u_,p_)∈^n×(/)
satisfying (S').





See <cit.> for the proof.






Under the condition (<ref>) and (<ref>), for G∈ Q^*,
there exists a unique solution (u_,p_)∈^n× Q satisfying (PP').





Proof.
Using the Lax–Milgram theorem,
since Q× Q∋(p,ψ)↦∇ p·∇ψ∈
is a continuous and coercive bilinear form,
p_∈ is uniquely determined from the second and fourth equations of (PP').
Then u_∈^n is also uniquely determined 
from the first and third equations, again using the Lax–Milgram theorem.






Under the condition (<ref>) and (<ref>), 
for >0 and G∈ Q^*,
there exists a unique solution (u_,p_)∈^n×
satisfying (ES').





This is a generalization of Theorem 2.6 in <cit.>.
See Appendix <ref> for the proof.


From now on, let the solutions of (S'), (PP') and (ES') be denoted by
(u_,p_),(u_,p_) and (u_,p_), respectively.
We show their properties in connection with a pressure error 
on the boundary Γ.





Suppose that p_∈, ⊂ Q and 
⟨ G,ψ⟩=
 -( F)ψ for all ψ∈.
Then there exists a constant c>0 independent of  such that

    [                 u_-u__^n                        ≤ cγ_0 p_-γ_0 p__H^1/2(Γ),;                 u_-u__^n                        ≤ cγ_0 p_-γ_0 p__H^1/2(Γ). ]

In particular, if γ_0p_=γ_0 p_,
then (u_,p_)=(u_,p_)=(u_,p_) holds for all >0.





This is a generalization of Proposition 2.7 in <cit.>.
See Appendix A for the proof.


Since ⊄/, Proposition <ref> does not apply directly
for the case of the Neumann boundary condition (<ref>).
However, we add natural assumptions, then it leads to (<ref>).





Suppose that p_∈, Q=/.
If G∈ Q^*=(/)^* is such that G∈^* and
⟨ G,ψ⟩= g_bψ-( F)ψ 
for all ψ∈, then we have (<ref>).





Proof.
Since G∈^* satisfies 
⟨ G,ψ⟩= g_bψ-( F)ψ for all ψ∈,
it holds that

    ∇ p_·∇ψ=-( F)ψ

for all ψ∈ from the second equation of (PP'). 
Hence, it leads the second equation of (<ref>).
Using the proof of Proposition <ref>, we obtain (<ref>).






§ LINKS BETWEEN (ES) AND (PP)




as guaranteed by Theorem <ref>, <ref> 
and <ref>.
In this section, we show that (u_,p_) converges
to (u_,p_) strongly in ^n× 
as →∞.
We also treat the case of the regular perturbation asymptotics
by exploring the structure of the lower order terms and their effect
on the convergence rate.





 §.§ Convergence as →∞




We use the following Lemma <ref> for the proofs of the theorems 
in this section.





Let h∈ Q^* and (v_,q_)∈^n× Q satisfy
  
    {[        ∇ v_:∇φ + (∇ q_)·φ=0                       φ∈^n,; ∇ q_·∇ψ +( v_)ψ
        =⟨ h,ψ⟩                        ψ∈ Q ].

  for an arbitrarily fixed >0. Then there exists a constant c>0 such that

    v__^n≤c/h_Q^*,    q__≤c/h_Q^*.






Proof.
Putting φ:=v_ and ψ:=q_ and
adding two equations of (<ref>), we obtain

    ∇ v_^2_^n× n
    +∇ q_^2_^n≤h_Q^*∇ q__^n.

where we have used
∇ q_· v_=-( v_)q_.
Thus

    ∇ q__^n≤1/h_Q^*.


In addition, from the first equation of (<ref>) by putting φ:=v_,
we have

    [                          ∇ v_^2_^n
      = ∇ v_:∇ v_
      =	-(∇ q_)· v_≤∇ q__^nv__^n;                                      ≤                    c∇ q__^n∇ v__^n× n, ]
and then

    ∇ v__^n≤ c∇ q__^n≤c/h_Q^*.



Using Lemma <ref>, we obtain Theorem <ref>.





There exists a constant c>0 independent of >0 such that

    u_-u__^n≤c/ u__Q^*,  p_-p__≤c/ u__Q^*.

for all >0. In particular, we have

    u_-u__^n→ 0, p_-p__→ 0
       as →∞.






Proof.
Combining (PP') and (ES'), we obtain

    {[ ∇ v_:∇φ
          +∇ q_·φ =0                    φ∈^n,; ∇ q_·∇ψ+( v_)ψ
    	=-( u_)ψ                    ψ∈ Q, ].

where v_:=u_-u_,q_:=p_-p_ and h:= u_.
By Lemma <ref>, we conclude the proof.






If u_ satisfies u_=0,
then u_=u_ and p_=p_
hold for all >0. Furthermore,
u_=u_=u_ and p_=[p_]=[p_] hold for all >0.






 §.§ Regular Perturbation Asymptotics









By Theorem <ref>, we have that 
(u_-u_)_^n≤ c and (p_-p_)_≤ c
for all >0.
It implies that there exists a subsequence of
((u_-u_),(p_-p_)) which converges weakly to
(v^(1),q^(1))∈^n× Q if →∞.
The next theorem states properties of the limit functions v^(1) and q^(1).





  Let v^(1)_:=(u_-u_)∈^n,q^(1)_:=(p_-p_)∈ Q
  and let (v^(1),q^(1))∈^n× Q satisfy

    {[ ∇ v^(1):∇φ+ (∇ q^(1))·φ=0                     φ∈^n,;      ∇ q^(1)·∇ψ = -( u_)ψ                     ψ∈ Q. ].

Then there exists a constant c>0 independent of  such that

    v^(1)_-v^(1)_^n≤c/ v^(1)_Q^*,    q^(1)_-q^(1)_≤c/ v^(1)_Q^*.






Proof.
The existence and the uniqueness of the pair (v^(1),q^(1))∈^n× Q
as a solution to (<ref>) follows from Theorem <ref>. 
As in (<ref>), we have

    {[            ∇ v^(1)_:∇φ + (∇ q^(1)_)·φ =0                                    φ∈^n,; ∇ q^(1)_·∇ψ+1/( v^(1)_)ψ=
          - ( u_)ψ                                    ψ∈ Q. ].

Subtracting (<ref>) from (<ref>), it holds that

    {[ ∇ (v^(1)_-v^(1)):∇φ
          + (∇ (q^(1)_-q^(1)))·φ =0                                               φ∈^n,;           ∇ (q^(1)_-q^(1))·∇ψ
          +1/( v^(1)_)ψ=0                                               ψ∈ Q. ].

Hence,

    {[         ∇ v_:∇φ
          +∇ q_·φ =0                            φ∈^n,; ∇ q_·∇ψ
          +( v_)ψ=-( v^(1))ψ                            ψ∈ Q, ].

where v_:=v^(1)_-v^(1), q_:=q^(1)_-q^(1) and h:=- v^(1).
By Lemma <ref> , we have

    v^(1)_-v^(1)_^n≤c/ v^(1)_Q^*,  q^(1)_-q^(1)_≤c/ v^(1)_Q^*

for all >0.


Next, we generalize Theorem <ref> to the following theorem:





  Let k∈ℕ be arbitrary (k≥ 1) and let v^(0):=u_.
  If functions v^(1),v^(2),⋯,v^(k)∈^n and
  q^(1),q^(2),⋯,q^(k)∈ Q satisfy
  
    {[  ∇ v^(i):∇φ+ (∇ q^(i))·φ=0                      φ∈^n,; ∇ q^(i)·∇ψ = -(  v^(i-1))ψ                      ψ∈ Q, ].

  for all 1≤ i≤ k, then there exists a constant c>0
independent of  satisfying
  
    [ u_
              -(u_+1/v^(1)
              +⋯+(1/)^kv^(k))_^n≤c/^k+1  v^(k)_Q^*,;   p_
              -(p_+1/q^(1)
              +⋯+(1/)^kq^(k))_≤c/^k+1  v^(k)_Q^*. ]






Proof.
Let (v^(i)_,q^(i)_)∈^n× Q (1≤ i≤ k) satisfy

    {[                            ∇ v^(i)_:∇φ+(∇ q^(i)_)·φ=0                                                 φ∈^n,; ∇ q^(i)_·∇ψ
          +1/( v^(i)_)ψ
          = -(  v^(i-1))ψ                                                 ψ∈ Q. ].

Subtracting (<ref>) from (<ref>), it holds that

    {[ ∇ (v^(i)_-v^(i)):∇φ
          + (∇ (q^(i)_-q^(i)))·φ =0                                               φ∈^n,;           ∇ (q^(i)_-q^(i))·∇ψ
          +1/( v^(i)_)ψ=0                                               ψ∈ Q. ].

Setting v_:=v^(i)_-v^(i),q_:=q^(i)_-q^(i) and 
h:=- v^(i), we obtain from Lemma <ref> that the estimates

    v^(i)_-v^(i)_^n≤c/  v^(i)_Q^*,  q^(i)_-q^(i)_≤c/  v^(i)_Q^*

hold for all >0.
In particular, putting i:=k, we obtain

    v^(k)_-v^(k)_^n≤c/  v^(k)_Q^*,


    q^(k)_-q^(k)_≤c/  v^(k)_Q^*

for all >0.
By the uniqueness of the solution to (ES') in Theorem <ref>,
it leads that 
v^(i+1)_=(v^(i)_-v^(i)),
q^(i+1)_=(q^(i)_-q^(i)) for all i=1,⋯,k-1,
and thus

    [                                                                                                              v^(k)_-v^(k);                                                            =                                     (v^(k-1)_-v^(k-1))-v^(k);                                                            =                            (v^(k-1)_-(
      v^(k-1)+(1/)v^(k)));                                                            =                                                            ⋯;                                                            =  ^k-1(v^(1)_-(
      v^(1)+⋯+ (1/)^k-2v^(k-1)
      +(1/)^k-1v^(k)));                                                            = ^k(u_-(
      u_+1/v^(1)
      +⋯+ (1/)^k-1v^(k-1)
      +(1/)^kv^(k))), ]


    [                                                                                                              q^(k)_-q^(k);                                                            =                                     (q^(k-1)_-q^(k-1))-q^(k);                                                            =                            (q^(k-1)_-(
      q^(k-1)+(1/)q^(k)));                                                            =                                                            ⋯;                                                            =  ^k-1(q^(1)_-(
      q^(1)+⋯+ (1/)^k-2q^(k-1)
      +(1/)^k-1q^(k)));                                                            = ^k(p_-(
      p_+1/q^(1)
      +⋯+ (1/)^k-1q^(k-1)
      +(1/)^kq^(k))). ]

Hence it holds that

    u_-(
      u_+1/v^(1)
      +⋯+(1/)^kv^(k))_^n   ≤c/^k+1  v^(k)_Q^*,
    p_-(
      p_+1/q^(1)
      +⋯+(1/)^kq^(k))_   ≤c/^k+1  v^(k)_Q^*.







Theorem <ref> can be interpreted from the operator theory.

Let t≥0,X:=^n× Q,Y:=^n× Q^* be equipped with norms

    [        (u,p)^2_X   :=u^2_^n+p^2_,;        (f,g)^2_Y :=f^2_^n+g^2_Q^* ]

for (u,p)∈ X,(f,g)∈ Y, and let A and B be

    [               A:                X                ⟶                Y;                               90∈              90∈;                             (u,p)                ⟼ (-Δ u+∇ p, Δ p), ]  [       B:        X        ⟶        Y;               90∈      90∈;             (u,p)        ⟼ (0,  u). ]

Then (u_,p_) and (u_,p_) satisfy

    A(u_,p_)=f,  (A+1/B)(u_,p_)=f,

where f=(F,G).
We have A+tB∈ Isom(X,Y) for an arbitrary t≥ 0
by the analogy of Theorem <ref> (t=0)
and Theorem <ref> (t=1/).
Equation (<ref>) states that 

    A(v^(i),q^(i))=-B(v^(i-1),q^(i-1))

for i=1,⋯,k, i.e.

    [                                     (v^(i),q^(i)) =-A^-1 B(v^(i-1),q^(i-1))
      =⋯=(-A^-1 B)^i(u_,p_);                                                                                  =A^-1(-BA^-1)^i f. ]

By Theorem <ref>, there exists a constant c>0 such that

    (A+1/B)^-1 f
    -A^-1∑^k_i=0(-1/BA^-1)^i f_X
    ≤c/^k+1(BA^-1)^k+1 f_Y

for all >0,f∈ Y.






§ CONVERGENCE OF (ES) TO (S)




In this section, we show that (u_,p_) converges
to (u_,p_) weakly in ^n×(/) 
as → 0.
Moreover, if p_∈, then (u_,p_) converges
to (u_,p_) strongly in ^n×(/) 
as → 0.

The outline of the proof of our convergence results 
(Theorem <ref>, <ref> and <ref>) is as follows.
First, we prove the boundedness of the sequence ((u_,p_))_>0
in ^n×(/).
By the reflexivity of ^n×(/), the sequence has a subsequence
converging weakly in ^n×(/).
In the end, we show that the limit pair of functions satisfies (S').

We start this section with a useful lemma. 





  If v∈^n,q∈ and f∈^n satisfy
  
    ∇ v:∇φ +⟨∇ q,φ⟩=⟨ f,φ⟩  φ∈^n,

  then there exists a constant c>0 such that
  
    q_/≤ c(∇ v_^n× n+f_^n).






Proof.
Let c be the constant from Theorem <ref>. Then we obtain

    [                                q_/≤ c∇ q_^n
      = csup_φ∈ S_n
      |⟨∇ q,φ⟩|;                                   ≤ csup_φ∈ S_n(|∇ v:∇φ|
        +|⟨ f,φ⟩|);                                   ≤                  c(∇ v_^n× n+f_^n). ]







There exists a constant c>0 independent of  such that

    u__^n≤ c,  p__/≤ c
      >0.

Furthermore, if C^∞_0(Ω)⊂ Q,
then we obtain

    u_⇀ u_ weakly ^n, 
    [p_]⇀ p_ weakly /   as → 0.






See Appendix A for the proof.

If we add a regularity assumption of p_, 
then (u_,p_) converges strongly in ^n×/




Suppose that p_∈. Then we obtain

    u_→ u_ strongly ^n, 
    [p_]→ p_ strongly /   as → 0.






See Appendix A for the proof.

Theorem <ref> does not give the convergence rate.
If Q=/ (corresponding to the Neumann boundary condition (<ref>)), 
then the convergence rate becomes √().





Suppose that Q=/ and p_∈. Then there exists a constant c>0
independent of  such that

    u_-u__^n≤ c√(), p_-p__≤ c√().






Proof.
We obtain from (ES') and (S') that

    {[ ∇ (u_-u_):∇φ + (∇(p_-p_))·φ =0                          φ∈^n,;          ∇ p_·∇ψ+( u_)ψ=⟨ G,ψ⟩                           ψ∈/. ].

Putting φ:=u_-u_∈^n and ψ:=p_-p_∈/, we get

    [                                                               ∇(u_-u_)^2_^n× n
    +∇ p_·∇(p_-p_);                                              = - (∇(p_-p_))·(u_-u_)
    -( u_)(p_-p_)
    +⟨ G,p_-p_⟩;                                              =     ( u_- u_)(p_-p_)
    -( u_)(p_-p_)
    +⟨ G,p_-p_⟩;                                              =                                    ⟨ G,p_-p_⟩.;                                                ]

Subtracting ∇ p_·∇(p_-p_) 
from both sides of (<ref>), we obtain

    [                                 ∇(u_-u_)^2_^n× n+∇ (p_-p_)^2_^n;                               =      -∇ p_·∇(p_-p_)
    +⟨ G,p_-p_⟩;                               ≤   (∇ p__^n+G_(/)^*)∇(p_-p_)_^n. ]

To clarify the following estimates, we set 
α:=∇(u_-u_)_^n× n,
β:=∇(p_-p_)_^n,a:=∇ p__^n+G_(/)^*.
The estimate (<ref>) reads as

    α^2+β^2≤ aβ, (α/√())^2+(β-a/2)^2
    ≤(a/2)^2.

Hence, α≤ a√()/2, i.e.,
∇(u_-u_)_^n× n≤
 (√()/2)(∇ p__^n+G_(/)^*).
By Lemma <ref>, we obtain

    [                           p_-p__ ≤ c∇(u__k-u_)_^n× n
    =cα≤ ca√()/2;                                        = c√()/2(∇ p__^n+G_(/)^*). ]







§ NUMERICAL EXAMPLES








For our simulations, we consider Ω=(0,1)× (0,1).
We take the following boundary conditions:

    u_b=(x(x-1),y(y-1))^T, g_b=(2,2)^T·ν

on Γ. The exact solutions for (PP_1) are 
u_=(x(x-1),y(y-1))^T and p_=2x+2y-2.
We solve the problems (PP_1), (ES_1) and (S') numerically by
using the finite element method with P2/P1 elements 
by the software FreeFem++ <cit.>. The numerical solutions
(u_,p_),(u_,p_) (=1,10^-2 or 10^-4)
and (u_,p_) to the problems (PP_1), (ES_1) and (S'), respectively,
are illustrated in Fig. <ref>–<ref>.
From these pictures we observe that (u_,p_) seems to 
converge to (u_,p_) as →∞ and
to (u_,p_) as → 0 
(as expected from Theorem <ref> and <ref>.) 







Next we compute the error estimate between the numerical solutions of 
(ES_1) and (PP_1). The numerical errors
u_-u__^n, ∇(u_-u_)_^n× n,
p_-p__ and ∇(p_-p_)_^n
are shown in Fig. <ref> and Fig. <ref>.
Based on these values, we have fitted a constant c such that
u_-u__^n∼ c/ and
p_-p__∼ c/ for  large.
Fig. <ref> and Fig. <ref> indicate that 
there exists a constant c such that
u_-u__^n≤ c/ and
p_-p__≤ c/,
as expected from Theorem <ref>.

We also compute the error estimate between the problems (ES_1) and (S') by numerical calculation.
The numerical error estimate
u_-u__^n, ∇(u_-u_)_^n× n ,
p_-p__ and ∇(p_-p_)_^n
are shown in Fig. <ref> and Fig. <ref>.
Based on these values, we have fitted a constant c such that
u_-u__^n∼ c and
p_-p__∼ c for  small.
Fig. <ref> and Fig. <ref> indicate that 
there exists a constant c̃ such that
u_-u__^n≤c̃√() and
p_-p__≤c̃√(),
as expected from Theorem <ref>.
















spmpsci      

   











§ APPENDIX









Theorem <ref>, <ref>, <ref>,
and Proposition <ref> are generalizations of several theorems
stated in <cit.>.
In this appendix, however, we give their proofs for the readers'
convenience.
We define a continuous coercive bilinear form depending on 
and prove Theorem <ref> by the Lax–Milgram Theorem.




Proof of Theorem <ref>.
We take arbitrary u_1∈^n with γ_0 u_1=u_b.
Since :^n→/ is surjective
<cit.>,
there exists u_2∈^n such that
u_2= u_1. We put 

    u_0:=u_1-u_2,

and note that γ_0 u_0=u_b and u_0=0.
To simplify the notation, we set
u:=u_-u_0∈^n,p:=p_-p_b∈ Q, and define 
f∈^n and g∈ Q^* by 

    [                     ⟨ f,v⟩ := Fv-∇ u_0:∇ v-(∇ p_b)· v                      v∈^n,;                     ⟨ g,q⟩         :=⟨ G,q⟩-∇ p_b·∇ q                      q∈ Q. ]

Then, (u_,p_) satisfies (ES') if and only if 
(u,p) satisfies

    {[ ∇ u:∇φ + (∇ p)·φ
          =⟨ f,φ⟩                          φ∈^n,;    ∇ p·∇ψ +( u)ψ
          =⟨ g,ψ⟩                          ψ∈ Q. ].

Adding the equations in (<ref>), we get

    (( [ u; p ]) ,
      ( [ φ; ψ ]) )_
      :=∇ u:∇φ + ∇ p·∇ψ
      +(∇ p)·φ +  ( u)ψ
      = ⟨ f,φ⟩ +⟨ g,ψ⟩.

We check that (· ,· )_ is a continuous coercive bilinear form
on ^n× Q. The bilinearity and continuity of (· ,· )_ are obvious.
The coercivity of (· ,· )_ is obtained in the following way.
Take (v,q)^T∈^n× Q. 
We have the following sequence of inequalities:

    [  (( [ v; q ]) ,
      ( [ v; q ]) )_
      = ∇ v:∇ v + ∇ q·∇ q
        + v·∇ q+ ( v)q;                                    =                        ∇ v^2_+∇ q^2_;                                    ≥              min{ 1,}(∇ v^2_+∇ q^2_);                                    ≥            c  min{ 1,}(v^2_^n+q^2_). ]

Summarizing, (· ,· )_ is a continuous coercive bilinear form and
^n× Q is a Hilbert space.
Therefore, the conclusion of Theorem <ref> follows from
the Lax–Milgram Theorem.


Let (u_,p_),(u_,p_) and (u_,p_) be 
the solutions of (S'), (PP') and (ES'), respectively,
as guaranteed by Theorem <ref>, <ref> 
and <ref>.
We show that the subtract p_-p_ satisfies

    Δ(p_-p_)=0

in distributions sense. 
The weak harmonicity is the key ingredient to proving Proposition <ref>.




Proof of Proposition <ref>.
First, we prove that there exists a constant c>0 independent of 
such that 
u_-u__^n≤ cγ_0 p_-γ_0 p__H^1/2(Γ),
and if γ_0(p_-p_)=0, then p_=p_.
Taking the divergence of the first equation of (S'), 
we obtain

    F=(-Δ u_+∇ p_)=-Δ ( u_)+Δ p_=Δ p_.

in distributions sense. 
Since p_∈ and C^∞_0(Ω) is dense in ,
it follows that

    ∇ p_·∇ψ=-( F)ψ

for all ψ∈.
Together with (S'), (PP') and ⊂ Q, we obtain

    {[ ∇ (u_-u_):∇φ
    =- (∇(p_-p_))·φ                        φ∈^n,;                ∇(p_-p_)·∇ψ=0                           ψ∈ ].

from the assumption ⟨ G,ψ⟩=∇ F·ψ.
Putting φ:=u_-u_∈^n in (<ref>), we get

    [       ∇(u_-u_)^2_^n× n = - (∇(p_-p_))·(u_-u_);                         ≤∇(p_-p_)_^nu_-u__^n. ]

Hence, 

    u_-u__^n≤ c_1∇(p_-p_)_^n

holds.
From the second equation of (<ref>), we obtain

    [                                                                ∇ (p_-p_-ψ)^2_^n;                                       = ∇ (p_-p_)^2_^n+∇ψ^2_^n
      -2∇ (p_-p_)·∇ψ;                                       =                  ∇ (p_-p_)^2_^n+∇ψ^2_^n;                                       ≥                          ∇ (p_-p_)^2_^n ]

for all ψ∈.
Thus we find

    ∇ (p_-p_)_^n≤inf_ψ∈^n(∇(p_-p_-ψ)_^n).

Since γ_0 is surjective and the space (γ_0)=, 
/ and H^1/2(Γ) are isomorphic,
there exists a constant c_2>0 such that
q_/≤ c_2γ_0 q_H^1/2(Γ) for all q∈.
Hence, we obtain

    [                 ∇ (p_-p_)_^n       ≤inf_ψ∈^n∇(p_-p_-ψ)_^n;                                         ≤inf_ψ∈^np_-p_-ψ_;                                                  =p_-p__/;                              ≤ c_2γ_0 p_-γ_0 p__H^1/2(Γ). ]

Together with (<ref>), we obtain
u_-u__^n≤ c_1c_2γ_0 p_-γ_0 p__H^1/2(Γ).
Moreover, if γ_0(p_-p_)=0, then p_=p_.

Next, we prove that there exists a constant c>0 independent of 
such that 
u_-u__^n≤ cγ_0 p_-γ_0 p__H^1/2(Γ),
and if γ_0(p_-p_)=0, then p_=p_.
Let w_:=u_-u_∈^n and r_:=p_-p_∈ Q.
By (S'), (PP') and (ES'), we obtain

    {[ ∇ w_:∇φ
        +(∇ r_)·φ
        =-(∇(p_-p_))·φ                                    φ∈^n,;                         ∇ r_·∇ψ+( w_)ψ=0                                    ψ∈ Q. ].

Putting φ:=w_ and ψ:=r_ 
and adding the two equations of (<ref>), we get

    ∇ w_^2_^n× n+∇ r_^2_^n≤∇(p_-p_)_^nw__^n

from (∇ r_)· w_=-( w_)r_.
Thus we find 

    w__^n≤ c_3∇(p_-p_)_^n.

Together with (<ref>), we obtain

    u_-u__^n
      =w__^n≤ c_2c_3γ_0 p_-γ_0 p__H^1/2(Γ).

Moreover, by (<ref>), we obtain 

    p_-p_^2_
      = r_^2_≤ c_4∇(p_-p_)_^nw__^n.

Hence, if γ_0(p_-p_)=0, then p_=p_.



We show that the sequence ((u_,p_))_>0 is bounded 
in ^n×(/).
By the reflexivity of ^n×(/),
the sequence ((u_,p_))_>0 has a subsequence
converging weakly to somewhere in ^n×(/).
It is sufficient to check that the limit satisfies (S').
Since the solution of (S') is unique,
the sequence ((u_,p_))_>0 converges weakly.




Proof of Theorem <ref>.
We take u_b∈^n,f∈^n and g∈ Q^*
as (<ref>) and (<ref>) in the proof of Theorem <ref>. 
We put ũ_:=u_-u_b∈^n,p̃_:=p_-p_b∈ Q.
Then we obtain

    {[ ∇ũ_:∇φ + (∇p̃_)·φ
          =⟨ f,φ⟩                           φ∈^n,;    ∇p̃_·∇ψ +(ũ_)ψ
          =⟨ g,ψ⟩                           ψ∈ Q. ].

Putting φ:=ũ_,ψ:=p̃_ and adding the two equations of
(<ref>), we get

    ∇ũ_^2_^n× n+∇p̃_^2_^n≤f_^n∇ũ__^n× n
    +g_Q^*∇p̃__^n

since (∇p̃_)·ũ_=-(ũ_)p̃_.
Hence, (ũ__^n)_0<<1 and
(√()p̃__)_0<<1 are bounded. 
Moreover, by Lemma <ref>, we obtain

    p̃__/≤c(∇ũ__^n× n+f_^n),

i.e., (p̃__/)_0<<1 is bounded.
By Theorem <ref>, (u__^n)_≥ 1 and
(p̃__/)_≥ 1 are bounded, and thus
(u__^n)_>0 and (p̃__/)_>0
are bounded.

Since ^n×(/) is reflexive and
(ũ_,[p̃_])_0<<1 is bounded in ^n×(/),
there exist (u,p)∈^n×(/) and a subsequence of pairs 
(ũ__k, p̃__k)_k∈⊂^n× Q
such that

    ũ__k⇀ u  weakly ^n, 
    [p̃__k]⇀ p  weakly /   as k→∞.

Hence, from (<ref>) with :=_k, taking k→∞,
we obtain

    {[ ∇ u:∇φ +⟨∇ p,φ⟩
    =⟨ f,φ⟩                    φ∈^n;                 ( u)ψ=0                   ψ∈ Q, ].

where we have used that

    |_k∇p̃__k·∇ψ|
    ≤√(_k)√()p̃__ψ_→ 0,


    ∇p̃__k·φ
    =-[p̃__k]φ→- pφ=⟨∇ p,φ⟩

as k→∞.
By (<ref>), the first equation of (<ref>) implies that

    ∇ (u+u_b):∇φ +⟨∇ (p+p_b),φ⟩
    = F·φ

for all φ∈^n.
From the second equation of (<ref>) and 
C^∞_0(Ω)⊂ Q, (u+u_b)=0 follows. 
Hence, we obtain that (u+u_b,p+[p_b]) satisfies (S'), i.e.,
u_=u+u_b and p_=p+[p_b]. Then we have

    u__k-u_=u__k-u-u_b=ũ__k-u_⇀ 0
    ^n,


    [p__k]-p_=[p__k-p-p_b]=[p̃__k]-p⇀ 0
    /

as k→∞.
Since any arbitrarily chosen subsequence of ((u_,[p_]))_0<<1
has a subsequence which converges to (u_,p_),
we can conclude the proof.



Using Theorem <ref> and the Rellich-Kondrachov Theorem,
it is easy to prove Theorem <ref>.




Proof of Theorem<ref>.
We have from (ES') and (S') that

    {[ ∇ (u_-u_):∇φ +(∇(p_-p_))·φ =0                         φ∈^n,;         ∇ p_·∇ψ+( u_)ψ=⟨ G,ψ⟩                         ψ∈ Q. ].

Putting φ:=u_-u_∈^n and p̃_:=p_-p_b∈, 
we get

    [                               ∇(u_-u_)^2_^n× n                           =-(∇(p_-p_))·(u_-u_);                                                =-(∇(p_-p_b))·(u_-u_)
        +(∇(p_-p_b))·(u_-u_);                                                          =(p_-p_b)(u_-u_)
        +(∇p̃_)·(u_-u_);                                                             =(p_-p_b) u_
        +(∇p̃_)·(u_-u_), ]

since 
-(∇(p_-p_b))·(u_-u_)=(p_-p_b)(u_-u_)
and u_=0. Thus,

    ∇(u_-u_)^2_^n× n
      =(p_-p_b) u_+(∇p̃_)·(u_-u_).

Putting ψ:=p_-p_b∈ Q, we have

    ∇ p_·∇(p_-p_b)+( u_)(p_-p_b)
      =⟨ G,p_-p_b⟩.

Hence,

    ∇ (p_-p_b)^2_^n
      =-∇ (p_-p_b)·∇ p_b
      -(p_-p_b) u_+⟨ G,p_-p_b⟩.

Together with (<ref>) and (<ref>), we obtain

    [                                                                ∇(u_-u_)^2_^n× n+∇ (p_-p_b)^2_^n;                                               =    ∇p̃_· (u_-u_)
    -∇ (p_-p_b)·∇ p_b
    +⟨ G,p_-p_b⟩;                                               ≤ ∇p̃__^nu_-u__^n
    +(∇ p_b_^n+G_Q^*)∇ (p_-p_b)_^n.;                                                 ]

By Theorem <ref> and the Rellich-Kondrachov Theorem,
there exists a sequence (_k)_k∈⊂
such that

    u__k→ u_^n
      k→∞.

Therefore,

    [                                                                                          ∇(u__k-u_)^2_^n× n;                                                     ≤ ∇p̃__^nu__k-u__^n
      +_k(∇p̃_b_^n+G_Q^*)∇ (p__k-p_)_^n;                                                     →                                                     0 ]

as k→∞. This implies that

    [p__k]-p__/=p__k-p__/≤ c∇(u__k-u_)_^n× n→ 0k→∞

by Lemma <ref>.
Since any arbitrarily chosen subsequence of ((u_,[p_]))_0<<1
has a subsequence which converges to (u_,p_),
we can conclude the proof.


