25 
J. mt. area res., Vol. 6, 2021 

   

         Journal of Mountain Area Research 
            

 

 

STABILIZED NUMERICAL METHODS FOR THE TWO KINDS OF 

PROBLEMS OF INCOMPRESSIBLE FLUID FLOWS 
Shahid Hussain1, * and Sajid Hussain2    

1. School of Mathematical Sciences, East China Normal University China 

2. School of Mathematical Sciences, Xian Jiaotong University Xian China 

 

ABSTRACT 

A mixed finite element method (MFEM) stabilized for the two kinds of problems related to the 

incompressible fluid flow is demonstrated. In the first kind, the Newtonian fluid flow is illustrated with 

the MFEM and considered discontinuous scheme. Initially, the model equations are considered 

nonlinear and un-stabilize. The model equations are solved for linear terms with the special technique 

first and then the model equation with the extra added term is utilized later to stabilize the model 

equations. A steady-state viscoelastic Oseen fluid flow model with Oldroyd-B type formulations was 

demonstrated in the second kind of problem with SUPG method. The nonlinear problems are 

linearized through the Oseen scheme. Numerical results for both the model equations are given and 

compared. The SUPG method is found more suitable and active. 

KEYWORDS: Finite Element Method, Lowest Equal Order Elements, DG method, SUPG technique. 

 

*Corresponding author: (Email: 52150601025@std.ecnu.edu.cn, Phone: 0086-19946036403) 

 

1. INTRODUCTION 

We are interested to discuss only the non-

steady state fluid flow problems under the  2-

dimensional bounded and connected 

domain. In this case, for the incompressible 

time independent model equations are 

standard under applied forces and stresses as 

follows [1, 2] 

.
total N

PI  


 
Where   represents the 

Wesenberg number which is well known in the 

given literature and the term 
( . )

a
g M 

 is 

defined as: 

1 1
( . ) ( . ) (( ) ( . ) )

2 2

T

a

a a
g M M M M     

 
      

 

 We can write the model equation for given 

forces f with the given parameter related to 

the materials  as: 
( . ) .

total
M M f  

 

while the Oldroyd model is defined by 

.
total N

PI  


 
 This denotes the total stress 

tensor where 


with N and without N  are the 

Newtonian and viscoelastic parts 

respectively. The model equation for these 

important issues are given below in detail 

formulation.

Full length article 

Vol. 6, 2021 

https://doi.org/10.53874/jmar.v6i0.90 

https://doi.org/10.53874/jmar.v6i0.85


Shahid Hussain et al., j. mt. area res., 06 (2021) 25-29

1.1 Model Problem

The model equations, without time dependent ter-
m under the open domain Ω is considered. The
Dirichlet boundary condition with homogeneous na-
ture is regarded for the fluid velocity M in the non-
newtonian form; The model equations are given as:

ρ + λ(M ·∇)ρ + λga(ρ,∇M) −β2D(M) (6)
= 0

−2(−β + 1)∇·D(M) −∇·ρ (7)
∇p = f inΩ,

∇·M = 0 inΩ, (8)
M = 0 onΓ, (9)

Let us consider pressure p is zero at the bound-
ary. The solution of the partial differential equa-
tions (7)-(9) require boundary conditions to be spec-
ified where appropriate [3]. The velocity compo-
nents are identified along the boundaries where the
assumptions of no-slip boundary conditions and no-
penetration conditions are applied so that both ve-
locity components vanish there. The inlet and out-
let boundaries requirements are applied smartly for
the vector fields [4]. So that for the calculation, no
boundary conditions are used for the stresses here.
For the literature review, the well-posedness of the
model equations or about the study of existence and
uniqueness of the equations (7)-(9), we would like to
motivate the reader to see [5] for further guidance.

In the analysis part, we are keen to note the lin-
earized form of the given equations. In the numerical
solution of the model equations section, we illustrate
the conditions
Problem(O): Solve the problem of (ρ,M,p) such
that

ρ + λ(M ·∇)ρ + λga(ρ,∇M) (10)
−2βD(M) = 0 inΩ,

∇p− 2(1 −β)∇·D(M) −∇·ρ (11)
= f inΩ,

∇·M = 0 inΩ,
M = 0 onΓ.(12)

We understand with the following assumption for
the additional term for velocity M(x) which can be
searched in the [6] for any positive constant M > 0.
This positive constant depends only on the smooth
domain, and also it is independent of the other con-
stants and grid size parameters,

M ∈ H10 (Ω), ∇·M = 0, || M ||∞ ≤ M,
|| ∇M ||∞≤ M < ∞.

1.2 Variational formulation

We are keen to introduce some basic required sym-
bols. These are classical for m ∈ N the norm is

related to the special type of Hilbert space known
as Sobolve spaces Wn,p(Ω) by || · ||Wn,q , with the
given special case Wm,2(Ω), this can be written as
[7,8]:

X = H10 (Ω)
2 = {v ∈ H1(Ω)2,v = 0 Hilbertspce},

Q = L2(Ω) = {q ∈ L2(Ω), },
Sρ = {ρ = (ρij); ρij = ρji : ρij ∈ L2(Ω); i,j = 1, 2}

∩{ρ = (ρij); M ·∇ρ ∈ L2(Ω)2×2,X}.

In order to find the appropriate weak formulation of
the given Problem (O). To find unknowns as:

(ρ,σ) + λ((M ·∇)ρ,σ) (13)
+λ(ga(ρ,∇M),σ) −β((σ,M)) = 0

∀σ ∈ Sρ, (14)
−(∇·v,p) + (1 −β)(T(M),T(v))+(ρ,D(v)) (15)

= (v,f)∀v ∈ X,
(∇·M,q) = 0 (16)

.

The solution generated is not a minimum in the giv-
en spaces, but is indeed a saddle point. The stan-
dard existence of this saddle point is satisfied by the
chosen finite elements or Lagrangian polynomials for
the velocity and pressure in discrete form, if it can
be proved that a compatibility condition for veloc-
ity v and pressure p is satisfied. This important
condition (inf-sup or LBB) states that [7,9]

sup
v∈X

(q,∇·v)
|| v ||1

≥ C || q || ∀ q ∈ Q,

It requires the roundedness of the variational oper-
ator and restricts the choices for the approximating
spaces.

For further detail formulation, The multiplica-
tion of β with the equation (16) and add together
with (14) deals with the bilinear form A and B:

A((ρ,M,p), (σ,v,q))
= (ρ,σ) + λ(ga(ρ,∇M),σ) −β(D(M),σ)
+β(ρ,D(v)) + 4α(1 −α)(D(M),D(v))
−α(∇·v,p) + β(q,∇·M), (17)
λB(M,ρ,σ) = λ((M ·∇)ρ,σ). (18)

By using the bilinear form A((·, ·, ·), (·, ·, ·)) and
B(·, ·, ·), the equations (14)

A((ρ,M,p), (σ,v,q)) + λB(M,ρ,σ) (19)
= β(f,v).

An equivalent formulation of (19) is given as

Ñ ((ρ,M,p), (σ,v,q)) = (20)

2β(f,v), ∀(σ,v,q) ∈ Sρ ×X ×Q,

J. mt. area res., Vol. 06 (2021) 26



Shahid Hussain et al., j. mt. area res., 06 (2021) 25-29

where

Ñ ((ρ,M,p), (v,σ, ,q))

= A((ρ,M,p), (σ,v,q)) + λB(M,ρ,σ).

2 Discontinuous FE

The DG method and SUPG method are mostly
utilized in viscoelastic fluid flow problems to find
the approximate solutions of discontinuous stress s-
train, In this work, the DG and SUPG methods are
used and compared. Let us consider the elements of
a triangulation in domain Ω is denoted as Th i.e.,
Ω̃ = {∪K : K ∈ Th}. Assume that rmin and rmax
are the maximum and minimum diameter of the ele-
ments of Th, respectively. This will allow us to sim-
plify notation, and use mesh size h to represent the
characteristic length of all the triangular elements
of Th [8]. Then there exist positive constants, i.e.,

rminh ≤ hK ≤ rmaxρK,

Accordingly, we define discrete subspaces for the
FE approximation of the equation (20)

Xh := {v ∈ X ∩C0(Ω̃)2;∀K ∈ Th},
Qh := {q ∈ Q∩C0(Ω̃); q|K ∈ P1(K);},
Shρ := {σ ∈ Sρ; σ|K ∈ P1(K)

2×2;},

We define ∂K−(M) = {x ∈ ∂K; M(x) · n(x) < 0}
where ∂K is the boundary of K ∈ Th and n, and

Γh = {∪∂K : K ∈ Th}\ Γ,
ρ±(M(x)) = lim

ε→0
ρ(x±εM(x)).

Also, for any (ρ,σ) ∈
∏

K∈Th
[H1(K)]4, we define

(ρ,σ) =
∑

K2∈Th
(ρ,σ)K,

〈ρ±,σ±〉 =
∑
K∈Th

∫
∂K−(M)

(ρ±(M),σ±(M))ds,

〈〈ρ±〉〉2h,M = 〈ρ
±,ρ±〉h,M,

|| ρ ||0,Γh = (
∑
K∈Th

| ρ |20,∂K)
1/2.

The term ((M ·∇)ρ,σ) is solved with an operator

Bh on (Xh,Shρ ,S
h
ρ ), which is stated in [9] by

Bh(M,ρh,σh) = ((M ·∇)ρh,σh)hMρh,σh) (21)
+(1/2)(∇· +〈ρh+

−ρh−,σh+〉h,M,
= −((M ·∇)σh,ρh)h (22)
−(1/2)(∇·Mσh,ρh) +
〈ρh−,σh− −σh+〉h,M,

= ((M ·∇)ρh,σh)h (23)
+〈ρh+ −ρh−,σh+〉h,M,
if ∇·M = 0.

Thus,

Bh(M,ρh,ρh) = (24)

(1/2)〈〈ρh+ −ρh−〉〉2h,M ≥ 0.

The main objective of this work is to find the solu-
tion of viscoelastic fluid flow problems with a mixed
FE method by applying the FE triples as a poly-
nomial space in discrete formulation due to the d-
eficiency of the inf-sup condition the scheme is not
stable more. Thus, it is important to add a stabi-
lization term to circumvent the inf-sup condition .
This idea was proposed and well-defined for the S-
tokes problem in the finite element method [10,11].
We are interested in working with the same idea
to find the approximate solution to the viscoelastic
fluid flow problems. We think that this idea is new
and very useful for viscoelastic fluid flow problem-
s. Specifically, this technique is new for the stable
solution of the viscoelastic fluid flow problems, and
in the existing research, there is no such technique
available. Indeed this stabilization method is not
expensive as the existing stabilization methods. For
more work, to ensure the well-posedness of the con-
firming finite element methods, we introduce asym-
metric, non-trivial, and penalty terms which add the
penalizing parameter. For further investigations and
brief study see the references we have cited [12,13]

H(ph,qh) = ((I − Π)ph, (I − Π)qh),

where Π : L2(Ω) → R0 is the standard piecewise
constant space R0. The reversed operator Π has the
following feature

|| Πp ||0 ≤ k || p ||0, (25)

|| p− Πp || ≤ kh || p || . (26)

In this work, k means as a positive constant, which
is free of any mesh size value.

J. mt. area res., Vol. 06 (2021) 27



Shahid Hussain et al., j. mt. area res., 06 (2021) 25-29

2.1 Analytical solution test

The verifications with convergence rates are demon-
strated by considering fluid flow across a square do-
main with a known solution. To verify the numerical
values of the new formulated scheme, some one can
considered the standard FE triples for the unknown-
s [14,15]. Many researchers used this experimental
pattern for the Stokes and the Navier-Stokes equa-
tion.

In the given example, the known function m(x)
is chosen to be the true values of the velocity M
[16,17]. However, the true values of the model e-
quations for velocity M = (u1,u2), pressure p and
polymeric stress ρ is given by




M =

(
−10(x4 − 2x3 + x2)(2y3 − 3y2 + y),
10(2x3 − 3x2 + x)(y4 − 2y3 + y2),

)
p = −10.0(2x− 1)(2y − 1),
ρ = 2βD(M).

The right-hand sides, initial and boundary condi-
tions are derived by model Problem(O) with the
parameters value a = 0, λ = 5.0 and

We illustrated the specific features of the low-
est equal order MFE method for the Oseen non-
newtonian. We included the H1-norm for fluid ve-
locity, L2-norm error for fluid pressure, and L2-norm
error for stress, respectively, with the variable s-
pacing such as h = 1/4, 1/8, 1/16, 1/64. In the
computations of the errors for the standard MFE
with P1b −P1 −P 1dc pairs. The table gives the DG
method. let’s say : ||u−uh||0 = A, ||u−uh||1 = B,
||p−ph||0 = C and ||τ − τh||0 = D

h A B C D
1/4 0.017 0.226 0.22 0.109
1/8 0.005 0.102 0.062 0.033
1/16 0.001 0.048 0.019 0.010

Table given to demonstrate the SUPG method

h A B C D
1/4 0.018 0.236 0.329 0.117
1/8 0.004 0.110 0.062 0.043
1/16 0.001 0.038 0.019 0.010

3 Conclusions

We have given the two methods for the stabilization
techniques by using finite element lowest equal poly-
nomial elements. Since the model equations were
non linear Partial Differential equations. We first
made these model equations linear with the help
of Oseen technique and formulated approximate nu-
merical solutions. Two different numerical methods

were demonstrated by considering the same finite
elements for the same stabilized formulation. From
the numerical studies, the results of the SUPG are
more accurate than the DG method. For results
and geometrical configuration, readers can see the
references given at the end of this manuscript.

Conflicts of interest

The authors declare no any conflict of interest.

Data availability

Not applicable.

Code availability

Available.

Authors contributions

Shahid hussain: Conceptualization, methodology,
analysis and writing original draft.
Sajid Hussain: Methodology editing and review-
ing.

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Received: 28 May 2021. Revised/Accepted: 26 August 2021.

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