# Comparison of SUPG with Bubble Stabilization Parameters and the Standard SUPG

Authors: Xiaowei Liu; Jin Zhang
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We study a streamline upwind Petrov-Galerkin method (SUPG) with bubble
stabilization coefficients on quasiuniform triangular meshes. The new algorithm is a
consistent Petrov-Galerkin method and shows similar numerical performances as the standard
SUPG when the mesh Péclet number is greater than 1. Relationship between the new algorithm and the standard SUPG will be explored. Numerical experiments support these results.


---

## Body

## 1. Introduction

We consider the reaction-convection-diffusion problem in 2D(1)-εΔu+b·∇u+cu=f,(x,y)∈Ω,u=0,(x,y)∈∂Ω,
where Ω is a bounded open domain in R2, with the boundary ∂Ω, b∈(W1∞(Ω))2,    c∈L∞(Ω) are given functions, and ε≪|b| is a small positive parameter. For simplicity, we only consider the case that Ω=(0,1)2. In the following, suppose that there is a constant μ0>0 such that
(2)c-12∇·b≥μ0>0,∀(x,y)∈Ω.
It is also assumed that f is sufficiently smooth. In the case of b=(b1,b2)T>(0,0)T (b1,b2≥β, where β is a positive constant), the solution of (1) typically has two exponential layers of width O(εln⁡(1/ε)) at the sides x=1 and y=1 of Ω. In the case of b=(b,0)T (b≥β>0), the solution of (1) typically has an exponential layer of width O(ε) near the outflow boundary at x=1 and two characteristic (or parabolic) layers of width O(ε) near the characteristic boundaries at y=0 and y=1.When the mesh Péclet numberPe is greater than 1, there exist global unphysical oscillations in numerical solutions of standard discretization schemes on general meshes. Hence, stabilized methods and/or a priori adapted meshes are widely used in order to get discrete solutions with satisfactory accuracy. An overview on these methods can be found in the survey [1, 2].One of the most famous stabilized finite element methods is the streamline upwind Petrov-Galerkin method (SUPG). The SUPG proposed by Hughes and Brooks [3] is known to provide good stability properties and high accuracy. However, there are several drawbacks in SUPG, such as lacking discrete maximum principle and involving second derivatives and difficulties in determining the stabilization coefficients. Driven by these problems, many researchers were devoted to improving the SUPG. A lot of numerical methods were proposed, such as SOLD [4], nonlinear residual [5], and LPS [6]. Also, relations of SUPG and other numerical formulation, like residual-free bubble method [7] and variational multiscale method [8], were studied to seek possible directions of improvement.By means of residual, the SUPG adds to the original bilinear form a term which introduces a suitable amount of artificial viscosity in the direction of streamlines. Also, the SUPG can be viewed as an inconsistent Petrov-Galerkin method, since its modified weighting function cannot apply to the diffusive term (see the details in Section4).For this reason, we are to analyze the SUPG with bubble stabilization coefficients in 2D and compare its numerical performance with the SUPG’s. From theoretical analysis and numerical results, we find that the new scheme is classified into the consistent Petrov-Galerkin formulation (CPGF) and behaves as well as SUPG. Also, the standard finite element method (FEM), which shows excellent performances forPe≤1, can be classified into the CPGF and viewed as a special “SUPG with bubble stabilization coefficients” by taking δK=0 in (7). Thus, the FEM and our new scheme could be viewed as two reference numerical methods in the CPGF. This provides possibilities of constructing new numerical schemes between them in the CPGF. In fact, in our forthcoming works, we obtain a linear maximum-principle-preserving stabilized method in the CPGF by means of the FEM and the SUPG with bubble stabilization coefficients, which shows better numerical performances than the standard SUPG. Thus, our results in this paper can be viewed as a starting point to construct new numerical schemes in the CPGF.The remainder of this paper is organized as follows. In Section2 we formulate the problem and introduce notation and mesh. Theoretical results including stability analysis and energy norm estimates can be found in Section 3. Section 4 is devoted to the relationship between SUPG, standard finite element method, and our method. Finally, numerical experiments that illustrate our theoretical results are presented in Section 5.


## 2. Mesh and Numerical Formulation

First we define a finite element space on triangular meshes(3)Vh:={vh∈C(Ω-):vh|∂Ω=0,vh|Kislinear,∀K∈Th},
where the term “linear” is to be understood in the usual isoparametric sense. Here we assume that the triangulations Th on Ω are quasiuniform:
(4)hK>α0h,αK>α0,
for any K∈Th, where α0 is a positive constant and hK,αK,h denote, respectively, the diameter of K, the smallest angle of K, and the maximum of all diameters of triangles in Th.Using the linear finite element spaceVh, we can state the standard Galerkin discretisation of (1) which reads as follows.Finduh∈Vh such that for all vh∈Vh,
(5)ah(uh,vh)=fh(vh),
where a(u,v)=ε(∇u,∇v)+(b·∇u+cu,v).The SUPG consists in adding to the original bilinear form a term which introduces a suitable amount of artificial viscosity in the direction of streamlines. In this case, the SUPG reads as follows.Finduh∈Vh such that for all vh∈Vh,
(6)ah(uh,vh)=fh(vh),
where
(7)ah(uh,vh)=a(uh,vh)+∑‍K(-εΔuh+b·∇uh+cuh,δKb·∇vh)K,fh(vh)=(f,vh)+∑‍K(f,δKb·∇vh)K.In (7) the term δK is defined as δK(x,y)≔δK=CK*hλ1λ2λ3,(x,y)∈K in which CK* is a constant to be determined and λi   (i=1,2,3) are the linear basis functions. Actually, δK is a bubble function. Moreover,
(8)λ1λ2λ3≤127inK,∫K‍λ1λ2λ3=S(K)60,
where S(K) represents the area of K.Finally, we define a special energy norm (SD norm) associated withah(·,·):
(9)|||uh|||2≔ε|uh|12+∑K∥δK1/2b·∇uh∥K2+∥uh∥2.We denote by∥·∥D the L2 norm in L2(D); that is,
(10)∥v∥D2=(v,v)D∀v∈L2(D).
If D=Ω, then we drop Ω from the notation.


## 3. Stability and Energy Estimates

Throughout this subsection, we assumeC is some positive constant.


### 3.1. Stability Analysis

The stability properties are a consequence of the following.Lemma 1.
Let the parameterCK* in δK satisfies
(11)CK*≤27μ0hCK2
for each K∈Th, where CK:=∥c∥L∞,K. Then the discrete bilinear form is coercive; that is,
(12)ah(vh,vh)≥A2|||vh|||2,
where A:=max⁡{μ0,1}.Proof.
By divergence theorem we obtain(13)ah(vh,vh)=ε|vh|12+∫Ω‍(c-12∇·b)vh2+∑‍K∥δK1/2b·∇vh∥K2+∑‍K∫K‍cvhδKb·∇vh.
Obviously the sum of first three terms is greater than A|||vh|||2 from the condition of (2), and we only need to estimate the last term. Using Hölder inequality, we have
(14)|∑‍K∫K‍cvhδKb·∇vh|≤∑‍K(∫K‍c2vh2δK)1/2(∫K‍(b·∇vh)2δK)1/2≤12∑‍K[∫K‍c2vh2δK+∫K‍(b·∇vh)2δK]≤μ02∥vh∥2+12∑K‍∥δK1/2b·∇vh∥K2≤A2|||vh|||2,
where (14) is based on the definition of δK, (8), and (11):
(15)∑‍K∫K‍c2vh2δK≤CK2∑‍K∫K‍CK*hλ1λ2λ3vh2≤∑‍K∫K‍μ0vh2=μ0∥vh∥2.
Then the proof of the lemma is finished.


### 3.2. Energy Norm Estimate

We denote byuI the nodal piecewise linear interpolant to u over Th. From Lemma 1 and the fact of ah(u-uh,uI-uh)=0 one gets
(16)μ02|||uI-uh|||2≤ah(uI-uh,uI-uh)=ah(uI-u,uI-uh).
Denote η:=uI-u,e≔uI-uh and estimate the right-hand side of (16) term by term:
(17)ε(∇η,∇e)≤ε1/2|η|1|||e|||,(18)(b·∇η+cη,e)=∫∂Ω‍b·nηe-∫Ω‍∇·bηe-∫Ω‍(b·∇e)η+∫Ω‍cηe=∫Ω‍(c-∇·b)ηe-∫Ω‍(b·∇e)η≤C(∫Ω‍η2)1/2(∫Ω‍e2)1/2+(∫Ω‍(b·∇e)2)1/2(∫Ω‍η2)1/2≤Ch2|||e|||+Ch-1/2|||e|||h2≤Ch3/2|||e|||,
where we have used the standard interpolation results ∥η∥≤Ch2 (see [9]) and the first inequality of (18) is obtained by
(19)∫Ω‍(b·∇e)2≤Cβ∫Ω‍(∇e)2=Ch-1β(∇e)2∑‍K∫K‍δK≤Ch-1∑‍K∫K‍δK(b·∇e)2≤Ch-1|||e|||2,∑‍K(εΔu+b·∇η+cη,δKb·∇e)≤∑‍K∫K‍(Cε+b·∇η+cη)δK1/2δK1/2b·∇e≤∑‍K∥(Cε+b·∇η+cη)δK1/2∥K∥δK1/2b·∇e∥K≤(∑‍K∥δK1/2(Cε+b·∇η+cη)∥K2)1/2(∑‍K∥δK1/2b·∇e∥K2)1/2≤Ch2|||e|||.
Combining all of these estimates, one gets
(20)|||u-uh|||≤Ch3/2.


## 3.1. Stability Analysis

The stability properties are a consequence of the following.Lemma 1.
Let the parameterCK* in δK satisfies
(11)CK*≤27μ0hCK2
for each K∈Th, where CK:=∥c∥L∞,K. Then the discrete bilinear form is coercive; that is,
(12)ah(vh,vh)≥A2|||vh|||2,
where A:=max⁡{μ0,1}.Proof.
By divergence theorem we obtain(13)ah(vh,vh)=ε|vh|12+∫Ω‍(c-12∇·b)vh2+∑‍K∥δK1/2b·∇vh∥K2+∑‍K∫K‍cvhδKb·∇vh.
Obviously the sum of first three terms is greater than A|||vh|||2 from the condition of (2), and we only need to estimate the last term. Using Hölder inequality, we have
(14)|∑‍K∫K‍cvhδKb·∇vh|≤∑‍K(∫K‍c2vh2δK)1/2(∫K‍(b·∇vh)2δK)1/2≤12∑‍K[∫K‍c2vh2δK+∫K‍(b·∇vh)2δK]≤μ02∥vh∥2+12∑K‍∥δK1/2b·∇vh∥K2≤A2|||vh|||2,
where (14) is based on the definition of δK, (8), and (11):
(15)∑‍K∫K‍c2vh2δK≤CK2∑‍K∫K‍CK*hλ1λ2λ3vh2≤∑‍K∫K‍μ0vh2=μ0∥vh∥2.
Then the proof of the lemma is finished.


## 3.2. Energy Norm Estimate

We denote byuI the nodal piecewise linear interpolant to u over Th. From Lemma 1 and the fact of ah(u-uh,uI-uh)=0 one gets
(16)μ02|||uI-uh|||2≤ah(uI-uh,uI-uh)=ah(uI-u,uI-uh).
Denote η:=uI-u,e≔uI-uh and estimate the right-hand side of (16) term by term:
(17)ε(∇η,∇e)≤ε1/2|η|1|||e|||,(18)(b·∇η+cη,e)=∫∂Ω‍b·nηe-∫Ω‍∇·bηe-∫Ω‍(b·∇e)η+∫Ω‍cηe=∫Ω‍(c-∇·b)ηe-∫Ω‍(b·∇e)η≤C(∫Ω‍η2)1/2(∫Ω‍e2)1/2+(∫Ω‍(b·∇e)2)1/2(∫Ω‍η2)1/2≤Ch2|||e|||+Ch-1/2|||e|||h2≤Ch3/2|||e|||,
where we have used the standard interpolation results ∥η∥≤Ch2 (see [9]) and the first inequality of (18) is obtained by
(19)∫Ω‍(b·∇e)2≤Cβ∫Ω‍(∇e)2=Ch-1β(∇e)2∑‍K∫K‍δK≤Ch-1∑‍K∫K‍δK(b·∇e)2≤Ch-1|||e|||2,∑‍K(εΔu+b·∇η+cη,δKb·∇e)≤∑‍K∫K‍(Cε+b·∇η+cη)δK1/2δK1/2b·∇e≤∑‍K∥(Cε+b·∇η+cη)δK1/2∥K∥δK1/2b·∇e∥K≤(∑‍K∥δK1/2(Cε+b·∇η+cη)∥K2)1/2(∑‍K∥δK1/2b·∇e∥K2)1/2≤Ch2|||e|||.
Combining all of these estimates, one gets
(20)|||u-uh|||≤Ch3/2.


## 4. Comparison of SUPG with Bubble Stabilization Coefficients and the Standard SUPG

Consider the bilinear form of SUPG:(21)ε(∇uh,∇vh)+(b·∇uh+cuh,vh)+∑K‍δK(-εΔuh+b·∇uh+cuh,b·∇vh)K=(f,vh)+∑K‍(f,δKb·∇vh)K,
where δK is constant in K.It can be rewritten in the form(22)ε(∇uh,∇vh)+(b·∇uh+cuh,vh+δ~b·∇vh)=(f,vh+δ~b·∇vh),
where δ~|K=δK.Notice that (22) does not correspond to a consistent Petrov-Galerkin formulation in general except in the case when b≡C and vh is linear. Clearly, when δK is a bubble function,
(23)ε(∇uh,∇(δ~b·∇vh))=∑K‍ε(∇uh,∇(δKb·∇vh))K=ε∑K‍∫∂K‍∇uh·nδKb·∇vh-ε∑K‍∫K‍(∇·∇uh)δKb·∇vh=ε∑K‍∫∂K‍∇uh·nδKb·∇vh≡0,
since δK vanishes on the boundary of K. Thus (22) can be written as
(24)ε(∇uh,∇(vh+δ~b·∇vh))+(b·∇uh+cuh,vh+δ~b·∇vh)=(f,vh+δ~b·∇vh).
Then SUPG with bubble stabilization coefficients is classified into the consistent Petrov-Galerkin formulation.Moreover, in general, piecewise constantsδK in SUPG make the test function vh+δKb·∇vh discontinuous. However, the test functions are continuous in the case of bubble stabilization coefficients and the consequent test space Wh is contained in H01(Ω). In this case, SUPG with bubble stabilization coefficients reads as follows.Finduh∈Vh, such that for all wh∈Wh,
(25)a(uh,wh)=(f,wh),
where
(26)Wh:={wh∈C(Ω-):wh|K=vh|K+δKb·∇vh|K,vh∈Vh}.On the other hand, SUPG with bubble stabilization coefficients is gradually close to the FEM in the same spaceH01(Ω) as CK*→0 for any ∈Th.In a word, SUPG with bubble stabilization coefficients inherits the advantages of SUPG and constructs the relation between FEM and SUPG in consistent Petrov-Galerkin formulations.


## 5. Numerical Experiments

In this section we give numerical results that appear to support our theoretical results. Errors and convergence rates for our numerical scheme are presented. All calculations are carried out by using Intel visual Fortran 11. The discrete problems are solved by using a version of Pardiso solver (see [10, 11]).For the computations we have chosenδK=60.0hλ1λ2λ3 in SUPG with bubble stabilization coefficients and δK=1.0h in SUPG. We set Ω=[0,1]2 and calculate the errors and convergence rates in the subdomain ΩC away from layers for Tables 1–16. For parabolic Problems 1 and 2, ΩC=[0,1/2]×[1/3,2/3]. For exponential Problems 3 and 4, ΩC=[0,1/2]×[0,1/2]. The errors in Tables 17–20 are calculated in the whole domain Ω. In the following we only list the results of the case of ε=10-2 and ε=10-8 since the comparison results between the two methods of other cases like ε=10-4, ε=10-6, and ε=10-10 are similar.Table 1
ε
=
1
0
-
8, Problem 1 (comparison of error).
N
L
∞ norm
L
2 norm
SD norm
Bubble SUPG
SUPG
Bubble SUPG
SUPG
Bubble SUPG
SUPG
6
4.10
×
1
0
-
2
4.13
×
1
0
-
2
2.59
×
1
0
-
2
2.60
×
1
0
-
2
2.59
×
1
0
-
2
2.6
×
1
0
-
2
12
6.30
×
1
0
-
3
6.35
×
1
0
-
3
3.32
×
1
0
-
3
3.34
×
1
0
-
3
3.33
×
1
0
-
3
3.34
×
1
0
-
3
24
3.53
×
1
0
-
4
3.45
×
1
0
-
4
1.32
×
1
0
-
4
1.33
×
1
0
-
4
1.32
×
1
0
-
4
1.33
×
1
0
-
4
48
1.14
×
1
0
-
4
1.13
×
1
0
-
4
2.00
×
1
0
-
5
2.00
×
1
0
-
5
2.00
×
1
0
-
5
2.01
×
1
0
-
5
96
3.11
×
1
0
-
5
3.10
×
1
0
-
5
4.74
×
1
0
-
6
4.75
×
1
0
-
6
4.75
×
1
0
-
6
4.75
×
1
0
-
6
192
8.15
×
1
0
-
6
8.13
×
1
0
-
6
1.16
×
1
0
-
6
1.16
×
1
0
-
6
1.16
×
1
0
-
6
1.16
×
1
0
-
6Table 2
ε
=
1
0
-
8, Problem 1 (comparison of convergence rate).
N
L
∞ norm
L
2 norm
SD norm
Bubble SUPG
SUPG
Bubble SUPG
SUPG
Bubble SUPG
SUPG
6
2.70
2.70
2.95
2.96
2.96
2.96
12
4.16
4.20
4.66
4.65
4.66
4.65
24
1.63
1.61
2.73
2.73
2.73
2.73
48
1.87
1.87
2.07
2.08
2.07
2.08
96
1.93
1.93
2.03
2.04
2.03
2.03Table 3
ε
=
1
0
-
2, Problem 1 (comparison of error).
N
L
∞ norm
L
2 norm
SD norm
Bubble SUPG
SUPG
Bubble SUPG
SUPG
Bubble SUPG
SUPG
6
3.63
×
1
0
-
2
3.63
×
1
0
-
2
2.75
×
1
0
-
2
2.73
×
1
0
-
2
3.02
×
1
0
-
2
3.01
×
1
0
-
2
12
9.03
×
1
0
-
3
9.07
×
1
0
-
3
4.63
×
1
0
-
3
4.64
×
1
0
-
3
6.23
×
1
0
-
3
6.23
×
1
0
-
3
24
9.72
×
1
0
-
4
9.85
×
1
0
-
4
4.71
×
1
0
-
4
4.74
×
1
0
-
4
1.36
×
1
0
-
3
1.36
×
1
0
-
3
48
3.96
×
1
0
-
4
3.98
×
1
0
-
4
1.63
×
1
0
-
4
1.64
×
1
0
-
4
6.40
×
1
0
-
4
6.39
×
1
0
-
4
96
2.58
×
1
0
-
4
2.59
×
1
0
-
4
8.24
×
1
0
-
5
8.25
×
1
0
-
5
3.20
×
1
0
-
4
3.20
×
1
0
-
4
192
1.47
×
1
0
-
4
1.47
×
1
0
-
4
4.19
×
1
0
-
5
4.19
×
1
0
-
5
1.61
×
1
0
-
4
1.61
×
1
0
-
4Table 4
ε
=
1
0
-
2, Problem 1 (comparison of convergence rate).
N
L
∞ norm
L
2 norm
SD norm
Bubble SUPG
SUPG
Bubble SUPG
SUPG
Bubble SUPG
SUPG
6
2.01
2.00
2.57
2.56
2.28
2.27
12
3.22
3.20
3.30
3.29
2.20
2.20
24
1.30
1.31
1.53
1.53
1.09
1.09
48
0.62
0.62
0.99
0.99
1.00
1.00
96
0.82
0.82
0.98
0.98
1.00
1.00Table 5
ε
=
1
0
-
8, Problem 2 (comparison of error).
N
L
∞ norm
L
2 norm
SD norm
Bubble SUPG
SUPG
Bubble SUPG
SUPG
Bubble SUPG
SUPG
6
4.99
×
1
0
-
3
5.01
×
1
0
-
3
5.80
×
1
0
-
3
5.82
×
1
0
-
3
8.81
×
1
0
-
3
9.23
×
1
0
-
3
12
6.04
×
1
0
-
4
6.04
×
1
0
-
4
5.63
×
1
0
-
4
5.64
×
1
0
-
4
1.33
×
1
0
-
3
1.42
×
1
0
-
3
24
2.37
×
1
0
-
4
2.36
×
1
0
-
4
5.54
×
1
0
-
5
5.54
×
1
0
-
5
2.93
×
1
0
-
4
3.16
×
1
0
-
4
48
6.59
×
1
0
-
5
6.59
×
1
0
-
5
1.43
×
1
0
-
5
1.43
×
1
0
-
5
1.02
×
1
0
-
4
1.10
×
1
0
-
4
96
1.73
×
1
0
-
5
1.73
×
1
0
-
5
3.65
×
1
0
-
6
3.65
×
1
0
-
6
3.60
×
1
0
-
5
3.88
×
1
0
-
5Table 6
ε
=
1
0
-
8, Problem 2 (comparison of convergence rate).
N
L
∞ norm
L
2 norm
SD norm
Bubble SUPG
SUPG
Bubble SUPG
SUPG
Bubble SUPG
SUPG
6
3.05
3.05
3.37
3.37
2.72
2.70
12
1.35
1.35
3.34
3.35
2.18
2.17
24
1.84
1.84
1.95
1.95
1.52
1.52
48
1.93
1.93
1.97
1.97
1.51
1.51Table 7
ε
=
1
0
-
2, Problem 2 (comparison of error).
N
L
∞ norm
L
2 norm
SD norm
Bubble SUPG
SUPG
Bubble SUPG
SUPG
Bubble SUPG
SUPG
6
6.67
×
1
0
-
3
6.69
×
1
0
-
3
6.12
×
1
0
-
3
6.13
×
1
0
-
3
9.36
×
1
0
-
3
9.75
×
1
0
-
3
12
1.32
×
1
0
-
3
1.32
×
1
0
-
3
9.23
×
1
0
-
4
9.25
×
1
0
-
4
1.94
×
1
0
-
3
2.03
×
1
0
-
3
24
1.72
×
1
0
-
4
1.72
×
1
0
-
4
1.07
×
1
0
-
4
1.07
×
1
0
-
4
4.55
×
1
0
-
4
4.69
×
1
0
-
4
48
1.09
×
1
0
-
4
1.09
×
1
0
-
4
4.56
×
1
0
-
5
4.57
×
1
0
-
5
1.98
×
1
0
-
4
2.01
×
1
0
-
4
96
7.14
×
1
0
-
5
7.14
×
1
0
-
5
2.45
×
1
0
-
5
2.45
×
1
0
-
5
9.28
×
1
0
-
5
9.39
×
1
0
-
5Table 8
ε
=
1
0
-
2, Problem 2 (comparison of convergence rate).
N
L
∞ norm
L
2 norm
SD norm
Bubble SUPG
SUPG
Bubble SUPG
SUPG
Bubble SUPG
SUPG
6
2.34
2.34
2.73
2.73
2.27
2.26
12
2.94
2.94
3.11
3.12
2.09
2.11
24
0.66
0.67
1.22
1.22
1.20
1.22
48
0.60
0.60
0.90
0.90
1.09
1.10Table 9
ε
=
1
0
-
8, Problem 3 (comparison of error).
N
L
∞ norm
L
2 norm
SD norm
Bubble SUPG
SUPG
Bubble SUPG
SUPG
Bubble SUPG
SUPG
6
2.70
×
1
0
-
1
2.70
×
1
0
-
1
5.66
×
1
0
-
2
5.66
×
1
0
-
2
3.27
×
1
0
-
1
3.27
×
1
0
-
1
12
9.15
×
1
0
-
2
9.15
×
1
0
-
2
1.51
×
1
0
-
2
1.51
×
1
0
-
2
6.40
×
1
0
-
2
6.43
×
1
0
-
2
24
6.60
×
1
0
-
3
6.60
×
1
0
-
3
8.04
×
1
0
-
4
8.04
×
1
0
-
4
6.35
×
1
0
-
3
6.71
×
1
0
-
3
48
1.62
×
1
0
-
5
1.62
×
1
0
-
5
1.09
×
1
0
-
5
1.09
×
1
0
-
5
2.13
×
1
0
-
3
2.26
×
1
0
-
3
96
3.84
×
1
0
-
6
3.85
×
1
0
-
6
2.56
×
1
0
-
6
2.56
×
1
0
-
6
7.53
×
1
0
-
4
7.80
×
1
0
-
4
192
9.51
×
1
0
-
7
9.52
×
1
0
-
7
6.38
×
1
0
-
7
6.38
×
1
0
-
7
2.66
×
1
0
-
4
2.83
×
1
0
-
4Table 10
ε
=
1
0
-
8, Problem 3 (comparison of convergence rate).
N
L
∞ norm
L
2 norm
SD norm
Bubble SUPG
SUPG
Bubble SUPG
SUPG
Bubble SUPG
SUPG
6
1.56
1.56
1.91
1.91
2.35
2.35
12
3.79
3.79
4.23
4.23
3.33
3.26
24
8.67
8.67
6.21
6.21
1.58
1.57
48
2.08
2.07
2.08
2.08
1.50
1.50
96
2.01
2.02
2.01
2.01
1.50
1.50Table 11
ε
=
1
0
-
2, Problem 3 (comparison of error).
N
L
∞ norm
L
2 norm
SD norm
Bubble SUPG
SUPG
Bubble SUPG
SUPG
Bubble SUPG
SUPG
6
1.66
×
1
0
-
1
1.48
×
1
0
-
1
3.65
×
1
0
-
2
3.30
×
1
0
-
2
2.14
×
1
0
-
1
1.96
×
1
0
-
1
12
4.07
×
1
0
-
2
3.28
×
1
0
-
2
7.07
×
1
0
-
3
5.83
×
1
0
-
3
3.65
×
1
0
-
2
3.38
×
1
0
-
2
24
2.59
×
1
0
-
3
2.23
×
1
0
-
3
3.06
×
1
0
-
4
2.63
×
1
0
-
4
6.13
×
1
0
-
3
6.49
×
1
0
-
3
48
1.41
×
1
0
-
4
1.42
×
1
0
-
4
4.02
×
1
0
-
5
4.02
×
1
0
-
5
2.19
×
1
0
-
3
2.31
×
1
0
-
3
96
7.77
×
1
0
-
5
7.77
×
1
0
-
5
2.01
×
1
0
-
5
2.01
×
1
0
-
5
7.91
×
1
0
-
4
8.35
×
1
0
-
4
192
3.85
×
1
0
-
5
3.85
×
1
0
-
5
9.98
×
1
0
-
6
9.98
×
1
0
-
6
2.92
×
1
0
-
4
3.07
×
1
0
-
4Table 12
ε
=
1
0
-
2, Problem 3 (comparison of convergence rate).
N
L
∞ norm
L
2 norm
SD norm
Bubble SUPG
SUPG
Bubble SUPG
SUPG
Bubble SUPG
SUPG
6
2.03
2.18
2.37
2.50
2.55
2.54
12
3.97
3.88
4.53
4.47
2.57
2.38
24
4.20
3.97
2.93
2.71
1.49
1.49
48
0.86
0.87
1.00
1.00
1.47
1.47
96
1.01
1.01
1.01
1.01
1.44
1.44Table 13
ε
=
1
0
-
8, Problem 4 (comparison of error).
N
L
∞ norm
L
2 norm
SD norm
Bubble SUPG
SUPG
Bubble SUPG
SUPG
Bubble SUPG
SUPG
6
1.24
×
1
0
-
2
1.24
×
1
0
-
2
4.31
×
1
0
-
3
4.31
×
1
0
-
3
2.11
×
1
0
-
2
2.11
×
1
0
-
2
12
8.76
×
1
0
-
3
8.76
×
1
0
-
3
2.35
×
1
0
-
3
2.35
×
1
0
-
3
9.09
×
1
0
-
3
9.09
×
1
0
-
3
24
1.54
×
1
0
-
3
1.54
×
1
0
-
3
3.20
×
1
0
-
4
3.20
×
1
0
-
4
2.04
×
1
0
-
3
2.04
×
1
0
-
3
48
1.06
×
1
0
-
4
1.06
×
1
0
-
4
1.78
×
1
0
-
5
1.78
×
1
0
-
5
6.65
×
1
0
-
4
6.65
×
1
0
-
4
96
2.59
×
1
0
-
5
2.59
×
1
0
-
5
4.06
×
1
0
-
6
4.06
×
1
0
-
6
2.35
×
1
0
-
4
2.35
×
1
0
-
4Table 14
ε
=
1
0
-
8, Problem 4 (comparison of convergence rate).
N
L
∞ norm
L
2 norm
SD norm
Bubble SUPG
SUPG
Bubble SUPG
SUPG
Bubble SUPG
SUPG
6
0.50
0.50
0.87
0.87
1.22
1.22
12
2.51
2.51
2.88
2.88
2.16
2.16
24
3.85
3.85
4.16
4.16
1.62
1.62
48
2.04
2.04
2.13
2.13
1.50
1.50Table 15
ε
=
1
0
-
2, Problem 4 (comparison of error).
N
L
∞ norm
L
2 norm
SD norm
Bubble SUPG
SUPG
Bubble SUPG
SUPG
Bubble SUPG
SUPG
6
1.21
×
1
0
-
2
1.21
×
1
0
-
2
4.23
×
1
0
-
3
4.23
×
1
0
-
3
2.08
×
1
0
-
2
2.08
×
1
0
-
2
12
8.77
×
1
0
-
3
8.77
×
1
0
-
3
2.34
×
1
0
-
3
2.34
×
1
0
-
3
9.07
×
1
0
-
3
9.07
×
1
0
-
3
24
1.62
×
1
0
-
3
1.62
×
1
0
-
3
3.56
×
1
0
-
4
3.56
×
1
0
-
4
2.05
×
1
0
-
3
2.05
×
1
0
-
3
48
1.14
×
1
0
-
4
1.14
×
1
0
-
4
2.59
×
1
0
-
5
2.59
×
1
0
-
5
6.63
×
1
0
-
4
6.63
×
1
0
-
4
96
2.73
×
1
0
-
5
2.73
×
1
0
-
5
8.23
×
1
0
-
6
8.23
×
1
0
-
6
2.41
×
1
0
-
4
2.41
×
1
0
-
4Table 16
ε
=
1
0
-
2, Problem 4 (comparison of convergence rate).
N
L
∞ norm
L
2 norm
SD norm
Bubble SUPG
SUPG
Bubble SUPG
SUPG
Bubble SUPG
SUPG
6
0.47
0.47
0.85
0.85
1.20
1.20
12
2.44
2.44
2.72
2.72
2.14
2.14
24
3.82
3.82
3.78
3.78
1.63
1.63
48
2.06
2.06
1.65
1.65
1.46
1.46Table 17
ε
=
1
0
-
8, Problem 1 (comparison of global error).
N
L
∞ norm
L
2 norm
SD norm
Bubble SUPG
SUPG
Bubble SUPG
SUPG
Bubble SUPG
SUPG
6
3.18
×
1
0
-
1
3.19
×
1
0
-
1
3.72
×
1
0
-
1
3.73
×
1
0
-
1
1.44
1.44
12
3.31
×
1
0
-
1
3.32
×
1
0
-
1
2.64
×
1
0
-
1
2.64
×
1
0
-
1
1.42
1.42
24
3.38
×
1
0
-
1
3.38
×
1
0
-
1
1.86
×
1
0
-
1
1.86
×
1
0
-
1
1.41
1.41
48
3.41
×
1
0
-
1
3.41
×
1
0
-
1
1.31
×
1
0
-
1
1.31
×
1
0
-
1
1.41
1.41
96
3.42
×
1
0
-
1
3.42
×
1
0
-
1
9.29
×
1
0
-
2
9.29
×
1
0
-
2
1.41
1.41
192
3.43
×
1
0
-
1
3.43
×
1
0
-
1
6.54
×
1
0
-
2
6.54
×
1
0
-
2
1.41
1.41Table 18
ε
=
1
0
-
8, Problem 2 (comparison of global error).
N
L
∞ norm
L
2 norm
SD norm
Bubble SUPG
SUPG
Bubble SUPG
SUPG
Bubble SUPG
SUPG
6
6.60
×
1
0
-
2
6.60
×
1
0
-
2
3.13
×
1
0
-
2
3.14
×
1
0
-
2
6.61
×
1
0
-
2
7.02
×
1
0
-
2
12
6.56
×
1
0
-
2
6.56
×
1
0
-
2
2.16
×
1
0
-
2
2.16
×
1
0
-
2
5.78
×
1
0
-
2
6.17
×
1
0
-
2
24
6.55
×
1
0
-
2
6.55
×
1
0
-
2
1.52
×
1
0
-
2
1.52
×
1
0
-
2
5.42
×
1
0
-
2
5.81
×
1
0
-
2
48
6.55
×
1
0
-
2
6.55
×
1
0
-
2
1.08
×
1
0
-
2
1.08
×
1
0
-
2
5.26
×
1
0
-
2
5.65
×
1
0
-
2
96
6.54
×
1
0
-
2
6.54
×
1
0
-
2
7.63
×
1
0
-
3
7.63
×
1
0
-
3
5.18
×
1
0
-
2
5.58
×
1
0
-
2Table 19
ε
=
1
0
-
8, Problem 3 (comparison of global error).
N
L
∞ norm
L
2 norm
SD norm
Bubble SUPG
SUPG
Bubble SUPG
SUPG
Bubble SUPG
SUPG
6
1.16
1.16
4.84
×
1
0
-
1
4.84
×
1
0
-
1
1.47
1.48
12
1.25
1.25
3.73
×
1
0
-
1
3.73
×
1
0
-
1
1.20
1.20
24
1.28
1.28
2.66
×
1
0
-
1
2.66
×
1
0
-
1
1.41
1.41
48
1.29
1.29
1.88
×
1
0
-
1
1.88
×
1
0
-
1
1.62
1.62
96
1.30
1.30
1.32
×
1
0
-
1
1.32
×
1
0
-
3
1.75
1.75Table 20
ε
=
1
0
-
8, Problem 4 (comparison of global error).
N
L
∞ norm
L
2 norm
SD norm
Bubble SUPG
SUPG
Bubble SUPG
SUPG
Bubble SUPG
SUPG
6
2.53
×
1
0
-
2
2.53
×
1
0
-
2
1.26
×
1
0
-
2
1.26
×
1
0
-
2
7.82
×
1
0
-
2
7.82
×
1
0
-
2
12
2.97
×
1
0
-
2
2.97
×
1
0
-
2
1.27
×
1
0
-
2
1.27
×
1
0
-
2
7.18
×
1
0
-
2
7.18
×
1
0
-
2
24
2.87
×
1
0
-
2
2.87
×
1
0
-
2
9.44
×
1
0
-
3
9.44
×
1
0
-
3
6.66
×
1
0
-
2
6.66
×
1
0
-
2
48
2.79
×
1
0
-
2
2.79
×
1
0
-
2
6.60
×
1
0
-
3
6.60
×
1
0
-
3
6.43
×
1
0
-
2
6.43
×
1
0
-
2
96
2.76
×
1
0
-
2
2.76
×
1
0
-
2
4.64
×
1
0
-
3
4.64
×
1
0
-
3
6.34
×
1
0
-
2
6.34
×
1
0
-
2Problem 1.
One has(27)-εΔu+(2-x)ux+32u=f(x,y)∈Ω,u=0(x,y)∈∂Ω,
where f is chosen such that the solution u is
(28)u(x,y)=(sinπ2x-e-((1-x)/ε)-e-(1/ε)1-e-(1/ε))×(1-e-(y/ε))(1-e-((1-y)/ε))1-e-1/ε.Problem 2.
One has(29)-εΔu+ux+u=x(1-x)+y(1-y),(x,y)∈Ω,u=0(x,y)∈∂Ω.Problem 3.
One has(30)-εΔu+2ux+3uy+u=f(x,y)∈Ω,u=0(x,y)∈∂Ω,
where f is chosen such that the solution u is
(31)u(x,y)=2sinx(1-exp⁡(-2(1-x)ε))×y2(1-exp⁡(-(1-y)ε)).Problem 4.
One has(32)-εΔu+2ux+3uy+u=x(1-x)+y(1-y)(x,y)∈Ω,u=0(x,y)∈∂Ω.From the above tables it is shown that the errors and convergence rates of SUPG with bubble stabilization coefficients and standard SUPG in the maximum norm, in theL2 norm, and in the SD norm are similar not only in the subdomain away from layers but also in the global sense. These results illustrate that SUPG with bubble stabilization coefficients also has good stability properties and high accuracy as standard SUPG.



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*Source: 364675-2014-06-02.xml*