Abstract and Applied Analysis

Volume 2014 (2014), Article ID 364675, 8 pages

http://dx.doi.org/10.1155/2014/364675

Research Article

Comparison of SUPG with Bubble Stabilization Parameters and the Standard SUPG

Xiaowei Liu¹ and Jin Zhang²

¹Department of Mathematics, Tongji University, Shanghai 200092, China

²School of Mathematical Sciences, Shandong Normal University, Jinan 250014, China

Received 23 February 2014; Accepted 13 May 2014; Published 2 June 2014

Academic Editor: Gerassimos Barbatis

Copyright © 2014 Xiaowei Liu and Jin Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. Introduction

We consider the reaction-convection-diffusion problem in 2D

where is a bounded open domain in , with the boundary , ,    are given functions, and is a small positive parameter. For simplicity, we only consider the case that . In the following, suppose that there is a constant such that

It is also assumed that is sufficiently smooth. In the case of (, where is a positive constant), the solution of (1) typically has two exponential layers of width at the sides and of . In the case of (), the solution of (1) typically has an exponential layer of width near the outflow boundary at and two characteristic (or parabolic) layers of width near the characteristic boundaries at and .

When the mesh Péclet number 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 Section 4).

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 for , can be classified into the CPGF and viewed as a special “SUPG with bubble stabilization coefficients” by taking 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 Section 2 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

where the term “linear” is to be understood in the usual isoparametric sense. Here we assume that the triangulations on are quasiuniform:

for any , where is a positive constant and denote, respectively, the diameter of , the smallest angle of , and the maximum of all diameters of triangles in .

Using the linear finite element space , we can state the standard Galerkin discretisation of (1) which reads as follows.

Find such that for all ,

where .

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.

Find such that for all ,

where

In (7) the term is defined as in which is a constant to be determined and    () are the linear basis functions. Actually, is a bubble function. Moreover,

where represents the area of .

Finally, we define a special energy norm (SD norm) associated with :

We denote by the norm in ; that is,

If , then we drop from the notation.

3. Stability and Energy Estimates

Throughout this subsection, we assume is some positive constant.

3.1. Stability Analysis

The stability properties are a consequence of the following.

Lemma 1. Let the parameter in satisfies

for each , where . Then the discrete bilinear form is coercive; that is,

where .

Proof. By divergence theorem we obtain

Obviously the sum of first three terms is greater than from the condition of (2), and we only need to estimate the last term. Using Hölder inequality, we have

where (14) is based on the definition of , (8), and (11):

Then the proof of the lemma is finished.

3.2. Energy Norm Estimate

We denote by the nodal piecewise linear interpolant to over . From Lemma 1 and the fact of one gets

Denote and estimate the right-hand side of (16) term by term:

where we have used the standard interpolation results (see [9]) and the first inequality of (18) is obtained by

Combining all of these estimates, one gets

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

Consider the bilinear form of SUPG:

where is constant in .

It can be rewritten in the form

where .

Notice that (22) does not correspond to a consistent Petrov-Galerkin formulation in general except in the case when and is linear. Clearly, when is a bubble function,

since vanishes on the boundary of . Thus (22) can be written as

Then SUPG with bubble stabilization coefficients is classified into the consistent Petrov-Galerkin formulation.

Moreover, in general, piecewise constants in SUPG make the test function discontinuous. However, the test functions are continuous in the case of bubble stabilization coefficients and the consequent test space is contained in . In this case, SUPG with bubble stabilization coefficients reads as follows.

Find , such that for all ,

where

On the other hand, SUPG with bubble stabilization coefficients is gradually close to the FEM in the same space as for any .

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 chosen in SUPG with bubble stabilization coefficients and in SUPG. We set and calculate the errors and convergence rates in the subdomain away from layers for Tables 1–16. For parabolic Problems 1 and 2, . For exponential Problems 3 and 4, . The errors in Tables 17–20 are calculated in the whole domain . In the following we only list the results of the case of and since the comparison results between the two methods of other cases like , , and are similar.

Table 1: , Problem 1 (comparison of error).

norm
norm
SD norm

Bubble SUPG
SUPG
Bubble SUPG
SUPG
Bubble SUPG
SUPG

6

12

24

48

96

192

Table 2: , Problem 1 (comparison of convergence rate).

norm
norm
SD norm

Bubble SUPG
SUPG
Bubble SUPG
SUPG
Bubble SUPG
SUPG

6

12

24

48

96

Table 3: , Problem 1 (comparison of error).

norm
norm
SD norm

Bubble SUPG
SUPG
Bubble SUPG
SUPG
Bubble SUPG
SUPG

6

12

24

48

96

192

Table 4: , Problem 1 (comparison of convergence rate).

norm
norm
SD norm

Bubble SUPG
SUPG
Bubble SUPG
SUPG
Bubble SUPG
SUPG

6

12

24

48

96

Table 5: , Problem 2 (comparison of error).

norm
norm
SD norm

Bubble SUPG
SUPG
Bubble SUPG
SUPG
Bubble SUPG
SUPG

6

12

24

48

96

Table 6: , Problem 2 (comparison of convergence rate).

norm
norm
SD norm

Bubble SUPG
SUPG
Bubble SUPG
SUPG
Bubble SUPG
SUPG

6

12

24

48

Table 7: , Problem 2 (comparison of error).

norm
norm
SD norm

Bubble SUPG
SUPG
Bubble SUPG
SUPG
Bubble SUPG
SUPG

6

12

24

48

96

Table 8: , Problem 2 (comparison of convergence rate).

norm
norm
SD norm

Bubble SUPG
SUPG
Bubble SUPG
SUPG
Bubble SUPG
SUPG

6

12

24

48

Table 9: , Problem 3 (comparison of error).

norm
norm
SD norm

Bubble SUPG
SUPG
Bubble SUPG
SUPG
Bubble SUPG
SUPG

6

12

24

48

96

192

Table 10: , Problem 3 (comparison of convergence rate).

norm
norm
SD norm

Bubble SUPG
SUPG
Bubble SUPG
SUPG
Bubble SUPG
SUPG

6

12

24

48

96

Table 11: , Problem 3 (comparison of error).

norm
norm
SD norm

Bubble SUPG
SUPG
Bubble SUPG
SUPG
Bubble SUPG
SUPG

6

12

24

48

96

192

Table 12: , Problem 3 (comparison of convergence rate).

norm
norm
SD norm

Bubble SUPG
SUPG
Bubble SUPG
SUPG
Bubble SUPG
SUPG

6

12

24

48

96

Table 13: , Problem 4 (comparison of error).

norm
norm
SD norm

Bubble SUPG
SUPG
Bubble SUPG
SUPG
Bubble SUPG
SUPG

6

12

24

48

96

Table 14: , Problem 4 (comparison of convergence rate).

norm
norm
SD norm

Bubble SUPG
SUPG
Bubble SUPG
SUPG
Bubble SUPG
SUPG

6

0.87

12

24

48

Table 15: , Problem 4 (comparison of error).

norm
norm
SD norm

Bubble SUPG
SUPG
Bubble SUPG
SUPG
Bubble SUPG
SUPG

6

12

24

48

96

Table 16: , Problem 4 (comparison of convergence rate).

norm
norm
SD norm

Bubble SUPG
SUPG
Bubble SUPG
SUPG
Bubble SUPG
SUPG

6

12

24

48

Table 17: , Problem 1 (comparison of global error).

norm
norm
SD norm

Bubble SUPG
SUPG
Bubble SUPG
SUPG
Bubble SUPG
SUPG

6

12

24

48

96

192

Table 18: , Problem 2 (comparison of global error).

norm
norm
SD norm

Bubble SUPG
SUPG
Bubble SUPG
SUPG
Bubble SUPG
SUPG

6

12

24

48

96

Table 19: , Problem 3 (comparison of global error).

norm
norm
SD norm

Bubble SUPG
SUPG
Bubble SUPG
SUPG
Bubble SUPG
SUPG

6

12

24

48

96

Table 20: , Problem 4 (comparison of global error).

norm
norm
SD norm

Bubble SUPG
SUPG
Bubble SUPG
SUPG
Bubble SUPG
SUPG

6

12

24

48

96

Problem 1. One has

where is chosen such that the solution is

Problem 2. One has

Problem 3. One has

where is chosen such that the solution is

Problem 4. One has

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 the 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.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was partially supported by Promotive Research Fund for Excellent Young and Middle-aged Scientists of Shandong Province (Grant no. BS2012SF012) and Independent Innovation Foundation of Shandong University (Grant no. 2012TS194).

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