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.
We consider the reaction-convection-diffusion problem in 2D
When the mesh Péclet number
One of the most famous stabilized finite element methods is the streamline upwind Petrov-Galerkin method (SUPG). The SUPG proposed by Hughes and Brooks [
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
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
The remainder of this paper is organized as follows. In Section
First we define a finite element space on triangular meshes
Using the linear finite element space
Find
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
In (
Finally, we define a special energy norm (SD norm) associated with
We denote by
Throughout this subsection, we assume
The stability properties are a consequence of the following.
Let the parameter
By divergence theorem we obtain
We denote by
Consider the bilinear form of SUPG:
It can be rewritten in the form
Notice that (
Moreover, in general, piecewise constants
Find
On the other hand, SUPG with bubble stabilization coefficients is gradually close to the FEM in the same space
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.
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 [
For the computations we have chosen
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SD norm | |||
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Bubble SUPG | SUPG | Bubble SUPG | SUPG | Bubble SUPG | SUPG | |
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SD norm | |||
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Bubble SUPG | SUPG | Bubble SUPG | SUPG | Bubble SUPG | SUPG | |
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SD norm | |||
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Bubble SUPG | SUPG | Bubble SUPG | SUPG | Bubble SUPG | SUPG | |
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SD norm | |||
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Bubble SUPG | SUPG | Bubble SUPG | SUPG | Bubble SUPG | SUPG | |
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SD norm | |||
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Bubble SUPG | SUPG | Bubble SUPG | SUPG | Bubble SUPG | SUPG | |
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SD norm | |||
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Bubble SUPG | SUPG | Bubble SUPG | SUPG | Bubble SUPG | SUPG | |
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SD norm | |||
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Bubble SUPG | SUPG | Bubble SUPG | SUPG | Bubble SUPG | SUPG | |
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SD norm | |||
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Bubble SUPG | SUPG | Bubble SUPG | SUPG | Bubble SUPG | SUPG | |
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Bubble SUPG | SUPG | Bubble SUPG | SUPG | Bubble SUPG | SUPG | |
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SD norm | |||
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Bubble SUPG | SUPG | Bubble SUPG | SUPG | Bubble SUPG | SUPG | |
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SD norm | |||
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Bubble SUPG | SUPG | Bubble SUPG | SUPG | Bubble SUPG | SUPG | |
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SD norm | |||
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Bubble SUPG | SUPG | Bubble SUPG | SUPG | Bubble SUPG | SUPG | |
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SD norm | |||
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Bubble SUPG | SUPG | Bubble SUPG | SUPG | Bubble SUPG | SUPG | |
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SD norm | |||
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Bubble SUPG | SUPG | Bubble SUPG | SUPG | Bubble SUPG | SUPG | |
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SD norm | |||
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Bubble SUPG | SUPG | Bubble SUPG | SUPG | Bubble SUPG | SUPG | |
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SD norm | |||
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Bubble SUPG | SUPG | Bubble SUPG | SUPG | Bubble SUPG | SUPG | |
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SD norm | |||
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Bubble SUPG | SUPG | Bubble SUPG | SUPG | Bubble SUPG | SUPG | |
6 |
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SD norm | |||
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Bubble SUPG | SUPG | Bubble SUPG | SUPG | Bubble SUPG | SUPG | |
6 |
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SD norm | |||
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Bubble SUPG | SUPG | Bubble SUPG | SUPG | Bubble SUPG | SUPG | |
6 |
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SD norm | |||
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Bubble SUPG | SUPG | Bubble SUPG | SUPG | Bubble SUPG | SUPG | |
6 |
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One has
One has
One has
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
The authors declare that there is no conflict of interests regarding the publication of this paper.
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).