We discuss several stabilized finite element methods, which are penalty, regular, multiscale enrichment, and local Gauss integration method, for the steady incompressible flow problem with damping based on the lowest equalorder finite element space pair. Then we give the numerical comparisons between them in three numerical examples which show that the local Gauss integration method has good stability, efficiency, and accuracy properties and it is better than the others for the steady incompressible flow problem with damping on the whole. However, to our surprise, the regular method spends less CPUtime and has better accuracy properties by using Crout solver.
In this paper, we will consider the following steady incompressible flow problem with damping: seek
These equations describe various physical situations such as porous media flow, drag or friction effects, and some dissipative mechanisms from the resistance to the motion of the flow. If the damping system is different, energy dissipation is different. So the application of these equations is very extensive in the daily life (see [
It is well known that it is very difficult to compute some PDEs directly while numerical method plays an important role in these problems, so developing an efficient and effective computational method for solving the incompressible flow problem has practical significance and has drawn the attention of many researchers (see [
In the analysis and practice of employing mixed finite element methods in solving the incompressible flow problems, the infsup condition has played an important role because it ensures stability and accuracy of the underlying numerical schemes. Pairs of finite element spaces that are used to approximate the velocity and the pressure unknown are said to be stable if they satisfy the infsup condition. Intuitively speaking, the infsup condition is something that enforces a certain correlation between two finite element spaces so that they both have the required properties when employed for the incompressible flow problems. However, due to computational convenience and efficiency in practice, some mixed finite element pairs which do not satisfy the infsup condition are also popular. Thus, much attention has been paid to the study of the stabilized method for the Stokes problem [
In the present years, studies have focused on stabilization techniques [
A brief outline of the rest of our paper is organized as follows: in Section
We will use the following Hilbert spaces:
Now, for convenience, we introduce the finite subspaces
There is a constant
There is a constant
In this section, we will give several stabilized mixed finite element algorithms to show their different aspects and several ways have been used to stabilize the lowest equalorder finite element space pair as follows (see [
Let
Let
Next we present four kinds of stabilized finite element method for the steady incompressible flow problem with damping.
For convenience, we let
So the corresponding discrete variational formulation of (
Here,
Given
In the nonconforming case, the discrete nonconforming formulation for the steady incompressible flow problem with damping is to seek
Note that (
Seek
Similar to (
It is well known that, if
By using the regularity assumptions and wellestablished techniques for velocity and pressure [
In this section, we will give three numerical tests to confirm the numerical theory developed in the previous section. In the given experiments, the pressure and velocity are approximated by the lowest equalorder finite element pairs defined with respect to the same uniform triangulation; that is, the mesh consists of triangular elements that are obtained by dividing
In this example, we consider the exact solution problem firstly. Let the domain
Numerical results for the penalty method with

CPUtime 





12  0.375 


—  — 
24  3.265 


1.0088  1.0023 
36  10.39 


1.0045  0.9990 
48  27.562 


1.0031  0.9968 
60  61.25 


1.0023  0.9942 
Results got from the regular method with

CPUtime 





12  0.282 


—  — 
24  1.641 


1.4344  1.6686 
36  4.657 


1.4628  1.5605 
48  10.313 


1.4550  1.5337 
60  21.094 


1.4431  1.5207 
Results got from the multiscale enrichment method with

CPUtime 





12  2.496  0.8924  1.1544  —  — 
24  12.746  0.6523  0.2823  0.4520  2.0319 
36  37.175  0.4853  0.2094  0.7293  0.7370 
48  83.648  0.3737  0.1713  0.9091  0.6977 
60  161.32  0.2996  0.1445  0.9900  0.7624 
Results got from the local Gauss integration method with

CPUtime 





12  0.406 


—  — 
24  4.578 


1.0353  1.6255 
36  8.5 


1.0488  1.1078 
48  16.328 


1.0493  1.0635 
60  27.39 


1.0414  1.1797 
Results got from the local Gauss integration method with the nonconforming element with

CPUtime 





12  0.297  3.6007 

—  — 
24  2.500  1.8851 

0.9336  1.8198 
36  5.735  1.2682 

0.9776  1.9217 
48  11.469  0.9543 

0.9887  1.9534 
60  21.719  0.7646 

0.9932  1.9681 
From this experiment, we can learn the following several points.
From Tables
Besides, from the convergence results on this example, we can see that regular and multiscale enrichment methods are not better than other methods. And the nonconforming local Gauss integration method shows the best numerical stability.
In this test, we test a popular benchmark problem, the liddriven flow. Let the computation be carried out in the region
In this example, we simulate the referred physics phenomena. In Figures
Velocity streamlines (a) and pressure level lines (b) for the penalty method with
Velocity streamlines (a) and pressure level lines (b) for the regular method with
Velocity streamlines (a) and pressure level lines (b) for the multiscale enrichment method with
Velocity streamlines (a) and pressure level lines (b) for the local Gauss integration method with
Velocity streamlines (a) and pressure level lines (b) for the local Gauss integration method with the nonconforming element with
In this example, we choose the exact solution for the velocity and pressure in the unit square as follows:
Numerical results for the penalty method with

CPUtime 





12  0.219 


—  — 
24  1.625 


1.0114  0.9478 
36  5.719 


1.0077  0.9682 
48  15.64 


1.0056  0.9771 
60  32.281 


1.0044  0.9820 
Results got from the local Gauss integration method with

CPUtime 





12  0.219 


—  — 
24  3.109 


1.22023  2.01511 
36  5.437 


1.15491  2.00432 
48  9.578 


1.18138  1.99853 
60  16.062 


1.23219  1.99177 
Numerical solutions for the penalty method (a) and pressure level lines for the local Gauss integration method (b) with
We have used several stabilized mixed finite element methods in solving the steady incompressible flow problem with damping based on the lowest equalorder pairs in this paper. We give some conclusions by comparing numerically as follows.
All of those methods’ stability and efficiency depends on their parameter values. In terms of the penalty method, the smaller its parameter value, the more stable the method. However, it can not be too small, otherwise, the condition number of the system matrix arising from this method will become too large to be solve. For the regular and multiscale enrichment methods whose performance heavily depends on the choice of the stabilization parameters, however, it is difficult to choose fine parameters in fact. What is more, a poor choice of these stabilization parameters can also lead to serious deterioration in the convergence rates. The local Gauss integration method is free of stabilization parameters and shows numerically the best performance among the methods considered for the given problem.
This work is in part supported by the NSF of China (nos. 11271313 and 61163027), the China Postdoctoral Science Foundation (nos. 2012M512056 and 2013M530438), the Key Project of Chinese Ministry of Education (no. 212197), the NSF of Xinjiang Province (no. 2013211B01), and the Doctoral Foundation of Xinjiang University (no. BS120102).