An adaptive finite element method is presented for the stationary incompressible thermal flow problems. A reliable a posteriori error estimator based on a projection operator is proposed and it can be computed easily and implemented in parallel. Finally, three numerical examples are given to illustrate the efficiency of the adaptive finite element method. We also show that the adaptive strategy is effective to detect local singularities in the physical model of square cavity stationary flow in the third example.
As we all know the stationary incompressible thermal flow occurs in many fields of the industrial sectors and natural world. The mathematical model of this flow can be described by a set of coupled equations, which consist of the conservation of mass, momentum, and energy equations. The stationary incompressible thermal flow problems constitute an important system of equations in atmospheric dynamics and dissipative nonlinear system of equations. The main difficulties for the numerical simulation of the stationary incompressible thermal flow problem include not only the incompressibility and strong nonlinearity, but also the the coupling between the energy equation and the equations governing the fluid motion. And the numerical methods for solving these problems have attracted the attention of many researchers. Many authors have worked hard to study for a great variety of efficient numerical schemes for these equations [
In the numerical simulation of stationary incompressible thermal flow problems, still a big challenge is how to increase the accuracy of the numerical approximations for the solutions. The overall accuracy of numerical approximations often deteriorates due to local singularities or large variations in small scales. Motivated by this phenomenon, a direct strategy is that the grids near the critical regions are refined adaptively to improve the quality of the approximate solutions. In 1978, Babuska and Rheinboldt developed a mathematical theory for a class of a posteriori error estimator of finite element solutions [
This paper is organized as follows. In Section
In this section, we consider the stationary incompressible thermal flow problems in twodimensions. The viscous incompressible flow and heat transfer for the fluid satisfy the incompressible NavierStokes equations coupled with the energy conservation equation under the Boussinesq hypothesis is as follows.
Find
We introduce the following Hilbert spaces
For the finite element discretization, let
With the previous notations, the Galerkin finite element discretization of (
For problem (
There exists a constant
Assuming
for all
for all
Under the assumption of
Under the assumption of
Before deriving the projection estimator, we define the orthogonal projection operator [
Now, based on the residual between the gradient of the finite element solutions velocity component
Then the global error estimator is given by
Based on orthogonal projection properties of operator
Our local projection error estimator can be computed more precisely and explicitly based on two local Gauss integrations technique presented in [
Before giving the global upper bound, we recall a lemma in [
There exists a positive constant C such that
This lemma was given in [
We note that
Additionally, under some mild assumption on
As a result, using the previous inverse inequality (
Then, this completes the proof of Lemma
For
There is a constant C depending on the smallest angle in the triangulation
We assume that there are two positive constants
Then, the theorem is proved.
In this section, we present three examples to illustrate the effectiveness of the adaptive method. In all experiments, the nonlinear systems are solved by the Newton iteration [
We make use of the Newton iteration scheme presented in [
For the sake of clarity, the main idea is offered about the refinement strategy presented in [
For simplicity, we choose
The computation adaptive strategy here is to choose a tolerance
If
Compute the
For convenience of presentation, we introduce the following notation:
For this problem the analytical solutions were taken to be
Tables
All in all, our projection a posteriori error estimator can assess the domain of rapid solution variation and refine them to improve the global accuracy with less meshes.
Uniform meshes for




err 


128  0.0403662  0.0127483  0.118008  0.125371  0.777592 
288  0.0179335  0.00576264  0.0553006  0.0584207  0.532955 
512  0.0100866  0.00326292  0.0317188  0.0334435  0.403894 
1152  0.00448271  0.00145755  0.0143002  0.015057  0.27133 
2048  0.00252149  0.000821408  0.00808489  0.00850871  0.204051 
2592  0.00199229  0.000649351  0.0063969  0.00673136  0.181512 
A series of adapted meshes for




err 


232  0.0202677  0.00817272  0.0752798  0.0783876  0.589817 
496  0.0157563  0.00659604  0.012231  0.0210087  0.331746 
906  0.00893535  0.00428475  0.00900528  0.0133901  0.237672 
1628  0.0077005  0.00420461  0.00351317  0.00945086  0.176478 
2987  0.003767  0.00197233  0.00270253  0.00503825  0.131168 
5714  0.00193568  0.000917294  0.00104072  0.00238147  0.0938836 
Uniform meshes for




err 


128  0.0404411  0.0127525  2.11996  2.12038  2.84188 
288  0.0179432  0.00576368  1.55127  1.55139  2.70273 
512  0.0100887  0.00326323  1.12379  1.12384  2.46931 
1152  0.00448294  0.00145758  0.631091  0.631109  1.99439 
1800  0.00286897  0.000934275  0.437859  0.43787  1.7078 
2592  0.00199231  0.000649354  0.31882  0.318827  1.48158 
4608  0.00112066  0.000365579  0.188543  0.188546  1.15963 
7200  0.000717221  0.000234067  0.123639  0.123641  0.947138 
8712  0.000592744  0.000193468  0.102968  0.10297  0.866675 
A series of adapted meshes for




err 


232  0.0202895  0.00817464  1.72925  1.72939  2.72944 
445  0.0182222  0.007264  0.388509  0.389004  1.63213 
1049  0.0182522  0.00735227  0.058678  0.0618895  0.693979 
2154  0.0129152  0.00591016  0.0356371  0.0383632  0.449907 
3745  0.00694219  0.00345662  0.0187145  0.0202577  0.340617 
6457  0.00600729  0.00300182  0.0123216  0.0140328  0.259971 
Exact solution for temperature.
Final adapted mesh (a) and the numerical solution for temperature (b) for
We choose the known solutions as follows,
Figure
To show the effectiveness of our adaptive method and the adaptive procedures based on the residual posteriori error estimator [
We present the numerical results for Example
Comparing Tables
Uniform meshes for




err 

Eff  CPU 

288  0.528492  5.54052  5.48443  7.81381  10.6499  1.36296  1.688 
512  0.388401  2.95819  2.90888  4.16693  10.8996  2.61573  2.844 
1152  0.177667  0.990161  0.954468  1.38672  8.21263  5.92234  6.594 
2048  0.0908647  0.573599  0.553251  0.802096  6.13502  7.64874  11.688 
4608  0.037012  0.304226  0.296276  0.426266  4.04822  9.49692  28.063 
8192  0.0202685  0.184336  0.180081  0.258495  3.03164  11.728  50.922 
10368  0.0159031  0.148582  0.145263  0.208401  2.69474  12.9306  83.172 
11858  0.0138586  0.131136  0.128251  0.183948  2.51991  13.699  79.578 
12800  0.0128167  0.122067  0.119403  0.171236  2.42552  14.1648  89.109 
A successive sequence of adapted meshes for




err 

Eff  CPU 

238  0.651433  6.32201  6.31274  8.95784  10.1411  1.13209  1.422 
427  0.161466  0.588001  0.577591  0.839898  6.03079  7.18038  2.531 
906  0.0891423  0.217318  0.20884  0.314305  2.37486  7.55591  5.375 
2074  0.0376693  0.13206  0.126842  0.186943  1.45756  7.79682  12.39 
3808  0.0143992  0.0623338  0.0619315  0.0890413  1.04981  11.7901  23.109 
6994  0.0115494  0.0437032  0.0428615  0.0622934  0.761153  12.2188  44.047 
A successive sequence of adapted meshes based on a residual error estimation for




err 

Eff  CPU 

238  0.651433  6.32201  6.31274  8.95784  74.4933  8.316  1.609 
286  0.167818  0.608207  0.594685  0.867023  8.75867  10.102  1.672 
669  0.0785628  0.279946  0.274727  0.400021  4.11085  10.2766  3.89 
1212  0.0425637  0.238712  0.237269  0.339252  3.31492  9.77129  7.156 
2309  0.041893  0.14484  0.138815  0.204947  2.12799  10.3831  13.672 
2962  0.0186839  0.0952816  0.0945824  0.135549  1.34394  9.91483  17.984 
5533  0.0194194  0.0628666  0.0620324  0.0904286  0.891394  9.85743  33.891 
7468  0.00787077  0.045213  0.0443621  0.0638291  0.618652  9.69232  46.922 
12045  0.00932973  0.0325655  0.0319025  0.0465331  0.450524  9.68181  77.109 
15180  0.00419643  0.022182  0.0221098  0.0315989  0.306211  9.69056  99.094 
Uniform meshes for




err 

Eff  CPU 

288  0.970876  5.40546  5.24747  7.59589  8.74174  1.15085  1.641 
512  0.663034  3.11391  2.97829  4.35961  9.4165  2.15994  2.875 
1152  0.329059  1.10894  1.00909  1.53502  7.4726  4.86808  6.641 
2048  0.169734  0.609773  0.548335  0.837439  5.6408  6.73578  11.797 
4608  0.0672255  0.313963  0.289187  0.432113  3.72585  8.62239  28.218 
8192  0.0364009  0.189996  0.176725  0.262021  2.79027  10.649  51.594 
10368  0.0284943  0.153168  0.142818  0.211351  2.48038  11.7358  70.953 
11858  0.0248071  0.1352  0.126205  0.186607  2.31957  12.4302  82.297 
12800  0.0229315  0.12586  0.117553  0.173739  2.23276  12.8512  86.297 
A successive sequence of adaptive meshes for




err 

Eff  CPU 

238  1.06895  6.04125  5.92896  8.53182  8.25774  0.967876  1.5 
417  0.132011  0.558502  0.539777  0.787851  5.4622  6.93303  2.453 
912  0.0502047  0.167527  0.153125  0.23245  2.01927  8.68689  5.469 
2168  0.0301542  0.0959719  0.0889449  0.13428  1.12367  8.3681  13.016 
4158  0.0137853  0.0438102  0.0398118  0.0607812  0.783331  12.8877  25.594 
7906  0.00823037  0.0287469  0.0265661  0.0399986  0.556596  13.9154  50.734 
A successive sequence of adaptive meshes based on a residual error estimation for




err 

Eff  CPU 

238  1.06895  6.04125  5.92896  8.53182  75.8374  8.8887  1.375 
253  0.138697  0.552635  0.537633  0.783385  8.51898  10.8746  1.469 
568  0.0835765  0.245759  0.224761  0.343365  3.69854  10.7714  3.234 
1045  0.0421669  0.180767  0.17622  0.255945  2.59402  10.1351  6.016 
2022  0.0471101  0.135906  0.124863  0.190475  1.9947  10.4723  11.89 
2441  0.0217445  0.0960252  0.0947848  0.136667  1.34754  9.86003  14.437 
4011  0.0222083  0.0678677  0.0615177  0.0942532  0.946418  10.0412  24.032 
5523  0.0109319  0.0484351  0.0486974  0.0695478  0.672826  9.6743  33.859 
8373  0.01082  0.0322648  0.0297083  0.0451738  0.45166  9.99828  52.172 
11033  0.0052382  0.0226784  0.0225007  0.0323733  0.317636  9.81166  69.656 
Exact solution for
Exact solution for
The successive adaptive refined meshes for
The successive adaptive refined meshes for
Numerical solution for
Numerical solution for
Comparing Tables
Combining all discussions above, we can derive a conclusion that these results suffice to show that our adaptive strategy obtains much better approximation with less meshes and CPU time. That is to say we can save lots of work by our adaptive procedures.
The last example is a physical model of square cavity stationary flow, which is a popular benchmark problem for testing numerical schemes. The side length of the square cavity and the boundary conditions are given in Figure
The left part of Figure
Physical model of the cavity flows.
Initial mesh (a). The first adaptation mesh (b).
The third adaptation mesh (a). The streamline of velocity numerical solutions after three levels of adaptation mesh refinement (b).
Numerical isobar of pressure solution (a) and numerical isotherm of temperature solutions (b) after three levels of adaptation mesh refinement.
In general, we cannot know the exact solutions of the stationary incompressible thermal flow problems. We take the numerical solutions using the
Vertical midline for
In this paper, we present an adaptive finite element method based on a posteriori error estimator for the stationary incompressible thermal flow problems. This error estimator is constructed by a projection operator and can be calculated explicitly and precisely by the difference of two Gauss integrations technique. The discussions and the numerical tests indicate that the projection error estimator is effective and efficient. Whereas there are still many questions in further analysis, such as the convergence and optimality of the adaptive finite element methods.
This work is supported by the National Natural Science Foundation of China (nos. 11001216, 11371288, 11371289, 11272251, and 11271298) and the China Scholarship Council (no. 201206285018).