<h1>Results</h1>

We can run the program with the following input file:
@code
# Listing of Parameters
# ---------------------
# Polynomial degree of the finite element to be used
set Finite element degree     = 2

# The number of energy different groups considered
set Number of energy groups   = 2

# Inner power iterations are stopped when the change in k_eff falls below this
# tolerance
set Power iteration tolerance = 1e-12

# Number of refinement cycles to be performed
set Refinement cycles         = 5
@endcode
The output of this program then consists of the console output, a file 
named ``convergence_table'' to record main results of mesh iteration, 
and the graphical output in vtu format.

The console output looks like this:
@code
Cycle 0:
   Numbers of active cells:       1156 1156 
   Numbers of degrees of freedom: 4761 4761 
Iter number:1 k_eff=319.375676634307 flux ratio=6.836246075631 max_thermal=1.433899030144
Iter number:2 k_eff=0.834072546055 flux ratio=5.204601882141 max_thermal=0.004630925876
Iter number:3 k_eff=0.862826188043 flux ratio=4.645051765984 max_thermal=0.005380396338
Iter number:4 k_eff=0.877887920967 flux ratio=4.318030683875 max_thermal=0.006005512201
Iter number:5 k_eff=0.887161559547 flux ratio=4.256596788174
max_thermal=0.006639443035
...
Iter number:69 k_eff=0.906841960370 flux ratio=4.384056022578 max_thermal=0.008466414246
Iter number:70 k_eff=0.906841960371 flux ratio=4.384056022582 max_thermal=0.008466414246
Cycle 1:
   Numbers of active cells:       1156 2380 
   Numbers of degrees of freedom: 4761 10667 
Iter number:1 k_eff=0.906838267472 flux ratio=4.385474405124 max_thermal=0.008463675976
Iter number:2 k_eff=0.906837892433 flux ratio=4.385486158840 max_thermal=0.008463675386
...
Cycle 11:
   Numbers of active cells:       11749 47074 
   Numbers of degrees of freedom: 50261 204523 
Iter number:1 k_eff=0.906805395149 flux ratio=4.384872231023 max_thermal=0.008464861813
...Iter number:32 k_eff=0.906834736551 flux ratio=4.384846081796 max_thermal=0.008465019607
Iter number:33 k_eff=0.906834736552 flux ratio=4.384846081800 max_thermal=0.008465019607
@endcode

We see that power iteration does converge faster after cycle 0 due to the initialization 
with solution from last mesh iteration. 
The contents of ``convergence_table'' are,
@code
0 4761 4761 0.906841960371 4.384056022582
1 4761 10667 0.906837901031 4.385489087760
2 4761 18805 0.906836075928 4.385466647499
3 6629 27301 0.906835500111 4.385404580865
4 12263 48095 0.906835001796 4.385381798734
5 17501 69297 0.906834858174 4.384853823414
6 19933 78605 0.906834824060 4.384850658788
7 23979 93275 0.906834787556 4.384848379257
8 30285 117017 0.906834761604 4.384846544947
9 40087 154355 0.906834746216 4.384846083191
10 45467 179469 0.906834740155 4.384846005044
11 50261 204523 0.906834736552 4.384846081800
@endcode
The meanings of columns are: number of mesh iteration, numbers of degrees of
 freedom of fast energy group, numbers of DoFs of thermal group, converged 
k-effective and the ratio between maximum of fast flux and maximum of thermal one.

The grids of fast and thermal energy groups at mesh iteration #9 look
as follows:

<img width="400" src="https://www.dealii.org/images/steps/developer/step-28.grid-0.9.order2.png" alt="">
&nbsp;
<img width="400" src="https://www.dealii.org/images/steps/developer/step-28.grid-1.9.order2.png" alt="">

We see that the grid of thermal group is much finer than the one of fast group. 
The solutions on these grids are, (Note: flux are normalized with total fission
source equal to 1)

<img width="400" src="https://www.dealii.org/images/steps/developer/step-28.solution-0.9.order2.png" alt="">
&nbsp;
<img width="400" src="https://www.dealii.org/images/steps/developer/step-28.solution-1.9.order2.png" alt="">

Then we plot the convergence data with polynomial order being equal to 1,2 and 3.

<img src="https://www.dealii.org/images/steps/developer/step-28.convergence.png" alt="">

The estimated ``exact'' k-effective = 0.906834721253 which is simply from last
mesh iteration of polynomial order 3 minus 2e-10. We see that h-adaptive calculations
deliver an algebraic convergence. And the higher polynomial order is, the faster mesh 
iteration converges. In our problem, we need smaller number of DoFs to achieve same
accuracy with higher polynomial order.
