Link to the source code for this example:

The console output of the program:

***************************************************************
* Running Example adjoints_ex6:
*   ./example-opt
***************************************************************

Started /home/jwpeterson/software/libmesh_build/opt_real/examples/adjoints/adjoints_ex6/.libs/example-opt
Reading in and building the mesh
Building system
Initializing systems
 Mesh Information:
  elem_dimensions()={2}
  spatial_dimension()=2
  n_nodes()=81
    n_local_nodes()=81
  n_elem()=20
    n_local_elem()=20
    n_active_elem()=16
  n_subdomains()=1
  n_partitions()=1
  n_processors()=1
  n_threads()=1
  processor_id()=0

 EquationSystems
  n_systems()=1
   System #0, "PoissonSystem"
    Type "Implicit"
    Variables="T"
    Finite Element Types="LAGRANGE"
    Approximation Orders="SECOND"
    n_dofs()=81
    n_local_dofs()=81
    n_constrained_dofs()=32
    n_local_constrained_dofs()=32
    n_vectors()=1
    n_matrices()=1
    DofMap Sparsity
      Average  On-Processor Bandwidth <= 13.4444
      Average Off-Processor Bandwidth <= 0
      Maximum  On-Processor Bandwidth <= 25
      Maximum Off-Processor Bandwidth <= 0
    DofMap Constraints
      Number of DoF Constraints = 32
      Average DoF Constraint Length= 0
      Number of Node Constraints = 0

  Nonlinear solver converged, step 0, residual reduction 9.99726e-11 < 1e-09
Adaptive step 0, we have 16 active elements and 49 active dofs.
Postprocessing:
The computed QoI 0 is 36.230655305114681
The relative error in QoI 0 is 2.9365219718146776
Computing the error estimate using the Adjoint Refinement Error Estimator

The flux QoI 0 is: -36.230655307215329

The computed relative error in QoI 0 is 2.3733244806489893
The effectivity index for the computed error in QoI 0 is 0.8082093386082686
Refined mesh to 64 active elements and 225 active dofs.
  Nonlinear solver converged, step 0, residual reduction 9.2613328277368742e-11 < 1.0000000000000001e-09
Adaptive step 1, we have 64 active elements and 225 active dofs.
Postprocessing:
The computed QoI 0 is 37.168893495603982
The relative error in QoI 0 is 3.8747601623039785
Computing the error estimate using the Adjoint Refinement Error Estimator

The flux QoI 0 is: -37.16889349635386

The computed relative error in QoI 0 is 3.8484173207715724
The effectivity index for the computed error in QoI 0 is 0.99320142655829768
Refined mesh to 256 active elements and 961 active dofs.
  Nonlinear solver converged, step 0, residual reduction 1.2147289955185059e-10 < 1.0000000000000001e-09
Adaptive step 2, we have 256 active elements and 961 active dofs.
Postprocessing:
The computed QoI 0 is 33.845391613283802
The relative error in QoI 0 is 0.55125827998379862
Computing the error estimate using the Adjoint Refinement Error Estimator

The flux QoI 0 is: -33.845391613994209

The computed relative error in QoI 0 is 0.55062439441880073
The effectivity index for the computed error in QoI 0 is 0.99885011148491676

***************************************************************
* Done Running Example adjoints_ex6:
*   ./example-opt
***************************************************************