The upper bound for the effective conductivity is 0.4272727

Notice that the finite element approximation always approximates the value of the energy from above.

The value of |U| seems to be approximately
|U(0,1,1/2)| = 0.32
|U(1/2,1/2,1/2)| = 0.225 * sqrt(3) = 0.39

Numerical results:

For cos(x)+cos(y)+cos(z)=0:
Cycle 0:
   Number of active cells:       512
   Number of degrees of freedom: 4913
   Energy norm:                  0.426537
Cycle 1:
   Number of active cells:       1597
   Number of degrees of freedom: 16207
   Energy norm:                  0.425996
Cycle 2:
   Number of active cells:       5328
   Number of degrees of freedom: 54775
   Energy norm:                  0.425677
Cycle 3:
   Number of active cells:       18453
   Number of degrees of freedom: 191223
   Energy norm:                  0.42555
Cycle 4:
   Number of active cells:       65045
   Number of degrees of freedom: 676326
   Energy norm:                  0.425478

For cos(pi*x)+cos(pi*y)+cos(pi*z)-0.150763*3*cos(pi*x)*cos(pi*y)*cos(pi*z)=0
(2-Nodal approximation of the P-surface)
Cycle 0: 0.427477
Cycle 1: 0.426971
Cycle 2: 0.426796
Cycle 3: 0.426524
Cycle 4: 0.426474

For 2*(cos(pi*x)+cos(pi*y)+cos(pi*z))-0.00956335*6*cos(pi*x)*cos(pi*y)*cos(pi*z)+0.00959441*(cos(2*pi*x)*cos(pi*y)+cos(2*pi*x)*cos(pi*z)+cos(2*pi*y)*cos(pi*x)+cos(2*pi*y)*cos(pi*z)+cos(2*pi*z)*cos(pi*x)+cos(2*pi*z)*cos(pi*y))-0.0181364*2*(cos(3*pi*x)+cos(3*pi*y)+cos(3*pi*z))-0.0292538*2*(cos(2*pi*x)*cos(2*pi*y)*cos(pi*z)+cos(2*pi*x)*cos(pi*y)*cos(2*pi*z)+cos(pi*x)*cos(2*pi*y)*cos(2*pi*z))=0
(5-Nodal approximation of the P-surface)
Cycle 0: 0.427477
Cycle 1: 0.426963
Cycle 2: 0.426797
Cycle 3: 0.426522
Cycle 4: 0.426477

For cos(pi*x)+cos(pi*y)+cos(pi*z)-2.26*cos(pi*x)*cos(pi*y)*cos(pi*z)=0
(which is quite different from the P-surface)
Cycle 0: 0.428237
Cycle 1: 0.427653
Cycle 2: 0.427387
Cycle 3: 0.42735
Cycle 4: 0.427267
At (0,1,1/2), I have 0.5+0.320242, and at (1/2,1/2,1/2), I have 0.5+0.223448

For 0.2260*2*(cos(pi*x)+cos(pi*y)+cos(pi*z))-0.0516*6*cos(pi*x)*cos(pi*y)*cos(pi*z)+0.0196*(cos(2*pi*x)*cos(pi*y)+cos(2*pi*x)*cos(pi*z)+cos(2*pi*y)*cos(pi*x)+cos(2*pi*y)*cos(pi*z)+cos(2*pi*z)*cos(pi*x)+cos(2*pi*z)*cos(pi*y))-0.0027*2*(cos(3*pi*x)+cos(3*pi*y)+cos(3*pi*z)) = 0
(improved nodal approx. of the P surface taken from http://www.bioms.de/schwarz/BicontinuousPhases/TPMS.html)
Cycle 0: 0.428166
Cycle 1: 0.427564
Cycle 2: 0.427228
Cycle 3: 0.427226
Cycle 4: 0.427151
At (0,1,1/2), I have 0.5+0.318147, and at (1/2,1/2,1/2), I have 0.5+0.225027