Pressure-gradient terms

The pressure-gradient terms:

1hξiδξip

contribute to the energy balance as follows:

N3k=1N2j=1N1+12i=12Ju11hξ1δξ1p=N3k=1N2j=1N1i=1Jp1Jδξ1(Jhξ1u1),N3k=1N212j=12N1i=1Ju21hξ2δξ2p=N3k=1N2j=1N1i=1Jp1Jδξ2(Jhξ2u2),N312k=12N2j=1N1i=1Ju31hξ3δξ3p=N3k=1N2j=1N1i=1Jp1Jδξ3(Jhξ3u3).

The sum of these three relations is

N3k=1N2j=1N1i=1Jp{1Jδξ1(Jhξ1u1)+1Jδξ2(Jhξ2u2)+1Jδξ3(Jhξ3u3)},

which is zero because the component inside the wavy parentheses is the incompressibility constraint.