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Correct minus sign before gv integral |
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Line 73:
:φ<sub>''v''</sub>(L''u'')=φ<sub>''v''</sub>(''g'')
for all ''v'' in V. From now on, we will use ∫<sub>T</sub> for the double integral ∫<sub>0</sub><sup>1</sup>∫<sub>0</sub><sup>1</sup>. One can see, via [[integration by parts]], and noting that because of the periodic boundary condition the non-integral term vanishes
<math>
:ψ(''u'',''v''):=-∫<sub>T</sub>(''u<sub>x</sub>v<sub>x</sub>''+''u<sub>y</sub>v<sub>y</sub>'') = -∫<sub>T</sub>''g·v'' = φ<sub>''v''</sub>(''g'')▼
\int_0^1 \int_0^1 v\left(u_{xx}+u_{yy}\right)dxdy = -\int_0^1\int_0^1 \left(v_x u_x + v_y u_y \right) dxdy.
</math>
Thus we find that the last equality is equivalent to:
▲:ψ(''u'',''v''):=-∫<sub>T</sub>(''u<sub>x</sub>v<sub>x</sub>''+''u<sub>y</sub>v<sub>y</sub>'') =
The function ψ of ''u'' and ''v'' is in fact [[bilinear]], and it is the bilinear form associated with L. The functions ''v'' are called ''test functions''.
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