3. Variational formulation of the problem

Let \(\Omega\) be an open of \({R}^{3}\), of border \(\partial \Omega ={\partial }_{1}\Omega \cup {\partial }_{2}\Omega \cup {\partial }_{3}\Omega \cup {\partial }_{4}\Omega\) such that,

For \(i\mathrm{\ne }j\) and \(i,j\mathrm{=}\mathrm{1,}\mathrm{...},4\), we have: \({\partial }_{i}\Omega \cap {\partial }_{j}\Omega \text{=}\varnothing\).

Let \(\psi\) still be a sufficiently regular function that is cancelled on \({\mathrm{\partial }}_{4}\Omega\): \(\psi \mathrm{\in }V\mathrm{=}\left\{y\text{régulière}\text{et}\psi {\mathrm{\mid }}_{{\mathrm{\partial }}_{4}\Omega }\mathrm{=}0\right\}\).

Multiply both sides of the equation [éq 2-1] by \(\psi\) and then integrate on \(\Omega\). An integration by parts then gives:

\(\begin{array}{ccc}\underset{\Omega }{\int }\mathrm{Qyd}\Omega & \text{=}\underset{\Omega }{\int }V\text{.}\text{grad}u(T)\mathrm{yd}\Omega -& \underset{\Omega }{\int }\text{div}(k(T)\text{grad}T)\mathrm{yd}\Omega \\ & \text{=}\underset{\Omega }{\int }V\text{.}\text{grad}u(T)\mathrm{yd}\Omega +& \underset{\Omega }{\int }k(T)\text{grad}T\text{.}\text{grad}\mathrm{yd}\Omega -\underset{\partial \Omega -{\partial }_{4}\Omega }{\int }(k(T)\frac{\partial T}{\partial n}y)\mathrm{dG}\end{array}\) eq 3-1

since \(\psi\) sucks on \({\partial }_{4}\Omega\).

Hence, taking into account the boundary conditions [éq 2-2], [éq 2-3] and [éq 2-4], the variational formulation of the reference problem which is given by the following equation:

\(\forall \psi \in V\)

\(\begin{array}{}\begin{array}{cc}\underset{\Omega }{\int }k(T)\text{grad}T\text{.}\text{grad}\mathrm{yd}\Omega & +\underset{\Omega }{\int }V\text{.}\text{grad}u(T)\mathrm{yd}\Omega +\underset{{\partial }_{1}\Omega }{\int }\mathrm{gTydG}-\underset{{\partial }_{3}\Omega }{\int }\alpha (T)\mathrm{ydG}\end{array}\\ =\underset{\Omega }{\int }\mathrm{Qyd}\Omega +\underset{{\partial }_{1}\Omega }{\int }{\gamma T}_{\text{ext}}\mathrm{ydG}+\underset{{\partial }_{2}\Omega }{\int }{q}_{0}\mathrm{yd}\Omega ,\end{array}\) eq 3-2