Strong formulation of the governing (partial differential equation) equation of steady-state transfer conduction
$ \nabla \cdot \mathbf{\mathbf{k} \nabla T} + Q = 0 $, (1)
in which $T$ is the temperature; $\mathbf{k} = diag(k_{xx}, k_{yy}, k_{zz})$ is the tensor of thermal conductivity; $Q$ is the heat sink/heat source.
Boundary conditions are given by
$T = \bar{T}$, on $\Gamma_{1}$: Dirichlet boundary, (2)
$(\mathbf{k} \nabla T ) \cdot \mathbf{n} = \bar{q}$, on $\Gamma_{2}$: Neumann boundary, (3)
$(\mathbf{k} \nabla T ) \cdot \mathbf{n} = h(T_{a} – T) $, on $\Gamma_{3}$: convection boundary, (4)
Here, $\bar{T}$ is the specified value of temperature, $\bar{q}$ is the prescribed heat flux, $\mathbf{n}$ is the outward unit normal vector of the boundary, $T_{a}$ is the ambient temperature.
Weak formulation can be obtained by multplying both sides of the governing equation (Eq. (1)) are then multiplied with a test function $\delta T$. Integration is then conducted over the entire domain. (Note: the test function $\delta T$ is arbitrary, infinitesimal, and it satifies the boundary conditions)
$\int \limits _{\Omega} \nabla \cdot (\mathbf{k} \nabla T) \delta T$ d $\Omega$ + $\int \limits _{\Omega} Q \delta T$ d $\Omega$ = 0, (5)
Integration by parts gives
$\int \limits _{\Omega} \nabla \cdot (\mathbf{k} \nabla T) \delta T$ d $\Omega$ = $\int \limits _{\Omega} \nabla \cdot \left( (\mathbf{k} \nabla T) \delta T \right)$ d $\Omega$ – $\int \limits _{\Omega} (\mathbf{k} \nabla T) \nabla \delta T $ d $\Omega$. (6)
Gauss theorem is then applied
$\int \limits _{\Omega} \nabla \cdot \left( (\mathbf{k} \nabla T) \delta T \right)$ d $\Omega$ = $\int \limits _{\Gamma} \delta T (\mathbf{k} \nabla T) \cdot \mathbf{n} $ d $\Gamma$, (7)
Substitution of Eq. (6) and (7) into Eq. (5) leads to
$\int \limits _{\Gamma} \delta T (\mathbf{k} \nabla T) \cdot \mathbf{n} $ d $\Gamma$ + $\int \limits _{\Omega} Q \delta T$ d $\Omega$ -$\int \limits _{\Omega} (\mathbf{k} \nabla T) \nabla \delta T $ d $\Omega$ = 0, (8)
Eq. (8) is the weak formulation of the governing equation. Note that the surface term (the first term in Eq. (8)) depends on the boundary conditions. Obviously, it vanishes on free boundary.
The weak formulation is essential for solving the governing equation using numerical methods, such as Finite Element Method (FEM), or Meshfree methods.