TITLE: Conceptual difference between strong and weak formulations
QUESTION [30 upvotes]: What are the conceptual differences in presenting a problem in strong or weak form?  For example for a 2D Poisson problem the strong form is:
\begin{split}- \nabla^2 u(\pmb{x}) &= f(\pmb{x}),\quad \pmb{x}\mbox{ in } \Omega,
\\
u(\pmb{x}) &= u_0(\pmb{x}),\quad \pmb{x}\mbox{ on } \partial \Omega\thinspace .\end{split}
where $\Omega$ is the spatial domain and $\partial\Omega$ is the boundary of $\Omega$.
The variational or weak formulation:
\begin{equation}    \int_{\Omega} \nabla u \cdot \nabla v \, \mathrm{d}x =
       \int_{\Omega} fv \, \mathrm{d}x
       \quad \forall v \in \hat{V}.\
\end{equation}
where $\hat{V}$ is the test space and $V$ is the trial space:
\begin{split}\hat{V} &= \{v \in H^1(\Omega) : v = 0 \mbox{ on } \partial\Omega\}, \\
 V      &= \{v \in H^1(\Omega) : v = u_0 \mbox{ on } \partial\Omega\}\thinspace .\end{split}
I know that the weak form is very useful in the Finite Element Method, but I don't understand why. The wikipedia page says that the problem requieres a solution in the sense of a distribution. What does this mean?
 Why are they called Strong and Weak? What is the intuition behind this formulations?
Thanks!

REPLY [4 votes]: Apologies for reviving an old question, but I feel the answers don't sufficiently appeal to physical intuition for what "weak form" is accomplishing. 
Consider systems in the real world where at an interface there can be abrupt changes to the material properties. These real physical systems can violate smoothness constraints for a PDE.  You could say that the classical PDE for a system is too strict when looking at a real, arbitrary, physical system.
One simple example is the temperature diffusion equation:
$$q(x)=-\partial_x T(x)$$
where $q$ is the heat flux for a temperature profile $T$. The conservation of heat flux is:
$$\partial_xq(x)=0$$
which includes the second derivative of $T$.  At a boundary where two materials have different coefficients of thermal conductivity, the first derivative of $T$ becomes discontinuous and the second derivative can't be found numerically. 
We turn the equation into an integral equation to make our solution less strict. We could integrate over the entire domain of $x$ to satisfy the conservation equation, but this isn't strict enough to match with reality (since in reality the integral equation must equal zero at every point, not just simply zero). 
Instead we split the integration region up into segments. 
$$\int_0^1\partial_xq(x)dx=0 \rightarrow \int_0^{.01}\partial_xq(x)dx=0, \int_{.01}^{.02}\partial_xq(x)dx=0,  ...\int_{.99}^1\partial_xq(x)dx=0 $$
Splitting this into more elements locks down the solution with more resolution. Essentially, the number of elements you have is now defining the strictness of your solution. In summary, the weak form allows us to represent and solve for real physical systems that aren't reliably represented but the strictness of the original PDE. 
For more details, I would recommend COMSOL's blog series, beginning with The Strength of the Weak Form, where they describe the weak form and how it's implemented in their software.