the weak formulation

Suppose a thin elastic membrane is pinned along a wire frame and sags under its own weight. The 'strong' way to describe its shape is a differential equation that must hold at EVERY single point: the curvature here, the load there, balanced exactly. But pointwise demands are fussy — they need the solution to be smooth enough to differentiate twice, and that is often more than physics actually guarantees. The weak formulation asks for less: instead of demanding the equation hold at every point, it demands that the equation hold on AVERAGE when weighed against every smooth 'test' shape.

Concretely, take a PDE like -u'' = f (a 1D model of the sagging membrane). You multiply both sides by an arbitrary test function v that vanishes where the value of u is prescribed, then integrate over the domain: integral of (-u'' v) = integral of (f v). Now integrate the left side by parts. The boundary term drops out (because v vanishes at the fixed ends), and one derivative moves from u onto v, leaving integral of (u' v') dx = integral of (f v) dx. This must hold for EVERY admissible v. The miracle is the trade: the new equation involves only first derivatives of u, not second, so it makes sense even for solutions with kinks — exactly the piecewise-polynomial functions a finite element method will use.

This reformulation is the doorway to the finite element method. Because the weak form only needs first derivatives and is stated as an integral identity, you can restrict v (and u) to a finite-dimensional space of simple shape functions and get a solvable linear system (the Galerkin method). It is honest to note that 'weak' does not mean 'inferior': for a wide class of problems the weak and strong solutions coincide when the strong solution exists, and the weak form can have a sensible solution even when no twice-differentiable strong solution does — which is a feature, not a bug, when modelling cracks, corners, or jumps in material.