Can anyone please explain the term 'duality argument', or the difference between this term and the weak formulation in PDE analysis? Or give some references?

Occasionally I see this term appears in papers on PDE(existence, uniqueness, continuity proofs of pdes), but I never found a detailed description of it. I hope to get a clear picture on this thing.

Here I am concerned with theoretical analysis in pde, not about numerical schemes (of cos please let me know if they are essentially connected).

Thanks.