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.



duality argument in PDE
Of course the "duality argument" may have different meaning. However a frequent use of the term is as follows. For typical evolution equations, the correct solvability of the Cauchy problem in some space $\Phi$ implies the uniqueness of its solution in the dual space $\Phi'$. This approach gives precise uniqueness classes for equations and systems with constant coefficients. See


Gel'fand, I.M.; Shilov, G.E. Generalized functions. Vol. 3. Theory of differential equations. Academic Press, 1967.


