Can anybody help me understand the "method of weighted residuals" ?

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Pankaj K Mishra

Awesome with simple solved exaplmes. Download here.

Five methods of weighted residuals are applied to a very simple problem and compared.

http://www.me.ua.edu/me611/f02/pdf/mwr.pdf

Mohammad Naisipour

Read Introduction to finite element method of Dr. hinton. It's simple and clear.

Suvabrata Das

You may also follow the first chapter of Boundary elements—an introductory course, by CA Brebbia and J Dominguez.