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In the method of weighted residual applied to boundary value problems, is it necessary for the basis function to satisfy all of the boundary conditions? Will it work even if it does not satisfy all of the boundary conditions?

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The test functions must come from the space of variations, i.e., if your solution has to satisfy $$ u|_{\partial \Omega}=g $$ then your test functions have to satisfy $$ v|_{\partial \Omega}=0. $$

This is a question everyone has trouble with at first, which is why I expand on it in quite some detail in lecture 21.5 here.

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  • $\begingroup$ Could you please explain a bit more on the 'space of variations'? If $u|_{\partial \Omega}=0$, can $v|_{\partial \Omega}\neq0$? $\endgroup$
    – adipro
    Apr 2 '15 at 13:25
  • $\begingroup$ No. You'll understand from the video. $\endgroup$ Apr 2 '15 at 17:36

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