I Metric for knowing when numerical BC is "good"

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I think that one usually tries to restrict the solution such that the boundary conditions are exactly (up to round-of errors) satisfied.
For example, if you have an expansion
$$f(x)=\sum_{k=1}^n a_k \phi_k(x)$$ and a condition ##f(0)=0##, you can select the basis functions such that the condition is satisfied.
However, I'm not sure if it is possible in your case.
 

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