Dr. Bangerth, 

Thanks so much for your reply that pointed out my problem.
Quadrature is applied to the residual expression, which includes both 
analytical and approximation.
"right-hand-side" showed me that approximation can be obtained at 
quadrature point, before getting Jacobians and weighted.
This is wonderful! I will keep up my work.

Happy Holidays!
Best,
Judy
On Wednesday, December 22, 2021 at 12:27:59 AM UTC-5 Wolfgang Bangerth 
wrote:

> On 12/21/21 8:59 AM, Judy Lee wrote:
> > 
> > Can I make an expression u(x) for 1D analytical solution, that can take 
> > *q_index* for evaluating u(@ *q_index*), while skip obtaining *q_index* 
> > locations/points value?
>
> What is 'q_index'? It's usually the index (an integer) of a quadrature 
> point 
> on a cell. But u(x) is a function of a spatial variable (a point in the 
> domain). You can't ask for u(q_index). You have to say what that point is 
> where you want to evaluate the function.
>
> If I understand right then all you need is the ability to evaluate the 
> exact 
> solution at the location of a quadrature point. There are many examples of 
> how 
> this is done in the example programs (just search for where a right hand 
> side 
> is evaluated, for example). For evaluating the error u-uh, you can also 
> take a 
> look at step-7.
>
> Best
> WB
>
> -- 
> ------------------------------------------------------------------------
> Wolfgang Bangerth email: bang...@colostate.edu
> www: http://www.math.colostate.edu/~bangerth/
>
>

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