Dear Kyusik,

Firstly (and just so that we're all clear on what you're trying to 
implement), was I correct that your comment that I linked contained the 
description of the integral that you are trying to compute?
 

> As far as I know fe_values2.shape_value(i,q_index) is vector-valued shape 
> function
>

No, this is not true. Here is this function's documentation 
<https://www.dealii.org/8.5.0/doxygen/deal.II/classFEValuesBase.html#abe4de48ff59778bb82a0ec13037804aa>.
 
This is different from what is returned 
<https://www.dealii.org/8.5.0/doxygen/deal.II/classFEValuesViews_1_1Vector.html#a9e2686feec1a56451b674431b63c92d1>
 
if you use a FEValuesExtractor::Vector 
<https://www.dealii.org/8.5.0/doxygen/deal.II/structFEValuesExtractors_1_1Vector.html>
.
 

> sol_grad is also vector
>

Yes, since your original solution field is a scalar I think that this is 
correct. 

Is it now clear as to why you're seeing this error?

J-P

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