The inner product is defined as follows, if you take the inner product over a complex field, does the test function have to be a conjugate function, is the inner product defined the same way?
This is the L2 inner product for real functions. For complex functions, one would use a conjugate function.
Hello, in FreeFem++, does the programming language
int2d(Th)(u*v)
represent real-valued functions. But I see an example of a complex function, which is also
int2d(Th)(u*v),
which in fact should not be
int2d(Th)(u*conj(v))?
Yes but the test function is real so no conj .
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I see. Thank you very much for your reply. 
Good morning, if the test function is a real function, then the stiffness matrix is real, but why define matrix complex>A=va (Xh,Xh) rather than matrix A=va(Xh,Xh)?
because the you get matrix coefficient as complex and not real.
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Thank you for your reply. The problem has been solved.