Feimi,
u^h \cdot grad{w^h} \cdot laplacian{u^h} for SUPG and
> grad{q^h} \cdot laplacian{u^h} for PSPG.
>
> Where u^h is the trial function and w^h, q^h refer to the velocity and
> pressure test function, respectively.
> It seems impossible to calculate these terms with linear elements (because
> the linear shape functions does not have 2nd derivatives, or they are
> zero). I also checked some literatures but they did not mention it.
> Can anyone provide me some idea or some reference? I really appreciate it!
>
These terms are simply zero if you consider Q1 elements for the velocity.
In particular, the MINI-element is equivalent to the PSPG stabilization for
the Q1/Q1-pair which does not include the laplacian of the velocity.
You might want to have a look at Verfürth's script "Computational Fluid
Dynamics" <http://www.ruhr-uni-bochum.de/num1/files/lectures/CFD.pdf>. In
particular, the proofs for a general PSPG stabilization (in the script for
the Stokes problem) also hold if the pressure is continous
or the velocity is piecewise linear.
Best,
Daniel
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