Hi,
I'm working on a solver to solve Navier-Stokes equations using the
Streamline Upwind Petrov-Galerkin (SUPG) and Pressure Stabilization
Petrov-Galerkin (PSPG) method, combined with Newton's iteration and FGMRES
iterative method. However, I had some difficulties with implementing the
formulation of the strong form residual in the SUPG and PSPG terms, mostly
about the viscous term, which is the laplacian of the velocity. As we know,
in the standard Galerkin we apply integrate-by-part to the viscous term and
we get a bilinear form. But for the viscous term in the SUPG part. It has a
weird form looking like:
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!
Thanks!
Feimi
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