the viscous term in SUPG stabilization terms

[Feimi Yu]

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

[Daniel Arndt]

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". 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

[Wolfgang Bangerth]

On 01/17/2018 11:59 AM, Daniel Arndt wrote:
> These terms are simply zero if you consider Q1 elements for the velocity. I

That's true only if you're on affine meshes. In general, the Laplacian
of a Q1 function on an arbitrary mesh is not zero.

That doesn't mean, though, that it may not be valid to just ignore the
issue.

But I think what you want to say is that the term is supposed to be
interpreted as a sum over cell interiors, not as an integral over the
entire domain. In the latter case, you'd have to worry about the fact
that Delta uh is singular on edges, whereas in the former you ignore
these contributions.

Best
W.

--
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Wolfgang Bangerth email: bang...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/

[Feimi Yu]

Thanks a lot, Daniel and Wolfgang!

What I understood was that since the SUPG and PSPG terms are integrated over the cell interiors so the laplacians can be ignored while using Q1/Q1 element without any singularity problems on the cell edges.

Thanks!
Feimi

[Bruno Blais]

I might be late to the party,

But I believe what Wolfgang is trying to say is that Q1-Q1 elements are bi-linear. Consequently, except if you mesh is perfectly alligned with the x and y axis, you will have a non-zero second derivative of the shape function.

Your shape function are of the form : a + b*x + c*y + d*x*y in your reference element. Thus if your Q1-Q1 element is slightly tilted you will get a non-zero second derivative.