### SPUG on CG Elements

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Hi,

I want to implement a SPUG Stabilization term in a 1D advection problem.

I'am reffering to this paper:

http://amasud.web.engr.illinois.edu/Papers/Stabilized_Methods-F-H-M-FEM-Book-2004.pdf

There a stabilization term is defined in equations 9-11.

As far as I understand the paper u,v would be functions in the normal continous garlerkin space, but the stabilization term is defined "cell-wise".

How do I define these stabilization terms cell-wise?

Best Regards,

Moritz

I want to implement a SPUG Stabilization term in a 1D advection problem.

I'am reffering to this paper:

http://amasud.web.engr.illinois.edu/Papers/Stabilized_Methods-F-H-M-FEM-Book-2004.pdf

There a stabilization term is defined in equations 9-11.

As far as I understand the paper u,v would be functions in the normal continous garlerkin space, but the stabilization term is defined "cell-wise".

How do I define these stabilization terms cell-wise?

Best Regards,

Moritz

Community: FEniCS Project

### 1 Answer

3

There is nothing special that needs to be done. The "dx" measure in FEniCS is already a sum of integrals over element interiors, so the stabilization terms can be integrated using it. This is usually equivalent to integrating over the entire domain. However, in the case of SUPG, there would be distributional contributions (Dirac deltas) when integrating the Laplacian of a function that has jumps in its first derivatives at element boundaries, so the stabilization terms are explicitly written as sums of integrals over element interiors, to emphasize that these contributions are not included. The "cell-wise" quantity \(h_K\) used to define \(\tau_K\) can be obtained by

`h = CellDiameter(mesh)`

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I will try to implement it and come back.