dolfin team mailing list archive

[Anders Logg <logg@xxxxxxxxx>]

On Thu, May 27, 2010 at 04:37:42PM +0100, Garth N. Wells wrote:
>
>
> On 27/05/10 16:34, Anders Logg wrote:
> >On Thu, May 27, 2010 at 04:22:33PM +0100, Garth N. Wells wrote:
> >>
> >>
> >>On 27/05/10 15:58, Anders Logg wrote:
> >>>Wouldn't it work to just to this:
> >>>
> >>>v = v + (h/2.0*vnorm)*dot(velocity, grad(v))
> >>>U = 0.5*(u0 + u)
> >>>F = v*(u - u0)*dx + 0.5*k*(v*dot(velocity, grad(U))*dx + c*dot(grad(v), grad(U))*dx) - k*v*f*dx
> >>>a = lhs(F)
> >>>L = rhs(F)
> >>>
> >>>That's my understanding of SUPG, but perhaps I've missed something?
> >>>
> >>
> >>Looks like the same thing (I'd have to check). I prefer to
> >>expressing the method as weighting the residual, e.g. Galerkin +
> >>weighted residual.
> >
> >I also think of it as weighting the residual, but with a modified test
> >function v. The above looks like a weighted (weak) residual to me.
> >
> >The difference from your implementation is that you multiply the
> >stabilization part of the test function with the strong residual for
> >which the Laplacian (for P1 elements) is zero. In the above case, the
> >stabilization also hits the nonzero integrated by parts grad-grad term.
> >
> >I don't know whether or not that is important.
> >
> >>The code I added can be made more compact using lhs/rhs, but I was
> >>having some trouble and initially wanted to make a bare bones
> >>implementation.
> >
> >ok.
> >
> >I might try to make some changes. I'd like to get a feeling for how
> >SUPG works.
> >
>
> OK, but just give me moment since I'm making a change to use lhs/rhs.
>
> Garth

On second thought, even if I like to think of it as modifying the test
function v --> v + delta * beta . grad(v), this is problematic when
delta * beta . grad(v) hits div(grad(u)) since we run into
difficulties of either defining div(grad(u)) for an H^1 function or
integrating by parts onto v which is already differentiated.

So I agree the best formulation is this:

  a(v, u_h) + sum_T <delta * beta . grad(v), R_T>_T = L(v)

where R_T is the element-wise residual (where we may differentiate u_h
twice and get zero). R_T will be zero for u so the method is consistent.

This is what is currently implemented in the advection-diffusion demo.

--
Anders

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