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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