Any jump operator will have a "preferred direction";
the sign of the jump changes if you swap the + and - cells.
These are all the same, it's just where you place the signs:
def normal_jump1(u, n):
return dot(u('+') - u('-'), n('+'))
def normal_jump2(u, n):
return dot(u('+'),n('+')) - dot(u('-'),n('+'))
def normal_jump3(u, n):
return dot(u('+'),n('+')) + dot(u('-'),n('-'))
Martin
On 10 June 2014 14:11, Anders Logg <[email protected]> wrote:
> On Tue, Jun 10, 2014 at 10:58:38AM +0200, Martin Sandve Alnæs wrote:
> > Maybe this term isn't quite like 'most DG formulations'?
> > But another alternative which gives you avg * jump:
> >
> > inner(avg(grad(u)), outer(jump(v), n('+'))))
>
> The problem with this one is that it seems to have a preferred
> direction (+) which it shouldn't. That's the beauty of defining the
> jump as something with n('+') + n('-'), that it does not have a
> preferred direction.
>
> --
> Anders
>
>
> > Based on:
> >
> > outer(v('+'), n('+')) + outer(v('-'), n('-'))
> > = outer(v('+'), n('+')) - outer(v('-'), n('+'))
> > = outer(v('+') - v('-'), n('+'))
> > = outer(jump(v), n('+'))
> >
> > Martin
> >
> >
> > On 10 June 2014 10:44, Anders Logg <[email protected]> wrote:
> >
> > On Tue, Jun 10, 2014 at 10:09:44AM +0200, Martin Sandve Alnæs wrote:
> > > To clarify, Kristian talks about the jump(f,n) version:
> > >
> > > jump(scalar,n) is vector-valued
> > > jump(vector,n) is scalar
> > >
> > > while I mixed up with the jump(f) version:
> > >
> > > jump(scalar) is scalar
> > > jump(vector) is vector-valued
> > >
> > > The jump(f) version gives the difference of the full value, while
> > > the jump(f,n) version gives the difference in normal component.
> >
> > I looked through the Unified DG paper that Kristian pointed to but
> > couldn't see any examples of vector-valued equations.
> >
> > I also don't see the point in defining jump(v, n) for vector valued u
> > as in the paper. If the result of jump(v, n) is a scalar quantity,
> > there is no way to combine the normal n with the thing it should
> > naturally be paired with, namely the flux (or grad(u)). It only works
> > out in the special case of scalar elements.
> >
> > But perhaps adding tensor_jump() is the best solution since that
> paper
> > is the standard reference for DG methods.
> >
> > (I'll ask Douglas Arnold about this, if he has not seen this
> > already. Maybe I am missing something obvious.)
> >
> > > However, if I didn't mess up some signs I think you can write your
> term
> > like
> > >
> > > 0.5*dot(jump(Dn(u)), jump(v))
> > >
> > > which seems much more intuitive to me (although I'm not that into
> DG
> > scheme
> > > terminology).
> >
> > Yes, that looks correct but I don't think it is intuitive since it
> > does not involve the avg() operator which is naturally paired with
> the
> > jump() operator in most DG formulations.
> >
> >
> >
> > > On 10 June 2014 09:05, Kristian Ølgaard <[email protected]>
> wrote:
> > >
> > >
> > > On 9 June 2014 20:58, Martin Sandve Alnæs <[email protected]>
> wrote:
> > >
> > >
> > > I object to changing definitions based on that it would
> work out
> > nicely
> > > for one particular equation. The current definition yields
> a
> > scalar
> > > jump for both scalar and vector valued quantities, and the
> > definition
> > > was chosen for a reason. I'm pretty sure it's in use.
> Adding a
> > > tensor_jump on the other hand wouldn't break any older
> programs.
> > >
> > > Maybe Kristian has an opinion here, cc to get his
> attention.
> > >
> > >
> > > I follow the list, but thanks anyway.
> > >
> > > The current implementation of the jump() operator follows the
> > definition
> > > often used in papers (e.g. UNIFIED ANALYSIS OF DISCONTINUOUS
> GALERKIN
> > > METHODS
> > > FOR ELLIPTIC PROBLEMS, arnold et al.
> http://epubs.siam.org/doi/abs/
> > 10.1137/
> > > S0036142901384162)
> > >
> > > where the jump of scalar valued function result in a vector,
> and the
> > jump
> > > of a vector valued function result in a scalar.
> > >
> > > Adding the tensor_jump() function seems like a good solution
> in this
> > case
> > > as I don't see a simple way of overloading the current jump()
> > function to
> > > return the tensor jump.
> > >
> > > Kristian
> > >
> > >
> > >
> > > Martin
> > >
> > > 9. juni 2014 20:16 skrev "Anders Logg" <[email protected]>
> > følgende:
> > >
> > >
> > > On Mon, Jun 09, 2014 at 11:30:09AM +0200, Jan Blechta
> wrote:
> > > > On Mon, 9 Jun 2014 11:10:12 +0200
> > > > Anders Logg <[email protected]> wrote:
> > > >
> > > > > For vector elements, the jump() operator in UFL is
> > defined as
> > > follows:
> > > > >
> > > > > dot(v('+'), n('+')) + dot(v('-'), n('-'))
> > > > >
> > > > > I'd like to argue that it should instead be
> implemented
> > like
> > > so:
> > > > >
> > > > > outer(v('+'), n('+')) + outer(v('-'), n('-'))
> > > >
> > > > This inconsistency has been already encountered by
> users
> > > > http://fenicsproject.org/qa/359/
> > > discontinuous-galerkin-jump-operators
> > >
> > > Interesting! I hadn't noticed.
> > >
> > > Are there any objections to changing this definition
> in UFL?
> > >
> > >
> > >
> > >
> > >
> >
> >
>
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