I greatly appreciate your comments. I would have missed these nuances
that you pointed out as I come from an engineering background and am trying
to slowly bring myself upto speed with the mathematical aspects of FEM and
CFD. So, all of this discussion is highly valuable and educational to me.
I do have a few questions about what all of this might imply from a
practical standpoint, that I hope you and the others may weigh in on.
-- How critical is this lack of conservation properties for general
application of CFD for transonic and supersonic aerodynamic analyses?
-- Is this apparant non-ideal conservation behavior a primary reason
for interest in DG vs continuous FEM?
-- The typical DG methods that I have seen do not add a stabilization
within the domain, but handle the boundary terms to ensure continuity of
flux across the discontinuous elements. However, a straight-forward
application of Galerkin method is known to be unstable. So, I am assuming
that the stabilization and non-oscillatory behavior comes from forgoing the
requirement of continuity and the treatment of boundary terms. However,
would this also not imply that if one continued to increase the element
size while increasing the polynomial order, the solution within the will at
some point become unstable/oscillatory?
-- My previous questions is movitvated by an interest to move to higher
order interpolation functions (read isogeometric methods) for applications
in design optimization of aerodynamic vehicles. There, I am interested in
really large element sizes (if possible). From what I understand,
the continuous FEM methods would be better suited than DG. But, I don't
know if the conservation properties that you pointed out might become an
I would appreciate your comments.
On Thu, Apr 18, 2013 at 10:30 AM, Jed Brown <jed@...> wrote:
> Manav Bhatia <bhatiamanav@...> writes:
> > Jed: I am curious about your comment on lack of conservation of the
> > GLS schemes. I did a bit of search and came across the following two
> > papers. They make a case for conservation properties of the methods. I
> > am curious what you think.
> Sure, I'm familiar with these papers.
> > Hughes, T. J. R., Engel, G., Mazzei, L., & Larson, M. G. (2000). The
> > Continuous Galerkin Method Is Locally Conservative. Journal of
> > Computational Physics, 163(2), 467–488. doi:10.1006/jcph.2000.6577
> > Abstract: We examine the conservation law structure of the continuous
> > Galerkin method. We employ the scalar, advection–diffusion equation as
> > a model problem for this purpose, but our results are quite general
> > and apply to time-dependent, nonlinear systems as well. In addition to
> > global conservation laws, we establish local con- servation laws which
> > pertain to subdomains consisting of a union of elements as well as
> > individual elements. These results are somewhat surprising and
> > contradict the widely held opinion that the continuous Galerkin method
> > is not locally conser- votive.
> This paper changes the definition of local conservation. I wouldn't say
> it's "surprising" at all because it is exactly the conservation
> statement induced by the choice of test space. In essence, the
> continuous Galerkin conservation statement is smeared out over the width
> of one cell where as the DG or finite volume conservation statement has
> no such smearing. On coarse grids, one cell can be mighty big.
> > Hughes, T. J. R., & Wells, G. N. (2005). Conservation properties for
> > the Galerkin and stabilised forms of the advection–diffusion and
> > incompressible Navier–Stokes equations. Computer Methods in Applied
> > Mechanics and Engineering, 194(9-11),
> > 1141–1159. doi:10.1016/j.cma.2004.06.034
> > Abstract: A common criticism of continuous Galerkin finite element
> > methods is their perceived lack of conservation. This may in fact be
> > true for incompressible flows when advective, rather than
> > conservative, weak forms are employed. However, advective forms are
> > often preferred on grounds of accuracy despite violation of
> > conservation. It is shown here that this deficiency can be easily
> > remedied, and conservative procedures for advective forms can be
> > developed from multiscale concepts. As a result, conservative
> > stabilised finite element procedures are presented for the
> > advection–diffusion and incompressible Navier–Stokes equations.
> This paper is specific to incompressible flow, but it's mostly
> investigating the "advective" form
> v \cdot \nabla v
> as compared to the divergence form
> \nabla \cdot (v \otimes v)
> With stabilization, they are able to make a weak conservation statement
> ("smeared" as in the other paper) using the advective form. Note that
> when using the identity
> \nabla \cdot (u \otimes a) = a \cdot \nabla u + u (\nabla\cdot a)
> where 'a' is a discrete velocity field, we rarely have that 'a' is
> exactly divergence free. Indeed, it is generally only weakly
> divergence-free unless we use a stable element pair with a discontinuous
> pressure. Their analysis assumes that 'a' is exactly divergence-free
> and still only makes a weak conservation statement.
> Again, the difference between strong and weak conservation is more
> significant on coarse grids. With agglomeration-based multigrid (FV or
> DG), a coarse-grid cell satisfies exactly the same conservation
> statement as the corresponding agglomerated fine-grid cells.
> As an aside, we can see mass conservation problems already for Stokes.
> We need only choose a discontinuous body force (as in Rayleigh-Taylor
> initiation) or discontinuous viscosity to find a velocity field that has
> serious non-conservative artifacts on coarse grids when using stabilized
> finite elements. This is why finite element methods for problems like
> subduction must use stable finite element pairs with discontinuous
> pressure. (Some use almost-stable Q1-P0, but these still have