Manav Bhatia <[email protected]> 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
problems.)
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