Re: [Libmesh-devel] Pyramids

[Derek G. <fri...@gm...>]

For now... I would vote for "do nothing"... maybe print a warning in debug
mode.  Having the library try to interpret what you "really" want might be
trouble.
Derek

On Wed, Nov 12, 2008 at 3:24 PM, John Peterson <jwp...@gm...> wrote:

> On Thu, Nov 6, 2008 at 11:18 AM, John Peterson <jwp...@gm...>
> wrote:
> > On Thu, Nov 6, 2008 at 11:03 AM, Roy Stogner <roy...@ic...>
> wrote:
> >> John Peterson wrote:
> >>> Anyone know anything about the
> >>> accuracy of quadrature for functions which are ratios of polynomials?
> >>
> >> We can derive custom quadrature rules which would integrate a mass
> matrix
> >> exactly... but would they then also integrate, say, a Laplacian matrix
> >> exactly?  The answer is an obvious "yes" for polynomial bases but I'd
> expect
> >> a "no" for pyramids.  That could be bad.
> >>
> >> What are we doing for them now?
> >
> > The current quadrature rules have accuracies like you would expect for
> > 1D elements, since they are conical products of Gauss-like rules.  So,
> > for example, a 2x2x2 rule will integrate exactly all monomials of the
> > form x^a y^b z^c, a+b+c <= 3.  I have no idea what will happen when we
> > try to integrate the rational basis functions...
>
> Just a quick update on the quadrature over pyramids stuff.
>
> After checking it with Maple, it appears that the "standard"
> 2nd/3rd-order quadrature rule can exactly integrate the Pyramid5 mass
> matrix.  The laplace matrix, however, is a different story.  I needed
> to go up to 6/7th-order quadrature before I could get 9-10 digits of
> precision from LibMesh.
>
> At first, this seems a little paradoxical since the Laplace matrix is
> usually the easier of the two, but with rational basis functions, the
> more derivatives you take the more poles you get in the denominator,
> and the harder it is to integrate the functions.  Since the default
> quadrature rule is currently selected by the FEType without regard to
> the geometric element type, it's not immediately obvious how we should
> ensure the user gets accurate quadrature on pyramids.  A couple
> options...
>
> 1.) Just remember that higher-than-normal order quadrature on pyramids
> is required and your answer may be inaccurate.  AKA "do nothing" :-)
> 2.) Redefine, within the pyramid quadrature rules, the meaning of
> order.  I.e. return a rule several orders higher than what the user
> requests.
> 3.) Research quadrature rules for rational functions.  I have a few
> papers on this but haven't looked into it too much yet.  Phillipe
> Devloo may be doing something special in his library, so I will check
> there as well...
>
> --
> John
>
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