Subject: [FIAT-dev] trapezoidal quadrature with FIAT?

From: "Shi Jin" <[email-masked]>
Date: Tue, 8 Jan 2008 15:22:48 -0700

Hi there,

I have been using FIAT as a component from PETSc and I love it. Thank you.
Recently I have the need to integrate a discontinuous function in a
tetrahedral finite element (integrate only inside a sphere instead of the
whole domain) and the Gaussian quadrature rules does not work very well,
even if I put lots of quadrature points. My naive idea is that if I use the
trapezoidal quadrature rules, the accuracy of integration is guaranteed when
the number of quadrature points is large enough. Therefore, I am wondering
if it is possible to create trapezoidal quadrature rules using FIAT.

Note that I am not very familiar with the mathematics of quadratures. I
originally thought the one created by PETSc using FIAT is default to
Gaussian quadratures. But then I read the FIAT manual and it says  "FIAT
implements arbitrary-order collapsed quadrature, as discussed in Karniadakis
and Sherwin [], for the simplex of dimension one, two, or three." I am not
sure what "collapsed quadrature" is. Does it include Gaussian and
Trapezoidal quadratures? How do I choose from them? I would greatly
appreciate it if some guidance can be provided. Also, could more information
be provided about the Karniadakis and Sherwin paper, such as title, journal
and year? Maybe better quadrature rules exist for the integration of
discontinuous functions and I would love to hear some advice.

Thank you very much.

Shi Jin

---

From: "Robert Kirby" <[email-masked]>
Date: Tue, 8 Jan 2008 22:01:52 -0600

1.) Integration rules tend to do poorly on discontinuous functions.  I'd
recommend breaking up the domain onto regions where the integrand is
continuous.2.) The collapsed coordinates are thoroughly described in
Karniadakis & Sheriwn's "hp and spectral methods for CFD" -- it's a book (I
believe from Oxford press).  The get quadrature rules (and polynomials) on
simplices via singular mappings from rectangles.
3.) Create whatever kinds of rules you like for FIAT -- the QuadratureRule
class is pretty simple for the reference element -- just provide the points
and quadrature weights.

On Jan 8, 2008 4:22 PM, Shi Jin <[email-masked]> wrote:

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