## Re: [Libmesh-devel] Pyramids

 Re: [Libmesh-devel] Pyramids From: John Peterson - 2008-11-12 22:25:12 ```On Thu, Nov 6, 2008 at 11:18 AM, John Peterson wrote: > On Thu, Nov 6, 2008 at 11:03 AM, Roy Stogner 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 ```