Prof. Bangerth,
Ok, so my initial understanding of deal.II was accurate. It all makes sense
now. Although FE_DGP is spanning {1,x,y} in reference space, it is not spanning
those bases in physical space. Instead, it is spanning some combination of
{1,x,y,xy} with 3 degrees of freedom, that only spans {1,x,y} in physical space
if the mapping is affine. As a result, loss of convergence order entails since
it cannot properly represent linear functions in physical space.
On the other hand, a bilinear basis on the reference space can indeed represent
physical linear functions if a bilinear mapping is used! Hence why FE_DGQXXXX
work.
If we were working with triangles, the Legendre basis {1,x,y} would be able to
represent linear functions in physical space since the mapping is automatically
affine for straight sided elements, hence my wrong assumption that Legendre
polynomials should produce optimal orders of convergence.
Thank you for the response, it was immensely helpful.
Doug
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