Marie and I just discussed some different options for choosing the interpolation and quadrature degree.
Consider this simple form which can be the right-hand side in Poisson:
L = v*f*dx
Making some simple assumptions, we can assume that the order of
convergence for the finite element method is h^{q+1} if we use
Lagrange elements of degree h^q.
If f is approximated with its interpolant into some suitable space,
there are two questions to answer: what should the degree be for the
interpolation and what should it be for the quadrature.
Marie suggests this strategy:
1. Interpolate f into P_{q+1}
2. Integrate the resulting integrand exactly
The integrand will here be a polynomial of degree q + q + 1 = 2q + 1.
The error will be something like h^{q+1} + h^{q+2}.
My suggestion is this:
1. Interpolate f into P_q
2. Integrate the resulting integrand exactly
The integrand will here be a polynomial of degree q + q = 2q.
The error will be something like h^{q+1} + h^{q+1}.
I think Marie's suggestion is overkill, but she think my suggestion is
unsafe.
Any opinions on what the default behavior should be?
--
Anders
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