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[Anders Logg <logg@xxxxxxxxx>]

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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