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Ok, here comes the answer promised in the comment section. Let's start the other way round. In grid methods, one basically selects a number of $N+1$ gridpoints $\{x_k\}_{k=0}^{N}$. As basis functions, one can use Lagrange polynomials constructed over these nodes, each one having polynomial order $N$. For this setup, using Newton-Cotes-like quadrature weights ...

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