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I've recently learned about using weighted residuals with the Galerkin method to numerically approximate even-order differential equations (for linear elements, I'm still a beginner). It seems for odd-order differential equations you cannot use integration by parts to reduce the order of the third-order differential operator.

I've read that the Petrov-Galerkin method can approximate these solutions, but I cannot find cogent material on it or examples of it in action. Can someone please explain?

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    $\begingroup$ Third derivatives are a bit odd, and don't appear too often. But first derivatives as in the advection equation are quite common. Learn how these are solved and you will have a good starting point for knowing how to solve 3rd order problems. $\endgroup$
    – Wolfgang Bangerth
    Commented Apr 9, 2020 at 23:47
  • $\begingroup$ Have you checked this answer? $\endgroup$
    – nicoguaro ♦
    Commented Apr 10, 2020 at 18:47

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