#TIL how Fourier transforms and #pseudospectral methods are intimately related to #FEM. What was missing from the picture is how a tool for numerical integration could be used for numerical differentiation.
If you think about it, Fourier transforms are also expressed as integration over an interval in space. When you differentiate a function it only affects the basis. For example:
exp(ikx) -> ik * exp(ikx)
Similarly, in FEM the basis function is modified. But in this case it would be a Lagrange basis function (GL, GLL, Chebyshev etc) and its derivative. Recall Lagrange interpolation? Interpolation nodes are chosen to be the same as quadrature nodes.
Another interesting consequence is all PDEs look analogous to simple harmonic oscillators with mass and stiffness block matrices.
The only oddity that I notice in Fluids <-> SH is that the viscous term <-> stiffness and advection term <-> damping.
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