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I tried to solve a one dimensional biharmonic equation with finite elements. I wanted to use a conforming approach (as I simply do not know a lot about other approaches) and therefore was looking for 1D $C^1$ finite elements. I found a bit of literature on 2D $C^1$ finite elements, however, I could not find anything useful on the 1D case, neither theoretical nor any implementation in free software. Does anybody know any references or I am overlooking sth and there is a good reason, why I cannot find anything about these types of elements? And if so, how else could I approach the problem?

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You might find more information by looking for approaches to model "Euler-Bernoulli beams" which corresponds to $u''''=f$ when material parameters and cross sections are constant. $C^1$ elements are quite easy to construct in 1D since you can use the derivative as a degree-of-freedom at nodes, e.g., cubic Hermite elements. This leads directly to $C^1$ conformity.

I maintain Python implementation, here is the corresponding documentation: https://scikit-fem.readthedocs.io/en/latest/listofexamples.html#example-34-euler-bernoulli-beam

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