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I'm an engineer who uses a lot of finite-element software for doing multiphyics modeling. I've been learning a lot about the foundations of the finite-element method, but I have a particulary nagging question that I havn't been able to get a satisfying answer to: Why do most finite-element programs use relatively low order lagrange elements?

From my understanding, are more advanced elements, with higher order that have many advantages like continous spatial derivatives. For some applications like solid-mechanics or heat transfer, shouldn't more advanced elements be desirable? If the field variable is the displacements, than the strains are formed from the derivatives. If the field variable is temperature, then the heat flux is determined according to the gradient. Since Lagrange elements are only C0 continuous (derivatives are discontinous between elements) strain and heat flux may be discontinous between elements. Shouldn't we use fancier C1 continous elements? Like Argyris elements? Wouldn't this also benifit the accuracy of convective terms?

I know that they can be more computationally expensive because higher order elements require more quadurature points and shape funtions to evaluate. Obviously there must be some significant downside or else they would be common.

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# Why are advanced finite-element methods not more popular?

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