# Problem solving fourth-order PDE

**URL:** <https://fenicsproject.discourse.group/t/problem-solving-fourth-order-pde/16789>\
**Category:** General\
**Created:** [January 19, 2025, 6:23pm UTC](https://fenicsproject.discourse.group/t/problem-solving-fourth-order-pde/16789 "2025-01-19T18:23:10Z")\
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**Author:** ![Stein](https://yyz2.discourse-cdn.com/free1/user_avatar/fenicsproject.discourse.group/stein/32/8751_2.png) [@Stein](https://fenicsproject.discourse.group/u/Stein)\
**Post date:** [January 20, 2025, 10:15am UTC](https://fenicsproject.discourse.group/t/problem-solving-fourth-order-pde/16789/4 "2025-01-20T10:15:49Z")

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Indeed, the function-space is comprised of _piecewise_ third-order polynomials. So a third-order polynomial on each element. Within each element, you can take as many derivatives as you want, as a polynomial is infinitely differentiable. However, across element boundaries, the functions are “only” continuous. The first derivative of a function in your `Q` jumps from element to element. That is causing an integration inconsistency. The keywords to look for here are H1-sobolev space, C0 continuity, and FEM for higher-order PDEs. Most classical textbooks cover this material.

If I recall correctly, I touch upon that in this lecture:

[![](https://global.discourse-cdn.com/free1/uploads/fenicsproject1/original/2X/8/8e01ae6a95c35342da9105086b4e6d52b6d92157.jpeg "09 - Functional analysis - Differential operators") ](https://www.youtube.com/watch?v=8V1oFHOO_L8)

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