Lagrange elements of any polynomial describe piecewise continuous functions. Typically, those functions are differentiable.

Mixed finite element methods use vector fields of even less continuity, such as normal continuity. With some great oversimplication, one might argue that discontinuous Galerkin methods, hybridized methods, etc, completely toss out contininuity concerns.

What are the major applications for finite elements with higher smoothness? I am thinking here of standard textbook elements such as the Argyris, Hermite, or Morley elements. While I can see that one might be tempted to use them for higher-order partial differential equations, they do not seem to dominate the literature on those topics.

Are finite elements with $C^1$ continuity or higher widely adopted? What are their main applications in real life?


This paper by Kirby and Mitchell describes the implementation of $C^1$ elements in the Firedrake package*. One of the main use cases is biharmonic problems, which show up in the elastic deformation of thin plates, or other higher-order PDE like Cahn-Hilliard. My impression from reading the paper and talking to Rob Kirby is that $C^1$ elements are better but used relatively less often because they're difficult to implement. For example, one of the main technical innovations of that paper was around the scheme used to map the physical triangle to the reference triangle. For Lagrange elements this is easy -- you just use an affine transformation. For $H(\textrm{div})$-conforming elements you need to use a Piola transformation, and for Argyris and related elements the transformations are even more unusual. One of the findings of that paper (see figure 15) was that $C^1$ elements have produce a more favorable sparsity pattern that takes less time to factor than non-conforming methods based on polynomials of the same order. So basically you get the same accuracy at less cost compared to using a penalty or DG formulation with polynomials of the same order.

*Disclaimer, I contribute to Firedrake and develop an application based on it for my job.


$C^1$ elements are mostly a historic relic. In the finite element method, the traditional view is that the best methods are "conforming", i.e., methods where the finite element space $V_h$ is a subspace of the space $V$ in which the solution lies. For second-order elliptic equations, $V=H^1$ and functions that are continuous and piecewise polynomial (but not necessarily continuously differentiable) are a subspace of $V$.

But that is not the case for fourth-order equations such as the biharmonic equation. There, $V=H^2$ which contains only functions that are continuous differentiable (i.e., $C^1$). So in that case, the usual Lagrange elements are not a subspace of $V$ and it is not clear how to implement the bilinear form with these elements. So people, going back to the 1960s, developed elements that are $C^1$ and for which consequently $V_h \subset V$. This works, but the elements are quite difficult to implement for non-conforming meshes and they just don't quite fit into the systematic view we have of elements today, the paper by Kirby and Mitchell mentioned in one of the other comments notwithstanding.

But starting in the 1990s, we learned how to use non-conforming elements more efficiently -- first in the form of discontinuous Galerkin methods for elliptic equations and then also how to use Lagrange elements for biharmonic equations. I would specifically refer you to the 2005 paper by Sue Brenner and Sung on the $C^0$ Interior Penalty ("C0IP") method for biharmonic problems that is also used in the step-47 tutorial program of deal.II and that shows how relatively easy it is to solve these kinds of problems with just the usual elements. (Disclaimer: I'm one of the authors of deal.II and of step-47 in particular.)

Now, it is true that the paper by Kirby and Mitchell shows that the $C^1$ elements have advantages regarding condition numbers and solver speeds. But at least in my opinion, I don't think this outweighs the very substantial pain to implementing them on unstructured meshes and meshes that potentially contain hanging nodes. I had a long discussion with Rob Kirby about that paper at some point and have to admit that he's one of my heros for undertaking this kind of project -- he's the only person I know who had the gumption to implement $C^1$ elements in the last 20 years, and I think I know a substantial fraction of the people who implement finite elements :-)


Two libraries that I know of, besides Firedrake, that use $C^1$ elements are:

Another application where I found it was in Magnetohydrodynamics, in the following paper:

Jardin, S. C. (2004). A triangular finite element with first-derivative continuity applied to fusion MHD applications. Journal of Computational Physics, 200(1), 133-152.

But I could not make it work even for the Laplace equation.


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