Stable $C^1$-conforming finite element methods for a class of nonlinear fourth-order evolution equations

We propose some finite element schemes to solve a class of fourth-order nonlinear PDEs, which include the vector-valued Landau--Lifshitz--Baryakhtar equation, the Swift--Hohenberg equation, and various Cahn--Hilliard-type equations with source and convection terms, among others. The proposed numerical methods include a spatially semi-discrete scheme and two linearised fully-discrete $C^1$-conforming schemes utilising a semi-implicit Euler method and a semi-implicit BDF method. We show that these numerical schemes are stable in $\mathbb{H}^2$. Error analysis is performed which shows optimal convergence rates in each scheme. Numerical experiments corroborate our theoretical results.