We propose \(C^1\) -conforming finite element methods for a class of nonlinear fourth-order partial differential equations arising in continuum physics. Our primary motivation is the Landau–Lifshitz–Baryakhtar equation in micromagnetics, although the framework also applies to other fourth-order models such as the convective Swift–Hohenberg equation and Cahn–Hilliard-type equations with source and convection terms. The proposed methods consist of a spatially semi-discrete scheme and two linearly implicit fully discrete schemes based on the semi-implicit Euler and BDF2 time discretisations. For all schemes, we establish unconditional \(\mathbb {H}^2\) -stability and optimal-order error estimates. Numerical experiments are provided to corroborate the theoretical results.
