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

Abstract We propose $$C^1$$ 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 $$\mathbb {H}^2$$ H 2 -stability and optimal-order error estimates. Numerical experiments are provided to corroborate the theoretical results.