Energy-Stable Finite Element Approximation of the Landau–Lifshitz–Bloch Equation Below the Curie Temperature

Abstract The Landau–Lifshitz–Bloch (LLB) equation is a micromagnetic model which describes the time evolution of the magnetisation vector field in a ferromagnet at elevated temperatures. In this work, we develop an energy-stable finite element scheme for solving the LLB equation in the regime below the Curie temperature. In this setting, the LLB equation takes the form of a vector-valued quasilinear PDE with a singular term. To overcome the challenges associated with this singular term, we adopt a ‘regularise-then-discretise’ strategy. Specifically, we introduce a regularised version of the equation whose solution converges to that of the original LLB equation. We then design an energy-stable, fully discrete Galerkin finite element method based on implicit Euler time discretisation to solve the regularised problem. Under suitable regularity assumptions on the exact solution, we prove that the scheme converges at an optimal rate. Numerical experiments are presented to corroborate the theoretical results.