The Landau–Lifshitz–Bloch equation on polytopal domains: unique existence and finite element approximation

Abstract The Landau–Lifshitz–Bloch equation (LLBE) describes the evolution of the magnetic spin field in ferromagnets at high temperatures. In this paper we study the numerical approximation of the LLBE in the regime above the Curie temperature on bounded polytopal domains in $\mathbb{R}^{d}$, with $d\in \{1,2,3\}$, allowing for nonconvex domains when $d=2$. We first establish the existence and uniqueness of strong solutions to the LLBE and propose a linear, fully discrete, conforming finite element scheme for its approximation. While this scheme is shown to converge, the obtained rate is suboptimal. To address this shortcoming we introduce a viscous (pseudo-parabolic) regularization of the LLBE, which we call the $\epsilon $-LLBE. For this regularized problem we prove the unique existence of strong solutions and establish a rate of convergence of the solution $\boldsymbol{u}^\epsilon $ of the $\epsilon $-LLBE to the solution $\boldsymbol{u}$ of the LLBE as $\epsilon \to 0^{+}$. Furthermore, we propose a linear, fully discrete, conforming finite element scheme to approximate the solution of the $\epsilon $-LLBE. Given sufficiently smooth initial data error analysis is performed to show stability and uniform-in-time convergence of the scheme. Finally, several numerical simulations are presented to corroborate our theoretical results.