Unconditional superconvergence analysis of an energy-stable C–N fully discrete scheme for the nonlinear magnetic diffusion model with memory

Abstract

The unconditional superconvergence behavior of an energy-stable Crank–Nicolson (C–N) fully discrete scheme with nonconforming EQ1rot finite element method (FEM) is investigated for the nonlinear integro-differential system associated with the diffusion of the magnetic field. The stability of the numerical solution in the broken H1-norm is proved by the Gagliardo–Nirenberg (G–N) inequality, and the unique solvability is demonstrated by the Brouwer fixed point theorem. In which the boundness of the L∞-norm and the inverse inequality are eliminated. The combination of the Green formula and two special properties of EQ1rot element leads to the unconditional superconvergence estimate without any restriction between the time step and space partition size. Finally, the validity of the theoretical findings is supported by some numerical results.