A Gradient Recovery Technique for Enhancing the Convergence of Demagnetizing Field Based on PDE Approach

Abstract

The PDE approach is a popular technique for the demagnetizing field calculation due to the flexibility in handling complex domains. However, it faces a challenge on delivering desired accuracy due to the suboptimal convergence of the function gradient and singularity on the boundary. In this work, a robust gradient recovery technique is applied and analysed for fixing such an issue. An $L^2$ error estimate of the finite element approximation for the demagnetizing field is derived, which consists of two parts, i.e., of finite element discretization error and boundary approximation error. A gradient recovery method based on the polynomial preserving recovery technique is applied to enhance the accuracy of the finite element approximation, and a superconvergence result is established. An idea of locally refined surface meshes is applied to resolve the singularity in the boundary conditions, thereby reducing boundary approximation errors. Extensive numerical tests are provided to verify our theoretical findings and the efficiency of our proposed method. The results indicate that the proposed method achieves second-order convergence for the approximate demagnetizing field, positioning it as a highly competitive technique to develop optimal algorithms in computational micromagnetics.