Numerical analysis of the A-Formulation of Maxwell variational inequality

¹ School of Mathematics and Statistics and Xi’an Key Laboratory of Scientific Computation and Applied Statistics, Northwestern Polytechnical University, Xi’an 710129, Shaanxi, P.R. China.
² MOE Key Laboratory for Complexity Science in Aerospace, Northwestern Polytechnical University, Xi’an 710129, Shaanxi, P.R. China
³ School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, P.R. China
⁴ LSEC, NCMIS, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, School of Mathematical Science, University of Chinese Academy of Sciences, Beijing 100190, P.R. China

^* Corresponding author: zhangxiaodi@lsec.cc.ac.cn

Received: 25 July 2024
Accepted: 11 October 2025

Abstract

In this work, we explore a Maxwell variational inequality arising from Bean’s critical-state model in type-II superconductivity. Instead of the coupled electric and magnetic fields, we study an equivalent model based on the magnetic vector potential. The well-posedness of this transformed Maxwell variational inequality comes naturally from the hyperbolic Maxwell (quasi-)variational inequality arguments. We establish error estimates for both the spatially semi-discrete scheme and the fully discrete scheme. We propose two kinds of methods, the projection method and the regularization method, to solve the variational inequality problem. Two numerical experiments are reported to verify the theories and to investigate the physics of type-II superconductivity.