A-ϕ finite element method with composite grids for time-dependent eddy current problem

We investigate the A-ϕ finite element method with the global coarse grids and the local fine grids (composite grids) for solving a time-dependent eddy current problem, which can improve the accuracy of the coarse grid solutions in some subdomains of interest in the case when properly increasing computational costs. Meanwhile, to decrease calculation complexity and avoid dealing with a saddle-point problem from the traditional A-ϕ scheme, we design an iteration which combines the composite grid method with classic steepest descent such that the solutions of the unknowns A and ϕ for the global coarse grids domain are decoupled in two separate equations. We prove it converges with a bounded rate independent of the mesh sizes.