Finite Element Iterative Algorithm Based On Anderson Acceleration Technique For Incompressible MHD Equations

Incompressible MHD equations have strong nonlinearity,multi-physics coupling and a large number of equations.Iterative methods,such as Stokes-type iteration,Newton-type iteration and Oseen-type iteration are used for linearization.When the calculation scale becomes larger or the physical parameters of the equations system become larger,the iterative method converges slowly or does not converge.Therefore,for the steady incompressible MHD equations,a finite element iterative algorithm based on Anderson’s acceleration technique is designed.In the MHD discretization process,we use Taylor-Hood mixed element space to approximate the flow field subproblem,and the Nedelec edge element to approximate the magnetic field subproblems.For the study on the acceleration algorithm of steady incompressible MHD equations with large Hartmann numbers,existing literature[1]mainly focuses on using a robust solver and Picard-Krylov.This paper mainly designs Anderson’s acceleration algorithm for the Oseen-type finite element iterative scheme of the incompressible MHD equations.The acceleration convergence theory under different acceleration depths m is given.Numerical experiments of the smooth solution,singular solution and physical model are carried out to verify the correctness and effectiveness of the algorithm.Numerical results show that:for the Anderson accelerated algorithm,when the depth is m=0,the convergence speed is the same as that of the iterative format currently used,when the depth is m=1,the convergence speed is faster,and when the depth is m=2,the convergence rate is faster than that of the case of m=1.