Structure-Preserving Finite Element Methods For Stochastic Maxwell Equations

Maxwell’s equations,proposed by James Clerk Maxwell in the 1860s,describe the relationship between electric and magnetic fields,charge density,and current density,forming the foundation of electromagnetics with broad applications.To describe the electromagnetic phenomena in random media,stochastic terms need to be added to the Maxwell’s equations to obtain the stochastic Maxwell equations.Generally,stochastic Maxwell equations are nonlinear partial differential equations and can be solved numerically due to the difficulty of obtaining analytical solutions.Researchers have proposed various numerical methods,including semi-implicit Euler schemes,stochastic symplectic Runge-Kutta methods,multiple symplectic numerical methods,energy-conserving methods,discontinuous Galerkin methods,and finite element approximations.This thesis proposes a fully discrete format for the stochastic Maxwell equations with multiplicative It? noise.Based on the existing time semi-discrete format[Chen C,Hong J,Ji L.SIAM J.Numer.Anal.,57(2019),728-750],a mixed finite element method is used for spatial discretization.This format maintains energy conservation and magnetic Gauss law in the expected sense.Using the projection-based quasi-interpolation operator,the optimal-order error estimation is obtained for the fully discrete format in the energy norm.Numerical experiments further validate the theoretical results.The fully discrete stochastic Maxwell equations form a saddle-point system with a 2 × 2 block structure.As the grid size decreases,the ill-posedness of the coefficient matrix increases,making some classical iterative methods inefficient in solving this problem.To address this issue,this thesis employs V-cycle multigrid methods and two efficient iterative methods for solving the problem.The experimental results show that both iterative methods do not significantly increase the number of iterations as the grid step decreases.