Two-grid Method And Discontinuous Galerkin Method For Maxwell's Equation Analysis In Two Kinds Of Media

This paper mainly studies the following two problems.In the first past,we develop the superconvergence analysis of two-grid algorithm by Crank-Nicolson finite element discrete scheme with the lowest Nedelec element for nonlinear power-law conductivity in Maxwell's problems.Our main contribution will have two parts.On one hand,in order to overcome the difficulty of misconvergence of classical two-grid method by the lowest Nedelec element,we employ the Newton-type Taylor expansion at the superconvergent solutions for the nonlinear terms on coarse mesh,which is different from the numerical solution on the coarse mesh classically.On the other hand,we push the two-grid solution to high accuracy by the postprocessing interpolation technique.Such a design can improve the computational accuracy in space and decrease time consumption simultaneously.Based on this design,we can obtain the convergent rate O(?t2+h2+H5/2)in three-dimension space,which means that the space mesh size satisfies h=O(H5/4).We also present two examples to verify our theorem.In the second past,we establish the discontinuous Galerkin method for Maxwell's problems in nonlocal electromagnetic meta-materials.The semi-discrete format is given,and give the proof of energy dissipation with time.In the case of spatial dispersion,the upwind numerical flux is adopted,and in the case of time dispersion,the convergence order of O(?t4+hN+1)under L2-norm is obtained by using the low-storage five-stage fourth-order Runge-Kutta method,where N is the degree of the basis function polynomial.We also present example for transverse electric(TE)case in two-dimension to verify our theorem.