Post-processing technique of two-grid algorithm for wave propagation with Debye polarization in nonlinear dielectric materials

摘要

In this paper, we develop the superconvergence analysis of the implicit second-order two-grid discrete scheme with the lowest Nedelec element for wave propagation with Debye Polarization in nonlinear Dielectric materials. Our main contribution will have two parts. On one hand, in order to overcome the difficulty of misconvergence of classical two-grid algorithm by the lowest Nedelec elements, we employ the Newton-type Taylor expansion at the superconvergent solutions for the nonlinear terms on coarse mesh, which is different from the classical numerical solution on the coarse mesh. On the other hand, we push the two-grid solution to high accuracy by the interpolation post-processing technique. Such a design can both improve the computational accuracy in spatial and decrease time consumption simultaneously. Based on this design, we can obtain the convergent rate O(τ~2 +h~2+ H~3), and the spatial convergence can be obtained by choosing the mesh size h = O(H~(3/2)). At last, one numerical experiment is illustrated to verify our theoretical results.