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Multigrid methods for Maxwell’s equations and a fourth-order curl problem based on Hodge decomposition

In this work, we investigate the numerical solution for two-dimensional Maxwell’s equations on graded meshes. The approach is based on the Hodge decomposition. The solution u of Maxwell’s equations is approximated by solving standard second order elliptic problems. The quasi-optimal error estimates for both u and curl of u in the L2norm are obtained on graded meshes. We then prove the uniform convergence of the W-cycle and full multigrid algorithms for the resulting discrete problems. The performance of these methods is illustrated by numerical results. Similar numerical approach can also be applied to solve a fourth order curl problem. We will report some preliminary results for the multigrid
algorithms.