An Edge-based cascadic multigrid method for H(curl) problems

An efficient extrapolation cascadic multigird (EXCMG) method is developed to solve large linear systems resulting from edge element discretizations of 3D H(curl)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H(\textbf{curl})$$\end{document} problems on rectangular meshes. By treating edge unknowns as defined on the midpoints of edges, following the similar idea of the nodal EXCMG method, we design a new prolongation operator for 3D edge-based discretizations, which is used to construct a high-order approximation to the finite element solution on the refined grid. This good initial guess greatly reduces the number of iterations required by the multigrid smoother. Furthermore, the divergence correction technique is employed to further speed up the convergence of the multigrid method. Numerical examples including problems with high-contrast discontinuous coefficients are presented to validate the effectiveness of the proposed EXCMG method. The edge-based EXCMG method is more efficient than the auxiliary-space Maxwell solver (AMS) for definite problems in the considered geometrical configuration, and it can also efficiently solve large-scale indefinite problems encountered in engineering and scientific fields.