Multilevel Methods for $p$ -Adaptive Finite Element Analysis of Electromagnetic Scattering

In p-adaptive finite element analysis, the large, sparse matrix that arises can be block structured according to the hierarchical level of the unknowns. A multilevel preconditioner for the matrix is a V-cycle that starts by applying Gauss-Seidel to the highest level, then the next level down, and so on. On the other side of the V, Gauss-Seidel is applied in the reverse order. At the bottom of the V is the lowest order system, which typically is solved exactly with a direct solver. However, for a complex geometry even the lowest order system may be too large for direct factorization. Here an alternative is proposed: to continue the V-cycle downwards, first into a set of auxiliary, node-based spaces, then through a series of progressively smaller matrices generated by an algebraic multigrid method. The smallest matrix is solved by factorization. The method is applied to p-adaptive analysis of a five-resonator iris filter, a split-ring resonator loaded waveguide, a “buckyball” metallic frame surrounding a conducting sphere, and a noncommensurate frequency selective surface. Tetrahedral elements up to fourth order are used. The largest matrix has over 12 million rows and 0.6 billion nonzero entries.