Combined direct-iterative matrix solvers for hierarchal vector finite elements

A preconditioned conjugate gradient solver works reasonably well for the matrix equations obtained when hierarchal vector finite elements are applied to problems in three-dimensional electromagnetics. By adding to the elements a redundant subspace of constant gradient functions, considerably faster convergence is achieved. Further improvement is possible by combining the iterative solver with direct approaches: by employing a frontal solver for the lowest order degrees of freedom and by eliminating the interior degrees of freedom from each element. With these methods, the number of iterations is virtually unchanged up to fourth order, and during p-adaptive analysis the time taken for matrix solution grows with the number of degrees of freedom at about the same rate as it does in h-adaption, despite the reduction in sparsity.