In this talk, we report on some new results on two iterative substructuring methods of Neumann-Neumann and FETI type for some edge element approximations of Maxwell equations in two dimensions.
Iterative substructuring methods provide powerful preconditioners for the solution of linear systems arising from the finite element approximation of partial differential equations. In these methods, the computational domain is partitioned into non-overlapping subdomains and a preconditioner is built by solving local problems on the subdomains.
The Neumann-Neumann and FETI preconditioners that we consider have many algorithmic components in common, such as the solution of local Dirichlet and Neumann problems, a set of local functions needed to build a coarse space correction, and a set of scaling functions that employ the values of the coefficients on the subdomains and ensure that the condition number of the resulting linear system is independent of the jumps of the coefficients. In addition, FETI methods are more easily generalizable to approximations on non-matching grids.
For both methods, we show bounds for the condition number of the corresponding preconditioned systems which grow logarithmically with the number of unknowns in each subdomain. Such bounds are independent of possibly large jumps of the coefficients across the boundaries of the subdomains. For a Neumann-Neumann and a FETI method, we present some numerical results for conforming approximations. We will also show some results for a FETI method for a mortar approximation on non-matching grids.