Abstract:

Iterative substructuring methods, also known as Schur complement methods, form an important family of domain decomposition algorithms. They are preconditioned conjugate gradient methods where solvers on local subregions and a solver on a coarse mesh are used to construct the preconditioner. For conforming finite element approximations of $H^1$, it is known that the number of conjugate gradient steps required to reduce the residual norm by a fixed factor is independent of the number of substructures and that it grows only as the logarithm of the dimension of the local problem associated with an individual substructure. In this paper, the same result is established for similar iterative methods for low--order N{\'e}d{\'e}lec finite elements, which approximate $\Hcurl$ in two dimensions. Results of numerical experiments are also provided.