Abstract:

The iterative substructuring methods, also known as Schur complement methods, form one of two important families of domain decomposition algorithms. They are based on a partitioning of a given region, on which the partial differential equation is defined, into non-overlapping substructures. The preconditioners of these conjugate gradient methods are then defined in terms of local problems defined on individual substructures and pairs of substructures, and, in addition, a global problem of low dimension. An iterative method of this kind is introduced for the lowest order Raviart-Thomas finite elements in three dimensions and it is shown that the condition number of the relevant operator is independent of the number of substructures and grows only as the square of the logarithm of the number of unknowns associated with an individual substructure. The theoretical bounds are confirmed by a series of numerical experiments.