After an introduction, we consider the extension of iterative substructuring algorithms to the equations of linear elasticity. In order to be able to handle almost incompressible materials, we choose to work with mixed finite element methods; the resulting systems have much in common with finite element models for Stokes' equation. We explore a number of different algorithmic ideas, formulate new theoretical results, and report on some numerical experiments. Our work is joint with Luca Pavarino and principally concerns higher order methods of spectral element type.
In a second part, we discuss new domain decomposition algorithms for Helmholtz's equation. Our methods use overlapping subregions and the resulting local problems have boundary conditions of Sommerfeld type. Although the finite element approximation of the Helmholtz equation is continuous, the iterates can have jumps across the subdomain boundaries. Our algorithms generalizes a method developed and analyzed by Bruno Despres. Different variants are considered and results of numerical experiments are presented.
Our work is joint with Xiao-Chuan Cai, Mario Casarin, and Frank Elliott.