Boundary integral equation methods for Lipschitz domains in linear elasticity

A review of stable boundary integral equation methods for solving the Navier equation with either Dirichlet or Neumann boundary conditions in the exterior of a Lipschitz domain is presented. The conventional combined-field integral equation (CFIE) formulations, that are used to avoid spurious resonances, do not give rise to a coercive variational formulation for nonsmooth geometries anymore. To circumvent this issue, either the single layer or the double layer potential operator is composed with a compact or a Steklov–Poincaré type operator. The later can be constructed from the well-know elliptic boundary integral operators associated to the Laplace equation and Gårding's inequalities are satisfied. Some Neumann interior eigenvalue computations for the unit square and cube are presented for forthcoming numerical investigations.