Domain decomposition preconditioners for hp finite element approximations on geometrically refined boundary layer meshes.

We develop and analyze Neumann-Neumann and FETI methods for hp finite element approximations of scalar elliptic problems on geometrically refined boundary layer meshes in two and three dimensions. These are meshes that are highly anisotropic where the aspect ratio grows exponentially with the polynomial degree. The condition number is independent of the aspect ratio of the mesh and of potentially large jumps on the coefficients. In addition, it only grows polylogarithmically with the polynomial degree, as in the case of p approximations on shape-regular meshes. This is a joint work with X. Vasseur.