The author introduces and analyzes a parallel additive Schwarz method (ASM) preconditioner for the system of equations arising from the finite element discretizations of a fourth order elliptic problem with large jumps in coefficients on nonconforming meshes. More precisely, a domain decomposition method is presented in terms of ASM abstract framework, that is, the author introduces space decomposition into a coarse space and two types of local spaces and bilinear forms defined on these spaces.
Here, locally, a Morley nonconfirming element is used. The condition number estimate proved here is almost optimal (it grows polylogarithmically as the sizes of the meshes decrease), which is independent of the jumps of the coefficients, that is, the number of conjugate gradient iterations is proportional only to \((1 + \log(H/h))\), where \(H\) and \(h\) denote the subdomain sizes and mesh sizes, respectively.