Robust Two-Level Lower-Order Preconditioners for a Higher-Order Stokes Discretization with Highly Discontinuous Viscosities

The main goal of this paper is to present new robust and scalable preconditioned conjugate gradient algorithms for solving Stokes equations with large viscosities jumps across subregion interfaces and discretized on non-structured meshes. The proposed algorithms do not require the construction of a coarse mesh and avoid expensive communications between coarse and fine levels. The algorithms belong to the family of preconditioners based on non-overlapping decomposition of subregions known as balancing domain decomposition methods. The local problems employ two-level element-wise/subdomain-wise direct factorizations to reduce the size and the cost of the local Dirichlet and Neumann Stokes solvers. The Stokes coarse problem is based on subdomain constant pressures and on connected subdomain interface flux functions and rigid body motions. This guaranties scalability and solvability for the local Neumann problems. Estimates on the condition numbers and numerical experiments based on unstructured mesh parallel implementation are also discussed.