Parallel Auxiliary Space AMG Solver for H(div) Problems

In this paper we present a family of scalable preconditioners for matrices arising in the discretization of H(div) problems using the lowest order Raviart-Thomas finite elements. Our approach belongs to the class of “auxiliary space”-based methods and requires only the finite element stiffness matrix plus some minimal additional discretization information about the topology and orientation of mesh entities. We provide a detailed algebraic description of the theory, parallel implementation and different variants of this parallel auxiliary space divergence solver (ADS) and discuss its relations to the Hiptmair-Xu (HX) auxiliary space decomposition of H(div) [25] as well as the auxiliary space Maxwell solver AMS [27]. An extensive set of numerical experiments demonstrate the robustness and scalability of our implementation on large-scale H(div) problems with large jumps in the material coefficients. AMS subject classifications. 65F10, 65N30, 65N55