A Decoupled Iterative Method for Mixed Problems

where ΓD and ΓN are assumed to partition the boundary of Ω, and ~ν(~x) denotes the outward unit normal from Ω at ~x ∈ ΓN . Additionally we assume that each connected component of ΓN is simply connected. The numerical treatment of (1 3) involves the solution of usually very large indefinite linear equation systems. In this paper we describe a very efficient and practicable iterative method to solve these systems by decoupling the vector of velocities from the vector of pressures, resulting in a symmetric positive definite velocity system and a triangular pressure system. The crucial step in this approach is the construction of a basis for the divergence-free Raviart-ThomasNédélec elements (using results from Algebraic Topology and Graph Theory).