A bi-partitioned iterative algorithm for solving linear systems arising from incompressible flow problems

• Iterative algorithm to solve linear system in stabilized finite elements for fluids.
• Uses Schur complement to separately solve for velocity and pressure.
• Order of magnitude improvement compared to GMRES while maintaining stability.

A novel iterative algorithm, called the bi-partitioned method, is introduced for efficiently solving the system of linear equations that arises from the stabilized finite element formulation of the Navier–Stokes equations. The bi-partitioned algorithm is a Krylov subspace method designed for a matrix with separated momentum and continuity blocks. This structure allows for formation of the Schur complement to separately solve for the velocity and pressure unknowns. Hence, the bi-partitioned algorithm can also be applied to problems with similar matrix structure, involving the Schur complement. Two separate spaces are constructed iteratively from the velocity and pressure solution candidates and optimally combined to produce the final solution. The bi-partitioned algorithm calculates the final solution to a given tolerance, regardless of the approximation made in construction of the Schur complement. The proposed algorithm is analyzed and compared to the generalized minimal residual (GMRES) algorithm using two incompressible-flow and one fluid–structure-interaction example, exhibiting up to an order of magnitude improvement is simulation cost while maintaining excellent stability.