A Fast and Matrix-Free Segregated Solution Approach to Incompressible Navier-Stokes Equations Discretized by Structure-Preserving Splines on Cartesian Grids
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Fast formation and assembly techniques on Cartesian grids and tensor-product spaces have been proven to robustly reduce the computational cost by orders of magnitude. The application of such techniques in the context of Navier-Stokes, Navier-Stokes-Korteweg, Cahn-Hilliard and similar equations enables high resolution of the underlying physics and thus are of interest for future model and method development. In the current work we focus on the variational formulation of incompressible Navier-Stokes equations discretized by structure-preserving isogeometric Raviart-Thomas spline spaces with weak tangential boundary conditions. We propose a segregated solution approach based on the Schur complement of the system matrix. For the velocity step we develop a fast matrix-free and parallel direct solver with O(N) computational cost. For the pressure step, which is solved iteratively by the Conjugate Gradient method, we also propose a matrix-free parallel approach with O(N) computational cost and a suitable and cheap preconditioner. We validate the proposed method against benchmarks established in the literature and compare our performance to standard solvers in Fenics.
