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An Adaptive Finite Element Splitting Method for the Incompressible Navier-Stokes Equations

We present an adaptive finite element method for the incompressible Navier--Stokes equations based on a standard splitting scheme (the incremental pressure correction scheme). The presented method combines the efficiency and simplicity of a splitting method with the powerful framework offered by the finite element method for error analysis and adaptivity. An a posteriori error estimate is derived which expresses the error in a goal functional of interest as a sum of contributions from spatial discretization, time discretization and a term that measures the deviation of the splitting scheme from a pure Galerkin scheme (the computational error). Numerical examples are presented which demonstrate the performance of the adaptive algorithm and high quality efficiency indices. It is further demonstrated that the computational error of the Navier--Stokes momentum equation is linear in the size of the time step while the computational error of the continuity equation is quadratic in the size of the time step.