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

摘要

We present an adaptive finite element method for the incompressibleNavier--Stokes equations based on a standard splitting scheme (the incrementalpressure correction scheme). The presented method combines the efficiency andsimplicity of a splitting method with the powerful framework offered by thefinite element method for error analysis and adaptivity. An a posteriori errorestimate is derived which expresses the error in a goal functional of interestas a sum of contributions from spatial discretization, time discretization anda term that measures the deviation of the splitting scheme from a pure Galerkinscheme (the computational error). Numerical examples are presented whichdemonstrate the performance of the adaptive algorithm and high qualityefficiency indices. It is further demonstrated that the computational error ofthe Navier--Stokes momentum equation is linear in the size of the time stepwhile the computational error of the continuity equation is quadratic in thesize of the time step.