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

TL;DR

This paper introduces an adaptive finite element splitting method for the incompressible Navier-Stokes equations, combining efficiency with rigorous error control, and demonstrates its effectiveness through numerical experiments.

Contribution

It develops an a posteriori error estimate for a splitting finite element method, enabling adaptive refinement for the Navier-Stokes equations.

Findings

The adaptive algorithm achieves high efficiency indices.

Computational error in momentum equation is linear in time step size.

Computational error in continuity equation is quadratic in time step size.

Abstract

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.