Space–time coupled spectral/ hp least-squares finite element formulation for the incompressible Navier–Stokes equations

Abstract

We consider least-squares finite element models for the numerical solution of the non-stationary Navier–Stokes equations governing viscous incompressible fluid flows. The paper presents a formulation where the effects of space and time are coupled, resulting in a true space–time least-squares minimization procedure, as opposed to a space–time decoupled formulation where a least-squares minimization procedure is performed in space at each time step. The formulation is first presented for the linear advection-diffusion equation and then extended to the Navier–Stokes equations. The formulation has no time step stability restrictions and is spectrally accurate in both space and time. To allow the use of practical C 0 element expansions in the resulting finite element model, the Navier–Stokes equations are expressed as an equivalent set of first-order equations by introducing vorticity as an additional independent variable and the least-squares method is used to develop the finite element model of the governing equations. High-order element expansions are used to construct the discrete model. The discrete model thus obtained is linearized by Newton's method, resulting in a linear system of equations with a symmetric positive definite coefficient matrix that is solved in a fully coupled manner by a preconditioned conjugate gradient method in matrix-free form. Spectral convergence of the L 2 least-squares functional and L 2 error norms in space–time is verified using a smooth solution to the two-dimensional non-stationary incompressible Navier–Stokes equations. Numerical results are presented for impulsively started lid-driven cavity flow, oscillatory lid-driven cavity flow, transient flow over a backward-facing step, and flow around a circular cylinder; the results demonstrate the predictive capability and robustness of the proposed formulation. Even though the space–time coupled formulation is emphasized, we also present the formulation and numerical results for least-squares space–time decoupled finite element models. The numerical results show that the space–time coupled formulation has superior predictive capabilities for flows demanding high space–time resolution, exemplified here by the transient flow over a backward-facing step.