[Paper Review] A Lagrange-Galerkin scheme with a locally linearized velocity for the Navier--Stokes equations

Masahisa Tabata, Shinya Uchiumi|arXiv (Cornell University)|May 25, 2015

Advanced Numerical Methods in Computational Mathematics17 references3 citations

TL;DR

This paper proposes a novel Lagrange--Galerkin finite element scheme for the Navier--Stokes equations that eliminates numerical quadrature errors by using a locally linearized velocity approximation in the backward Euler method to track particle trajectories. The scheme ensures exact implementation, leading to provably stable and optimally convergent results with $\ell^{\infty}(H^{1})\times\ell^{2}(L^{2})$ error estimates for both $\mathrm{P}_{2}/\mathrm{P}_{1}$ and $\mathrm{P}_{1}^{+}/\mathrm{P}_{1}$ elements.

ABSTRACT

We present a Lagrange--Galerkin scheme free from numerical quadrature for the Navier--Stokes equations. Our idea is to use a locally linearized velocity and the backward Euler method in finding the position of fluid particle at the previous time step. Since the scheme can be implemented exactly as it is, the theoretical stability and convergence results are assured. While the conventional Lagrange--Galerkin schemes may encounter the instability caused by numerical quadrature errors, the present scheme is genuinely stable. For the $\pk 2/\pk 1$- and $\mini$-finite elements optimal error estimates are proved in $\ell^\infty(H^1) imes \ell^2(L^2)$ norm for the velocity and pressure. We present some numerical results, which reflect these estimates and also show the genuine stability of the scheme.

Motivation & Objective

• To address the instability in conventional Lagrange--Galerkin schemes caused by numerical quadrature errors in integrating composite function terms.
• To develop a scheme that enables exact implementation of the method without relying on numerical integration.
• To prove optimal convergence rates in natural norms for velocity and pressure under realistic finite element discretizations.
• To demonstrate robustness for high Reynolds number flows, where standard schemes often fail due to oscillations.

Proposed method

• Introduce a locally linearized velocity approximation to replace the exact solution of the ODE system for particle trajectories.
• Use the backward Euler method to compute the foot of the characteristic curve at the previous time step using the locally linearized velocity.
• Construct the weak formulation of the Navier--Stokes equations using the method of characteristics with the approximate trajectory mapping.
• Implement the scheme using $\mathrm{P}_{2}/\mathrm{P}_{1}$ and $\mathrm{P}_{1}^{+}/\mathrm{P}_{1}$ finite elements for velocity and pressure, respectively.
• Ensure exact integration of all terms by leveraging the piecewise linear structure of the approximate velocity map.
• Derive and prove optimal error estimates in $\ell^{\infty}(H^{1})\times\ell^{2}(L^{2})$ norms for velocity and pressure.

Experimental results

Research questions

• RQ1 Can a Lagrange--Galerkin scheme be constructed that avoids numerical quadrature errors while maintaining optimal convergence rates?
• RQ2 Does the use of a locally linearized velocity in the backward Euler method yield a stable and accurate scheme for convection-dominated Navier--Stokes flows?
• RQ3 What is the convergence order of the proposed scheme in $\ell^{\infty}(H^{1})$ and $\ell^{2}(L^{2})$ norms for both $\mathrm{P}_{2}/\mathrm{P}_{1}$ and $\mathrm{P}_{1}^{+}/\mathrm{P}_{1}$ elements?
• RQ4 How does the scheme perform in high Reynolds number flows compared to standard Lagrange--Galerkin schemes with numerical quadrature?
• RQ5 Can the scheme maintain stability and accuracy without oscillations in benchmark problems like the lid-driven cavity flow?

Key findings

• The proposed scheme achieves optimal convergence rates of order $O(h^2)$ in $\ell^{\infty}(H^{1})$ for velocity and $\ell^{2}(L^{2})$ for pressure under $\Delta t = h^2$ for both $\mathrm{P}_{2}/\mathrm{P}_{1}$ and $\mathrm{P}_{1}^{+}/\mathrm{P}_{1}$ elements.
• For $\mathrm{P}_{1}^{+}/\mathrm{P}_{1}$ elements, the scheme achieves $O(h^2)$ convergence in $\ell^{\infty}(L^{2})$ for velocity under $\Delta t = h^2$.
• When $\Delta t = h^3$, the scheme achieves $O(h^3)$ convergence in $\ell^{\infty}(L^{2})$ for velocity in the $\mathrm{P}_{1}^{+}/\mathrm{P}_{1}$ case, confirming theoretical expectations.
• Numerical experiments confirm that the scheme remains stable and free of oscillations even at high Reynolds numbers ($\nu = 10^{-4}$ and $10^{-5}$), while the standard scheme exhibits spurious oscillations.
• The scheme shows robust performance in the lid-driven cavity flow problem, with no observed instability or oscillation in velocity components at $t^n = 8$ for $\nu = 10^{-4}$ and $10^{-5}$.
• The convergence orders observed in numerical results closely match the theoretical predictions, validating the error estimates.