A Lagrange-Galerkin scheme with a locally linearized velocity for the Navier--Stokes equations
This work addresses numerical instability in computational fluid dynamics simulations, offering a provably stable scheme for finite element methods.
The authors propose a Lagrange-Galerkin scheme for Navier-Stokes equations that eliminates numerical quadrature errors by using a locally linearized velocity and backward Euler method, achieving genuine stability and optimal error estimates in numerical tests.
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 $\ell^\infty(H^1)\times \ell^2(L^2)$ norm for the velocity and pressure. We present some numerical results, which reflect these estimates and also show the genuine stability of the scheme.