An exactly computable Lagrange–Galerkin scheme for the Navier–Stokes equations and its error estimates

Abstract:

We present a Lagrange–Galerkin scheme, which is computable exactly, for the Navier–Stokes equations and show its error estimates. In the Lagrange–Galerkin method we have to deal with the integration of composite functions, where it is difficult to get the exact value. In real computations, numerical quadrature is usually applied to the integration to obtain approximate values, that is, the scheme is not computable exactly. It is known that the error caused from the approximation may destroy the stability result that is proved under the exact integration. Here we introduce a locally linearized velocity and the backward Euler method in solving ordinary differential equations in the position of the fluid particle. Then, the scheme becomes computable exactly, and we show the stability and convergence for this scheme. For the $\mathrm {P}_{2}/\mathrm {P}_{1}$- and $\mathrm {P_{1}+}/\mathrm {P}_{1}$-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 robust stability for high Reynolds numbers in the cavity flow problem.