Spectral discretization of the time‐dependent vorticity–velocity–pressure formulation of the Stokes problem

Abstract

In this paper, we study the time‐dependent vorticity–velocity–pressure formulation of Stokes problem in two‐ and three‐dimensional domains provided with nonstandard boundary conditions, related to the normal component of the velocity and the tangential components of the vorticity. This problem is discretized by implicit Euler's scheme in time and spectral method in space. We prove an optimal error estimate for the three unknowns.

Spectral discretization of time-dependent vorticity–velocity–pressure formulation of the Navier–Stokes equations

“…Then we conclude that (ω n , u n ) is a solution of problem (10) for any w = θ + curlv ∈ V. Moreover, since the solution (ω n , u n ) is bounded by r, we obtain (16) by the same proof as for [12,Corollary 1].…”

“…IV, Corollary 1.1]), we conclude that problem (28) has a solution (ω n N , u n N ) in U N . Moreover, the solution (ω n N , u n N ) is bounded by r N , so we obtain (29) by the same proof as that of [12,Proposition 5)].…”

“…By passing to the limit in (21), it is readily checked that (ω n , u n ) is a solution of problem (10). Moreover, the solution (ω n , u n ) is bounded by r, then we obtain (20) by the same proof as for [12,Corollary 1].…”

“…We derive from the continuity of the trilinear form N in (13) that the function is continuous on the space U. Having the following inequality proved in [12]:…”

“…This type of problem has been handled in the stationary case by the finite element method [3,7] and by the spectral method [8,9]. However, the nonstationary case has been considered for a posteriori analysis of the finite discretization [10,11] and also for a spectral discretization of the linear Stokes problem [12]. Relying on the equivalent variational formulation of problem (2), we propose a semidiscrete scheme based on the backward Euler method.…”