Finite Element Approximation of the Nonstationary Navier-Stokes Problem. Part 2: Stability of Solutions and Error Estimates Uniform in Time

摘要

A finite element method approach using a stability theory for the solution of the general nonstationary Navier-Stokes equations based entirely on energy methods is presented. Solution stability assumptions are introduced in order to extend results local in time to ones which are global. If the solution of the initial boundary value problem is stable, then the error in its discrete approximation remains small uniformly in time. From the stability of a discrete solution, for a single sufficiently small choice of the mesh size, the global existence of a closely neighboring smooth solution can be inferred.