A Class Of Spectral Methods For Solving Partial Differential Equations

In this paper,we use a Legendre-spectral approximation method analysis the errormates for the Burgers equations and the unsteady,incompressible Navier-Stokes equations,for the time discretization we focus on a forward difference scheme.We provide some error estimates for the semidiscrete and fully discrecte solution which show that the velocity is first order accuracy in the proper mesh size.These estimates are proved assuming only a weak compatibility condition on the approximating spaces of velocity and pressure,which is statisfied by equal order interpolations.we provide a spectral scheme for N-S equations,what's more,the error estimation of the related variables of the Burgers equation and the incompressible N-S equation in this scheme is derived.Finally,by comparing with the classical Galerkin method,the superiority of the spectral method is obtained.we also proof the exactly of this scheme.