Spectral-Finite Element Method for Solving Three-Dimensional Unsteady Navier-Stokes Equations

This paper is devoted to the combined Fourier spectral and finite element approxima- tions of three-dimensional, semi-periodic, unsteady Navier-Stokes equations. Fourier spectral method and finite element method are employed in the periodic and non-periodic directions respectively. A class of fully discrete schemes are constructed with artificial compression. Strict error estimations are proved. The analysis shows also that the classical two-dimensional velocity- pressure elements can be readily extended to solving such three-dimensional semi-periodic prob- lems, provided they satisfy the two-dimensional "inf-suf" condition.