  A new type of systematic approach to study the incompressible Euler equations
numerically via the vanishing viscosity limit is proposed in this work. We show
the new strategy is unconditionally stable that the $L^2$-energy dissipates and
$H^s$-norm is uniformly bounded in time without any restriction on the time
step. Moreover, first-order convergence of the proposed method is established
including both low regularity and high regularity error estimates. The proposed
method is extended to full discretization with a newly developed iterative
Fourier spectral scheme. Another main contributions of this work is to propose
a new integration by parts technique to lower the regularity requirement from
$H^4$ to $H^3$ in order to perform the $L^2$-error estimate. To our best
knowledge, this is one of the very first work to study incompressible Euler
equations by designing stable numerical schemes via the inviscid limit with
rigorous analysis. Furthermore, we will present both low and high regularity
errors from numerical experiments and demonstrate the dynamics in several
benchmark examples.
