  We study how well the Gaussian approximation is valid for computing the
covariance matrices of the convergence power and bispectrum in weak
gravitational lensing analyses. We focus on its impact on the cosmological
parameter estimations by comparing the results with and without non-Gaussian
error contribution in the covariance matrix. We numerically derive the
covariance matrix as well as the cosmology dependence of the spectra from a
large set of N-body simulations performed for various cosmologies and carry out
Fisher matrix forecasts for tomographic weak lensing surveys with three source
redshifts. After showing the consistency of the power and bispectra measured
from our simulations with the state-of-the-art fitting formulas, we investigate
the covariance matrix assuming a typical ongoing survey across 1500 deg^2 with
the mean source number density of 30 arcmin^{-2} at the mean redshift z_s=1.0.
Although the shape noise contributes a significant fraction to the total error
budget and it mitigates the impact of the non-Gaussian error for this source
number density, we find that the non-Gaussian error degrades the cumulative
signal-to-noise ratio up to the maximum multipole of 2000 by a factor of about
2 (3) in the power (bi-) spectrum analysis. Its impact on the final
cosmological parameter forecast with 6 parameters can be as large as 15% in the
size of the one-dimensional statistical error. This can be a problem in future
wide and deep weak lensing surveys for precision cosmology. We also show how
much the dark energy figure of merit is affected by the non-Gaussian error
contribution and demonstrate an optimal survey design with a fixed
observational time.
