  We consider a fully discrete scheme for nonlinear stochastic partial
differential equations with non-globally Lipschitz coefficients driven by
multiplicative noise in a multi-dimensional setting. Our method uses a
polynomial based spectral method in space, so it does not require the elliptic
operator $A$ and the covariance operator $Q$ of noise in the equation commute,
and thus successfully alleviates a restriction of Fourier spectral method for
SPDEs pointed out by Jentzen, Kloeden and Winkel.
  The discretization in time is a tamed semi-implicit scheme which treats the
nonlinear term explicitly while being unconditionally stable. Under regular
assumptions which are usually made for SPDEs with additive noise, we establish
optimal strong convergence rates in both space and time for our fully discrete
scheme. We also present numerical experiments which are consistent with our
theoretical results.
