  We consider the numerical approximation of second-order semi-linear parabolic
stochastic partial differential equations interpreted in the mild sense which
we solve on general two-dimensional domains with a $\mathcal{C}^2$ boundary
with homogeneous Dirichlet boundary conditions. The equations are driven by
Gaussian additive noise, and several Lipschitz-like conditions are imposed on
the nonlinear function. We discretize in space with a spectral Galerkin method
and in time using an explicit Euler-like scheme. For irregular shapes, the
necessary Dirichlet eigenvalues and eigenfunctions are obtained from a boundary
integral equation method. This yields a nonlinear eigenvalue problem, which is
discretized using a boundary element collocation method and is solved with the
Beyn contour integral algorithm. We present an error analysis as well as
numerical results on an exemplary asymmetric shape, and point out limitations
of the approach.
