  Dynamical systems theory provides powerful methods to extract effective
macroscopic dynamics from complex systems with slow modes and fast modes. Here
we derive and theoretically support a macroscopic, spatially discrete, model
for a class of stochastic reaction-diffusion partial differential equations
with cubic nonlinearity. Dividing space into overlapping finite elements, a
special coupling condition between neighbouring elements preserves the
self-adjoint dynamics and controls interelement interactions. When the
interelement coupling parameter is small, an averaging method and an asymptotic
expansion of the slow modes show that the macroscopic discrete model will be a
family of coupled stochastic ordinary differential equations which describe the
evolution of the grid values. This modelling shows the importance of subgrid
scale interaction between noise and spatial diffusion and provides a new
rigourous approach to constructing semi-discrete approximations to stochastic
reaction-diffusion partial differential equations.
