  Coarse grained, macroscale, spatial discretisations of nonlinear
nonautonomous partial differential\difference equations are given novel support
by centre manifold theory. Dividing the physical domain into overlapping
macroscale elements empowers the approach to resolve significant subgrid
microscale structures and interactions between neighbouring elements. The
crucial aspect of this approach is that centre manifold theory organises the
resolution of the detailed subgrid microscale structure interacting via the
nonlinear dynamics within and between neighbouring elements. The techniques and
theory developed here may be applied to soundly discretise on a macroscale many
dissipative nonautonomous partial differential\difference equations, such as
the forced Burgers' equation, adopted here as an illustrative example.
