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Developments in dynamical systems theory provides new support for thediscretisation of \pde{}s and other microscale systems. By systematicallyresolving subgrid microscale dynamics the new approach constructsasymptotically accurate, macroscale closures of discrete models of the \pde.Here we explore reaction-diffusion problems in two spatial dimensions. Centremanifold theory ensures that slow manifold, holistic, discretisations exists,are quickly attractive, and are systematically approximated. Special couplingof the finite elements ensures that the resultant discretisations areconsistent with the \pde to as high an order as desired. Computer algebrahandles the enormous algebraic details as seen in the specific application tothe Ginzburg--Landau equation. However, higher order models in 2D appear torequire a mixed numerical and algebraic approach that is also developed. Beingdriven by the residuals of the equations, the modelling here may bestraightforwardly adapted to a wide class of reaction-diffusion differentialand lattice equations in multiple space dimensions.

Holistic discretisation ensures fidelity to dynamics in two spatial dimensions