Finite Element discretizations of coupled multi-physics partial differential equation models require the handling of composed function spaces. In this paper we discuss software concepts and abstractions to handle the composition of function spaces, based on a representation of product spaces as trees of simpler bases. From this description, many different numberings of degrees of freedom by multi-indices can be derived in a natural way, allowing to adapt the function spaces to very different data layouts, so that it opens the possibility to directly use the finite element code with very different linear algebra codes, different data structures, and different algebraic solvers.
A recurring example throughout the paper is the stationary Stokes equation with Taylor--Hood elements as these are naturally formulated as product spaces and highlight why different storage patterns are desirable.
In the second half of the paper we discuss a particular realization of most of these concepts in the dune-functions module, as part of the DUNE ecosystem.