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Description

Vector spaces of piecewise polynomial functions are a central concept in the
finite element method. However, in implementations, it is much more natural
to focus on bases of such spaces, rather than on the spaces themselves.
This abstraction leads to interfaces with clearer semantics.

The dune-functions module, which is a recent addition to the Dune eco-system,
offers a programmer interface for such function space bases. The interface
offers all functionality needed to conveniently implement finite element algorithms.
Besides scalar and vector-valued bases, a systematic composition mechanism
allows to construct advanced bases from simpler ones. Orderings of the basis
functions are configurable, and the individual functions can be indexed
using plain integers or different types of multi-indices.
Implementations of this interface for several relevant spaces are provided
as part of the module.

The clear separation between bases, coefficient vectors, and finite element
matrix assemblers makes the handling of finite element spaces easier. This is
even more true if these spaces involve time- or problem-dependent enrichment functions
like in the XFEM or cut-cell method. We demonstrate this with an example from
fracture mechanics, where the dune-functions basis interface helps to hide
a large part of the XFEM index book-keeping from the user.