The reduced basis element method is a new approach for approximating
the solution of problems described by partial differential equations.
The method takes its roots in domain decomposition methods and
reduced basis discretizations. The basic idea is to first decompose
the computational domain into a series of subdomains that are deformations
of a few reference domains (or generic computational parts).
Associated with each reference domain are precomputed solutions
corresponding to the same governing partial differential equation,
but solved for different choices of deformations of the reference
subdomains and mapped onto the reference shape.
The approximation corresponding to a new shape is then taken
to be a linear combination of the precomputed solutions, mapped
from the generic computational part to the actual computational part.
We extend earlier work in this direction to solve incompressible
fluid flow problems governed by the steady Stokes equations.
Particular focus is given to satisfying the inf-sup condition,
to a posteriori error estimation,
and to “gluing” the local solutions together in the multidomain case.