In this work we introduce a new class of lowest order methods for
diffusive problems on general meshes with only one unknown per
The underlying idea is to construct an incomplete piecewise affine
polynomial space with optimal approximation properties starting
from values at cell centers.
To do so we borrow ideas from multi-point finite volume methods,
although we use them in a rather different context.
The incomplete polynomial space replaces classical complete
polynomial spaces in discrete formulations inspired by discontinuous
Two problems are studied in this work:
a heterogeneous anisotropic diffusion problem, which is used
to lay the pillars of the method, and the incompressible
Navier-Stokes equations, which provide a more realistic
An exhaustive theoretical study as well as a set of numerical
examples featuring different difficulties are provided.