In this work we introduce a new class of lowest order methods for diffusive problems on general meshes with only one unknown per element. 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...

In this work we introduce a new class of lowest order methods for
diffusive problems on general meshes with only one unknown per
element.
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 the present work we introduce a new family of cell-centered Finite
Volume schemes for anisotropic and heterogeneous diffusion operators
inspired by the MPFA L method.
A very general framework for the convergence study of finite volume
methods is provided and then used to establish the convergence of the
new method.
Fairly general meshes are covered and a computable sufficient
criterion for coercivity is provided.
In order to guarantee consistency in the presence of heterogeneous
diffusivity,...

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