  In this paper, we propose a nonlinear positivity-preserving finite volume
element(FVE) scheme for anisotropic diffusion problems on quadrilateral meshes.
Based on an overlapping dual partition, the one-sided flux is approximated by
the iso-parametric bilinear element. A positivity-preserving nonlinear scheme
with vertex-centered unknowns is obtained by a new two-point flux technique,
which avoids the convex decomposition of co-normals and the introduction of
intermediate unknowns. The existence of a solution is proved for this nonlinear
system by applying the Brouwer's theorem. Numerical results show that the
proposed positivity-preserving scheme is effective on distorted quadrilateral
meshes and has approximate second-order accuracy for both isotropic and
anisotropic diffusion problems. Moreover, the presented scheme is applied on an
equilibrium radiation diffusion problem with discontinuous coefficients.
Numerical results show that the new scheme is much more efficient than the
standard FVE method.
