# A positivity-preserving finite volume element method for anisotropic diffusion problems on quadrilateral meshes

## TL;DR

This paper introduces a nonlinear finite volume element scheme that preserves positivity for anisotropic diffusion problems on quadrilateral meshes, demonstrating high accuracy and efficiency even on distorted meshes and with discontinuous coefficients.

## Contribution

A novel positivity-preserving nonlinear FVE scheme using a two-point flux technique for anisotropic diffusion on quadrilateral meshes, avoiding complex decompositions.

## Key findings

- Effective on distorted quadrilateral meshes
- Achieves approximate second-order accuracy
- More efficient than standard FVE methods

## Abstract

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.