In many application problems such as the electromagnetics, the unknowns are usually defined at the edges to satisfy the continuity requirement. This paper develops the first positivity-preserving edge-centred finite volume scheme for diffusion problems on general unstructured polygonal meshes. The edge-centred unknowns are primary and have associated finite volume equations. The cell-vertex and cell-centred unknowns are treated as auxiliary ones and are interpolated by the primary unknowns, making the final scheme purely edge-centred. The scheme has a fixed stencil due to the fixed decomposition of the co-normal, which makes the scheme very easy to implement. The positivity-preserving property is rigorously proved. Numerical experiments indicate that the scheme has second-order accuracy and positivity for heterogeneous and anisotropic problems on highly distorted meshes.

在许多应用问题（例如电磁学）中，未知数通常定义在边缘以满足连续性要求。本文针对一般非结构化多边形网格上的扩散问题开发了第一个保留正性的以边缘为中心的有限体积方案。以边缘为中心的未知数是主要的，并且具有相关的有限体积方程。单元顶点和单元中心未知数被视为辅助未知数，并由主要未知数进行插值，使最终方案纯粹以边缘为中心。由于共法线的固定分解，该方案具有固定的模板，这使得该方案非常容易实现。正性保持性质得到了严格证明。数值实验表明，该方案对于高畸变网格上的异质和各向异性问题具有二阶精度和积极性。