A new method for mimetic interpolation on polygonal meshes is described. This new method is based on harmonic function interpolation. Explicit formulas for harmonic functions on general polygons do not exist, so truncated harmonic polynomial expansions are used for computational efficiency. We show that the naive harmonic polynomial expansion, is not stable for arbitrary polygons. However, higher level truncations of harmonic interpolations are stable and accurate. This new method is shown to be a direct extension of the lowest order Raviart-Thomas finite elements to polygons. This method is also a direct extension of the finite volume MAC method to polygons. The accuracy of the interpolation is shown to be first-order irrespective of the polynomial truncation level. However the accuracy of vector Laplace equation solutions using this inner product is shown to be second-order accurate, in keeping with other lowest order Mimetic methods. The versatility of this numerical method is demonstrated on a multiphase incompressible flow problem with a density jump of 1000, on a moving polygonal mesh.

描述了一种在多边形网格上进行模拟插值的新方法。这种新方法基于谐波函数插值。通用多边形上的谐波函数的显式公式不存在，因此使用截断的谐波多项式展开来提高计算效率。我们表明，朴素的谐波多项式展开对于任意多边形而言是不稳定的。但是，谐波插值的高位截断是稳定且准确的。这种新方法被证明是最低阶Raviart-Thomas有限元到多边形的直接扩展。该方法还是有限体积MAC方法到多边形的直接扩展。插值的精度显示为一阶，与多项式截断级别无关。但是，与其他最低阶拟态方法保持一致，使用此内积的向量Laplace方程解的精度显示为二阶精度。该数值方法的多功能性在移动多边形网格上针对密度跃迁为1000的多相不可压缩流动问题进行了证明。