A mimetic method for polygons

Abstract 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.