Steady state heat conduction, diffusion or electrostatic problems are described by Piosson’s equation along with appropriate boundary conditions. Several numerical methods have been developed to solve this boundary value problem on regular and irregular nodal arrangements. We compare performance of three of them, namely the finite volume method, the virtual element method and the finite element method, applied on specific spatial discretization provided by Voronoi tessellation on random set of nuclei. The finite volume method advantageously employ perpendicularity of the faces and connections between nuclei.
The virtual element method provides correct integration scheme for polygonal finite elements because they otherwise suffer from imprecise integration of non-polynomial shape function. The last method under comparison is the finite element method based on polygonal elements created by static condensation of isoparametric triangles. The methods are compared on several patch tests and convergence studies are performed.
