An Adaptive Polygonal Finite Volume Element Method Based on the Mean Value Coordinates for Anisotropic Diffusion Problems (Communications in Computational Physics, 2025)
We propose a polygonal finite volume element method based on the meanvalue coordinates for anisotropic diffusion problems on star-shaped polygonal meshes.Because the convex cells with hanging nodes are always star-shaped, the computationon them is no longer a problem. Naturally, we apply this advantage of the new polygonal finite volume element method to construct an adaptive polygonal finite volume element algorithm. Moreover, we introduce two refinement strategies, called quadtree-based refinement strategy and polytree-based refinement strategy respectively, andthey all have great performance in our numerical tests. The new adaptive algorithmallows the use of hanging nodes, and the number of hanging nodes on each edge isunrestricted in general. Finally, several numerical examples are provided to show theconvergence and efficiency of the proposed method on various polygonal meshes. Thenumerical results also show that the new adaptive algorithm not only reduces the computational cost and the implementation complexity in mesh refinement, but also ensures the accuracy and convergence.
Keywords: Polygonal finite volume element method; mean value coordinates; hanging nodes; adaptive algorithm.