A Method for r Refinement of Polygonal Finite Element Based on Mean Value Coordinates

Local and anisotropic features are widely present in various physical phenomena. In order to enhance the accuracy of modeling equations for this phenomenon, a r-refinement method based on mean value coordinates for polygonal elements is proposed. The objective function is constructed using the error between the solution under the mean value coordinates based on quadratic serendipity element shape functions and the mean value coordinates shape functions. By optimizing the locations of sites in polygonal elements, the mesh is iteratively updated along with the solution surface of the equation, allowing regions with drastic surface characteristic changes to have more degrees of freedom. Numerical results of solving the 2D Poisson equation demonstrate that with the same number of the sites, the error can be reduced by over 50%, indicating that the proposed method effectively improves the accuracy of the solution.