Optimal L ² Error Quadratic Serendipity Finite Volume Element Methods for Diffusion Equations on Polygonal Meshes

This paper proposes a locally conservative corrected quadratic serendipity finite element solution for the diffusion equation on arbitrary convex polygonal meshes based on generalized barycentric coordinates. The construction of the solution is divided into two steps. First, a quadratic serendipity finite element scheme for the diffusion equation on polygonal meshes is constructed, and the finite element solution is obtained. Second, based on the finite element solution, a finite volume element scheme is derived by adding specially constructed bubble functions. Theoretically, it is proven that the resulting solution not only satisfies the most classical conservation properties of the finite volume element schemes but also achieves a theoretical analysis comparable to that of the finite element methods. To the best of our knowledge, this is the first locally conservative corrected quadratic serendipity finite element solution on polygonal meshes that is unconditionally stable and has optimal and errors. Finally, the theoretical analysis is validated through several typical numerical examples and compared with several existing finite volume element schemes.