A Quadratic Serendipity Finite Volume Element Method on Arbitrary Convex Polygonal Meshes

Abstract

Based on the idea of serendipity element, we construct and analyze the first quadratic serendipity finite volume element method for arbitrary convex polygonal meshes in this article. The explicit construction of quadratic serendipity element shape function is introduced from the linear generalized barycentric coordinates, and the quadratic serendipity element function space based on Wachspress coordinate is selected as the trial function space. Moreover, we construct a family of unified dual partitions for arbitrary convex polygonal meshes, which is crucial to finite volume element scheme, and propose a quadratic serendipity polygonal finite volume element method with fewer degrees of freedom. Finally, under certain geometric assumption conditions, the optimal $H^1$ error estimate for the quadratic serendipity polygonal finite volume element scheme is obtained, and verified by numerical experiments.