A Polygonal Finite Element Method for Stokes Equations

In this paper, we extend the Bernardi-Raugel element [1] to convex polygonal meshes by using the generalized barycentric coordinates. Comparing to traditional discretizations defined on triangular and rectangular meshes, polygonal meshes can be more flexible when dealing with complicated domains or domains with curved boundaries. Theoretical analysis of the new element follows the standard mixed finite element theory for Stokes equations, i.e., we shall prove the discrete inf-sup condition (LBB condition) by constructing a Fortin operator. Because there is no scaling argument on polygonal meshes and the generalized barycentric coordinates are in general not polynomials, special treatments are required in the analysis. We prove that the extended Bernardi-Raugel element has optimal convergence rates. Supporting numerical results are also presented.