On the application of polygonal finite element method for Stokes flow – A comparison between equal order and different order approximation

Abstract

In this work, we discuss the application of polygonal finite elements to the solution of Stokes equations in two dimensions. The proposed framework is based on the mixed finite element discretization that involves velocity and pressure as independent unknown field variables. We present two approaches: (a) same order of approximation for velocity and pressure with Pressure Stabilization Petrov-Galerkin method (PSPG) and (b) different order of approximation for velocity and pressure. The approximation functions over polygons are rational polynomials based on Wachspress coordinates and for numerical integration of the terms in the bilinear and the linear form, we employ the linear smoothing technique. The relative performance between the approaches, the convergence properties and the accuracy is presented for two dimensional numerical examples, which shows that the proposed framework yields accurate results and converges at optimal convergence rate.