Polygonal finite elements for incompressible fluid flow

We discuss the use of polygonal finite elements for analysis of incompressible flow problems. It is well-known that the stability of mixed finite element discretizations is governed by the so-called inf-sup condition, which, in this case, depends on the choice of the discrete velocity and pressure spaces. We present a low-order choice of these spaces defined over convex polygonal partitions of the domain that satisfies the inf-sup condition and, as such, does not admit spurious pressure modes or exhibit locking. Within each element, the pressure field is constant while the velocity is represented by the usual isoparametric transformation of a linearly-complete basis. Thus, from a practical point of view, the implementation of the method is classical and does not require any special treatment. We present numerical results for both incompressible Stokes and stationary Navier-Stokes problems to verify the theoretical results regarding stability and convergence of the method. Copyright (c) 2013 John Wiley & Sons, Ltd.