Talischi, Cameron; Pereira, Anderson; Paulino, Glaucio H.; Menezes, Ivan F. M.; Carvalho, Márcio S. Polygonal finite elements for incompressible fluid flow. (English) Zbl 1455.76095 Int. J. Numer. Methods Fluids 74, No. 2, 134-151 (2014). Summary: 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. Cited in 29 Documents MSC: 76M10 Finite element methods applied to problems in fluid mechanics 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 76D05 Navier-Stokes equations for incompressible viscous fluids 76D07 Stokes and related (Oseen, etc.) flows Keywords:polygonal finite elements; mixed variational problems; incompressible flow; Stokes and Navier-Stokes equations; Voronoi meshes PDFBibTeX XMLCite \textit{C. Talischi} et al., Int. J. Numer. Methods Fluids 74, No. 2, 134--151 (2014; Zbl 1455.76095) Full Text: DOI