Quadratic serendipity finite elements on polygons using generalized barycentric coordinates

Abstract

We introduce a finite element construction for use on the class of convex, planar polygons and show it obtains a quadratic error convergence estimate. On a convex n-gon, our construction produces 2n basis functions, associated in a Lagrange-like fashion to each vertex and each edge midpoint, by transforming and combining a set of n(n + 1)/2 basis functions known to obtain quadratic convergence. The technique broadens the scope of the so-called 'serendipity' elements, previously studied only for quadrilateral and regular hexahedral meshes, by employing the theory of generalized barycentric coordinates. Uniform a priori error estimates are established over the class of convex quadrilaterals with bounded aspect ratio as well as over the class of convex planar polygons satisfying additional shape regularity conditions to exclude large interior angles and short edges. Numerical evidence is provided on a trapezoidal quadrilateral mesh, previously not amenable to serendipity constructions, and applications to adaptive meshing are discussed.

The finite cell method for polygonal meshes: poly-FCM

“…Unfortunately, the implementation details are a little vague and no convergence studies have been conducted showing if optimal rates of convergence can be attained. Apart from their formulation, only Rand et al [128] and Sukumar [97] have developed quadratic serendipity shape functions. In [128] these basis functions are derived for planar and convex n-gons.…”

“…Apart from their formulation, only Rand et al [128] and Sukumar [97] have developed quadratic serendipity shape functions. In [128] these basis functions are derived for planar and convex n-gons. They state that although their construction is specific to quadratic elements the approach is in principle also suitable for the construction of higher order functions.…”

“…Sukumar [97] exploits Rand's approach to generate quadratic shape functions based on MAXENT coordinates. He maximizes the objective functional subject to the constraints for quadratic completeness proposed by Rand et al [128]. However, we have to bear in mind that using the presented methodology, the solution of an Note, that for all cut cells we also depict the quadtree partitioning being used for the numerical integration optimization problem is necessary.…”

Serendipity face and edge VEM spaces

“…However it should be clear from the very beginning that VEMs allow much more general geometries. For these more general geometries the comparison should actually be done between VEMs and other methods designed for polytopes, as for instance [11], [15], [19], [21], [23], [26], [27], [28], [31], [33], [34], [37]. The natural comparison, within Finite Elements, of our V f k,k−1,k−1 elements are clearly the BDM spaces as described in (2.20) for triangles (see Figure 1).…”

“…Similarly to other methods for polytopes (see e.g. [4], [15], [26], [34], [35], [36], [37]) they use finite dimensional spaces that, within each element, contain functions that are not polynomials.…”