Quadratic maximum-entropy serendipity shape functions for arbitrary planar polygons

• Quadratic serendipity shape functions for planar polygons are constructed.
• Relative entropy measure is minimized, subject to quadratic reproducing conditions.
• Maximal Poisson-sampling algorithm is used for polygonal mesh generation.
• A corrected integration scheme is used, which passes the quadratic patch test.
• Method delivers optimal convergence rates in Sobolev norms on polygonal meshes.

In this paper, we present the development of quadratic serendipity shape functions on planar convex and nonconvex polygons. Drawing on the work of Bompadre et al. (2012) [1] and Hormann and Sukumar (2008) [2], we adopt a relative entropy measure for signed (positive or negative) shape functions, with nodal prior weight functions that have the appropriate zero-set on the boundary of the polygon. We maximize the objective functional subject to the constraints for quadratic completeness proposed by Rand et al. (2013) [3]. Along an edge of a polygon, the approximation is identical to univariate Bernstein polynomials: the choice of the nodal prior weight function ensures that the shape functions satisfy a weak   Kronecker-delta property on each edge. The shape functions are well-defined for arbitrary planar polygons without self-intersections. On using a modified numerical integration scheme, we show that the quadratic patch test is passed on polygonal meshes with convex and nonconvex elements. Numerical tests for the Poisson equation on self-similar trapezoidal meshes and quasiuniform polygonal meshes are presented, which reveal the sound accuracy of the method, and optimal rates of convergence in the L2L2 norm and the H1H1 seminorm are established.