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Quadratic serendipity finite elements on polygons using generalized barycentric coordinates. (English) Zbl 1300.65091

The authors present a finite element construction for the use on the class of convex, planar polygons and show that it obtains a quadratic error convergence estimate. On a convex \(n\)-gon, the 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. This technique extends the scope of the so-called ‘serendipity’ elements. Some a priori error estimates are proved 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 and applications to adaptive meshing are discussed.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
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