Direct serendipity and mixed finite elements on convex polygons (Q2679833)

Direct serendipity and mixed finite elements are constructed on general planar, strictly convex polygons with optimal order of accuracy for any order. The shape functions are defined directly on the physical elements, i.e., without using a mapping from a reference element. A direct serendipity element has its function space of polynomials plus supplemental functions. The direct serendipity elements are the precursors of the mixed elements in a de Rham complex. Numerical results are presented from finite element numerical solutions of Poisson's equation. It is shown that the convergence rates are consistent with the theory. The authors observe that the mesh shape regularity is quite crucial in terms of the observed error. Short edges, lead to a poor (i.e., small) shape regularity parameter, which could also result in a poor approximation in that region of the mesh. Removing such edges greatly improve the approximation and convergence rates.