Serendipity Nodal VEM spaces

- Virtual Element Methods generalize Finite Elements to polytopal geometries.
- Serendipity VEMs imitate the Serendipity variants of Finite Elements.
- On triangles we reproduce FE, on quads we have a more robust version of them.
- On general polytopes, the Serendipity VEMs improve significantly the original ones.

We introduce a new variant of Nodal Virtual Element spaces that mimics the “Serendipity Finite Element Methods” (whose most popular example is the 8-node quadrilateral) and allows to reduce (often in a significant way) the number of internal degrees of freedom. When applied to the faces of a three-dimensional decomposition, this allows a reduction in the number of face degrees of freedom: an improvement that cannot be achieved by a simple static condensation. On triangular and tetrahedral decompositions the new elements (contrary to the original VEMs) reduce exactly to the classical Lagrange FEM. On quadrilaterals and hexahedra the new elements are quite similar (and have the same amount of degrees of freedom) to the Serendipity Finite Elements, but are much more robust with respect to element distortions. On more general polytopes the Serendipity VEMs are the natural (and simple) generalization of the simplicial case.