Serendipity Nodal VEM spaces

TL;DR

This paper introduces Serendipity Nodal Virtual Element spaces that significantly reduce internal degrees of freedom, improving computational efficiency and robustness, especially on distorted elements, while generalizing classical finite element methods.

Contribution

It develops a new variant of Nodal VEM spaces inspired by Serendipity FEMs, reducing degrees of freedom and enhancing robustness on general polytopes.

Findings

Reduces internal degrees of freedom in VEM spaces

Maintains classical FEM properties on triangles and tetrahedra

Improves robustness on distorted elements

Abstract

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.