General polytopal H(div) conformal finite elements and their discretisation spaces

TL;DR

This paper introduces a new class of H(div)-conformal finite elements adaptable to any polytope, combining virtual element flexibility with Raviart-Thomas divergence properties for improved discretisation spaces.

Contribution

The paper develops a general framework for H(div)-conformal elements on arbitrary polytopes, bridging virtual element flexibility with classical Raviart-Thomas properties.

Findings

Elements can be built on any polytope with divergence properties similar to Raviart-Thomas.

The framework allows easy customization of degrees of freedom for specific properties.

Basis functions are analyzed for particular 2D elements.

Abstract

We present a class of discretisation spaces and H(div)-conformal elements that can be built on any polytope. Bridging the flexibility of the Virtual Element spaces towards the element's shape with the divergence properties of the Raviart-Thomas elements on the boundaries, the designed frameworks offer a wide range of H(div)-conformal discretisations. As those elements are set up through degrees of freedom, their definitions are easily amenable to the properties the approximated quantities are wished to fulfill. Furthermore, we show that one straightforward restriction of this general setting share its properties with the classical Raviart-Thomas elements at each interface, for any order and any polytopial shape. Then, to close the introduction of those new elements by an example, we investigate the shape of the basis functions corresponding to particular elements in the two dimensional case.