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

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 fulfil. 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 polytopal 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.