A class of finite dimensional spaces and H(div) conformal elements on general polytopes

TL;DR

This paper introduces a versatile class of H(div)-conformal discretisation spaces and elements applicable to arbitrary polytopes, combining Virtual Element flexibility with Raviart-Thomas divergence properties for improved finite element methods.

Contribution

It develops a new framework for H(div)-conformal elements on general polytopes, bridging Virtual Element and Raviart-Thomas features, with easy-to-define degrees of freedom and shape function analysis.

Findings

Elements can be built on any polytope shape.

Classical Raviart-Thomas properties are recovered at interfaces.

Shape functions are analyzed in 2D cases.

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 polytopial shape. Then, we investigate the shape of the basis functions corresponding to particular elements in the two dimensional case.