On exact integration in the finite volume element method on simplicial meshes

This paper considers the technological aspects of the finite volume element method for the numerical solution of partial differential equations on simplicial grids in two and three dimensions. We derive new classes of integration formulas for the exact integration of generic monomials of barycentric coordinates over different types of fundamental shapes corresponding to a barycentric dual mesh. These integration formulas constitute an essential component for the development of high-order accurate finite volume element schemes. Numerical examples are presented that illustrate the effectiveness and validity of the technology.