High order geometric methods with exact conservation properties

Conservation laws in, for example, electromagnetism, solid and fluid mechanics, allow an exact discrete representation in terms of line, surface and volume integrals. In this paper, we develop high order interpolants, from any basis that constitutes a partition of unity, which satisfy these integral relations exactly. The resulting gradient, curl and divergence conforming spaces have the property that the conservation laws become completely independent of the basis functions. Hence, they are exactly satisfied at the coarsest level of discretization and on arbitrarily curved meshes. As an illustration we apply our approach to B-splines and compute a 2D Stokes flow inside a lid driven cavity, which displays, amongst others, a point-wise divergence-free velocity field.