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Conservation laws, in for example, electromagnetism, solid and fluidmechanics, allow an exact discrete representation in terms of line, surface andvolume integrals. We develop high order interpolants, from any basis that is apartition of unity, that satisfy these integral relations exactly, at celllevel. The resulting gradient, curl and divergence conforming spaces have theproperty that the conservation laws become completely independent of the basisfunctions. This means that the conservation laws are exactly satisfied even oncurved meshes. As an example, we develop high order gradient, curl anddivergence conforming spaces from NURBS - non uniform rational B-splines - andthereby generalize the compatible spaces of B-splines developed by Buffa etal.[1]. We give several examples of 2D Stokes flow calculations which result,amongst others, in a point wise divergence free velocity field.

High order gradient, curl and divergence conforming spaces, with an application to NURBS-based IsoGeometric Analysis