High order gradient, curl and divergence conforming spaces, with an application to NURBS-based IsoGeometric Analysis
For computational scientists using isogeometric analysis, this provides a framework to exactly enforce conservation laws in curved geometries, improving accuracy in fluid and electromagnetic simulations.
The paper develops high-order gradient, curl, and divergence conforming spaces from any partition-of-unity basis, ensuring exact satisfaction of conservation laws at the cell level even on curved meshes. Applied to NURBS, this generalizes compatible B-spline spaces and yields pointwise divergence-free velocity fields in 2D Stokes flow examples.
Conservation laws, in for example, electromagnetism, solid and fluid mechanics, allow an exact discrete representation in terms of line, surface and volume integrals. We develop high order interpolants, from any basis that is a partition of unity, that satisfy these integral relations exactly, at cell level. The resulting gradient, curl and divergence conforming spaces have the property that the conservation laws become completely independent of the basis functions. This means that the conservation laws are exactly satisfied even on curved meshes. As an example, we develop high order gradient, curl and divergence conforming spaces from NURBS - non uniform rational B-splines - and thereby generalize the compatible spaces of B-splines developed by Buffa et al.[1]. We give several examples of 2D Stokes flow calculations which result, amongst others, in a point wise divergence free velocity field.