OBELISK RESEARCH REPORT Volume 00, Number 0, Pages 000–000 S 0000-0000(XX)0000-0 II. SOME HERMITE INTERPOLATION FUNCTIONS FOR SOLENOIDAL AND IRROTATIONAL VECTOR FIELDS JONAS T. HOLDEMAN Abstract. Some remarkable new Hermite interpolation functions on rectan- gular Cartesian meshes in two dimensions are developed. The examples are cubic-complete for scalar fields and quadratic-complete for vector fields. These are extended to orthogonal curvilinear coordinate systems, and affine meshes in those systems. The pair provide a new paradigm for interpolating divergence- free vector fields. After revisiting the incompressible Navier-Stokes equation, the functions are used with the finite element method (FEM) to solve the equation of motion for incompressible flow. 1. Introduction In a previous paper [9], some Lagrange interpolation functions for vector fields were introduced, with the property that they exhibited a constant divergence or constant curl in Cartesian coordinates. They were extended to curvilinear coordi- nates, where they exhibited a common or consistent (non-constant) divergence or curl in each subdomain of the mesh. It was shown how strongly-solenoidal fields would result using a simple constraint on each subdomain of the mesh. In ap- plication to unsteady and nonlinear partial differential equations, the constraint equations have to be satisfied at each time step or nonlinear iteration. It would be more efficient to solve the constraint equation once, producing solenoidal basis functions to interpolate the vector field. This approach leads from Lagrange to Hermite interpolation functions. It was asserted that the Lagrange functions provided a basis from which scalar Hermite interpolation functions could be developed, which functions have sufficient continuity that they can interpolate divergence-free and irrotational vector fields in a strong or pointwise sense. In the following sections these Hermite interpolation functions will be introduced, and it will be shown how they can be derived. It will be shown how meshes can be generated to take advantage of the generalized form of these functions. After revisiting the equation of motion for incompressible flow, it will be shown how strongly solenoidal solutions to the equation of motion for incompressible flow may be found. Similar methods can be used with the three-dimensional Lagrange functions given in [9], and extensions thereof, to generate three-dimensional divergence-free Hermite interpolation functions. Extension of the methods, and the “solenoidal” Prepared January 27, 2003. 2000 Mathematics Subject Classification. Primary 65D05; Secondary 65N30, 76D05. c 2003 Jonas Holdeman 1