Title: Poly-Spline Finite Element Method,
Abstract: We introduce an integrated meshing and finite element method pipeline
enabling black-box solution of partial differential equations in the volume
enclosed by a boundary representation. We construct a hybrid
hexahedral-dominant mesh, which contains a small number of star-shaped
polyhedra, and build a set of high-order basis on its elements, combining
triquadratic B-splines, triquadratic hexahedra (27 degrees of freedom), and
harmonic elements. We demonstrate that our approach converges cubically under
refinement, while requiring around 50% of the degrees of freedom than a
similarly dense hexahedral mesh composed of triquadratic hexahedra. We validate
our approach solving Poisson's equation on a large collection of models, which
are automatically processed by our algorithm, only requiring the user to
provide boundary conditions on their surface.