A broken-FEEC framework for structure-preserving discretizations of polar domains with tensor-product splines

Abstract

We propose a novel projection-based approach to derive structure-preserving Finite Element Exterior Calculus (FEEC) discretizations using standard tensor-product splines on domains with a polar singularity. This approach follows the main lines of broken-FEEC schemes which define stable and structure-preserving operators in non-conforming discretizations of the de Rham sequence. Here, we devise a polar broken-FEEC framework that enables the use of standard tensor-product spline spaces while ensuring stability and smoothness for the solutions, as well as the preservation of the de Rham structure: A benefit of this approach is the ability to reuse codes that implement standard splines on smooth parametric domains, and efficient solvers such as Kronecker-product spline interpolation. Our construction is based on two pillars: the first one is an explicit characterization of smooth polar spline spaces within the tensor-product splines ones, which are either discontinuous or non square-integrable as a result of the singular polar pushforward operators. The second pillar consists of local, explicit and matrix-free conforming projection operators that map general tensor-product splines onto smooth polar splines, and that commute with the differential operators of the de Rham sequence.