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

TL;DR

This paper introduces a novel broken-FEEC framework that enables structure-preserving discretizations of polar domains using standard tensor-product splines, ensuring stability, smoothness, and de Rham structure preservation.

Contribution

It develops a polar broken-FEEC framework that allows the use of standard tensor-product splines on polar domains with singularities, maintaining stability and structure.

Findings

Enables reuse of existing spline codes on polar domains.

Provides explicit characterization of smooth polar spline spaces.

Constructs local, matrix-free projection operators that commute with differential operators.

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.