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

Yaman Güçlü, Francesco Patrizi, Martin Campos Pinto

TL;DR

This work introduces a projection-based polar broken-FEEC framework that enables structure-preserving FEEC discretizations on domains with polar singularities while using standard tensor-product splines. By explicitly characterizing conforming polar spline subspaces and constructing local, matrix-free commuting projections, the method preserves the de Rham sequence and allows reuse of existing spline code. The approach yields stable, high-order discretizations for Poisson and Maxwell problems on polar domains, with regularised mass matrices to handle non-$L^2$ behavior near the pole. Numerical results demonstrate robustness, numerical stability, and convergence comparable to conforming and $C^0$-based methods, validating the practical viability of the polar broken-FEEC framework for electromagnetism and diffusion problems.

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.

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

TL;DR

This work introduces a projection-based polar broken-FEEC framework that enables structure-preserving FEEC discretizations on domains with polar singularities while using standard tensor-product splines. By explicitly characterizing conforming polar spline subspaces and constructing local, matrix-free commuting projections, the method preserves the de Rham sequence and allows reuse of existing spline code. The approach yields stable, high-order discretizations for Poisson and Maxwell problems on polar domains, with regularised mass matrices to handle non- behavior near the pole. Numerical results demonstrate robustness, numerical stability, and convergence comparable to conforming and -based methods, validating the practical viability of the polar broken-FEEC framework for electromagnetism and diffusion problems.

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.

Paper Structure

This paper contains 28 sections, 15 theorems, 165 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Principles of broken-FEEC schemes on domains with a polar singularity
  3. Conforming discretizations of de Rham sequences in $\mathbb{R}^2$
  4. Projection approach for non-conforming FEEC spaces
  5. Domains with a polar singularity
  6. Singularity of the tensor-product spline de Rham sequence
  7. Construction of a polar broken-FEEC framework with tensor-product splines
  8. Work plan
  9. Characterization of the conforming spline spaces
  10. Geometric projections on the logical and polar domains
  11. Discrete conforming projections onto the $C^0$ sequence
  12. Discrete conforming projections onto the $C^1$ sequence
  13. Discrete derivative matrices
  14. Regularised mass matrices
  15. Proofs of Theorems \ref{['thm:Vseq']} and \ref{['thm:Useq']}
  16. ...and 13 more sections

Key Result

Proposition 2.5

Both the analytical mapping F_apol and the generic spline mapping F_spol (assuming detJpos) have a first order polar singularity in the sense of Definition def:singpol.

Figures (5)

  • Figure 1: Poisson problem: approximate solutions $\phi_h$ (top) and errors $\phi_h-\phi$ (bottom) on the discrete domains $\Omega_h$. The mapped cells defined by the spline breaking points are shown as solid lines on the top plots, clearly showing the pole ${\boldsymbol x}_0 = (D, 0)$.
  • Figure 2: Poisson problem: relative errors \ref{['err_phi']} in $L^2$ norm (left) and $H^1$ norm (right) are plotted for the three different methods described in the text ($C^1$ conforming, $C^0$ broken-FEEC and $C^1$ broken-FEEC) as a function of $N_s$ the number of cells along $s$ (with $N_\theta = 2N_s$), represented in abscissa. For each method different spline degrees are used, indicated by different line styles, namely $p = 2, 3, 4, 5$ indicated by solid, dashed, dotted and dash-dotted lines respectively.
  • Figure 3: Maxwell equations: propagation of a circular wave at three successive times ($t \approx 2.5, 5$ and $7.5$, from left to right) computed with the $C^1$ broken-FEEC scheme \ref{['eq:tMaxwell_hn']}. Three grids (represented on the left plots) are used, with $N_s = 8$, $16$ and $32$ cells along $s$, and $N_\theta = 2N_s$ along the periodic variable (from top to bottom).
  • Figure 4: Initial Bessel-Fourier solution (from left to right: $E_x$, $E_y$, $B$) with mode number $(m, n) = (2, 3)$ used for the convergence study of Maxwell's equations.
  • Figure 5: Maxwell's equations: relative $L^2$ errors for the ${\boldsymbol E}$ (left) and $B$ fields (right) obtained using the polar broken-FEEC scheme \ref{['eq:tMaxwell_hn']} with $C^0$ and $C^1$ projection operators $P^1 = P^1_V$ and $P^1_U$. As in Figure \ref{['fig:PoissonErr']}, errors are plotted versus $N_s$ the number of cells along $s$ (with $N_\theta = 2N_s$) and different spline degrees ($p = 2, 3, 4, 5$) are used, indicated by different line styles (solid, dashed, dotted and dash-dotted lines respectively).

Theorems & Definitions (37)

  • Remark 2.1
  • Remark 2.3
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • proof
  • Proposition 2.7
  • proof
  • Theorem 3.1: $H^1$-conforming polar spline sequence
  • ...and 27 more