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Error Estimates and Graded Mesh Refinement for Isogeometric Analysis on Polar Domains with Corners

Thomas Apel, Philipp Zilk

TL;DR

This work tackles the challenge of optimal isogeometric approximation on polar domains with corners by introducing a polar parameterization that collapses a boundary edge to the polar point. A graded, tensor-product Bézier mesh toward the corner provides true locality without breaking the tensor-product structure, facilitated by a modified polar spline space and a dedicated projector. The authors develop polar Sobolev and bent spaces to handle singularities and reduced regularity, derive error estimates under a simple grading condition $\mu < (\nu-q+1)/(s-q)$, and prove that graded polar meshes achieve optimal rates in $H^q$ norms. Numerical experiments on circular sectors and L-shaped domains confirm the theory and demonstrate practical efficiency of the approach. Overall, the paper provides a rigorous, implementable framework for local refinement in IGA on non-smooth polar geometries with corners.

Abstract

Isogeometric analysis (IGA) enables exact representations of computational geometries and higher-order approximation of PDEs. In non-smooth domains, however, singularities near corners limit the effectiveness of IGA, since standard methods typically fail to achieve optimal convergence rates. These constraints can be addressed through local mesh refinement, but existing approaches require breaking the tensor-product structure of splines, which leads to increased implementation complexity. This work introduces a novel local refinement strategy based on a polar parameterization, in which one edge of the parametric square is collapsed into the corner. By grading the standard mesh toward the collapsing edge, the desired locality near the singularity is obtained while maintaining the tensor-product structure. A mathematical analysis and numerical tests show that the resulting isogeometric approximation achieves optimal convergence rates with suitable grading parameters. Polar parameterizations, however, suffer from a lack of regularity at the polar point, making existing standard isogeometric approximation theory inapplicable. To address this, a new framework is developed for deriving error estimates on polar domains with corners. This involves the construction of polar function spaces on the parametric domain and a modified projection operator onto the space of $C^0$-smooth polar splines. The theoretical results are verified by numerical experiments confirming both the accuracy and efficiency of the proposed approach.

Error Estimates and Graded Mesh Refinement for Isogeometric Analysis on Polar Domains with Corners

TL;DR

This work tackles the challenge of optimal isogeometric approximation on polar domains with corners by introducing a polar parameterization that collapses a boundary edge to the polar point. A graded, tensor-product Bézier mesh toward the corner provides true locality without breaking the tensor-product structure, facilitated by a modified polar spline space and a dedicated projector. The authors develop polar Sobolev and bent spaces to handle singularities and reduced regularity, derive error estimates under a simple grading condition , and prove that graded polar meshes achieve optimal rates in norms. Numerical experiments on circular sectors and L-shaped domains confirm the theory and demonstrate practical efficiency of the approach. Overall, the paper provides a rigorous, implementable framework for local refinement in IGA on non-smooth polar geometries with corners.

Abstract

Isogeometric analysis (IGA) enables exact representations of computational geometries and higher-order approximation of PDEs. In non-smooth domains, however, singularities near corners limit the effectiveness of IGA, since standard methods typically fail to achieve optimal convergence rates. These constraints can be addressed through local mesh refinement, but existing approaches require breaking the tensor-product structure of splines, which leads to increased implementation complexity. This work introduces a novel local refinement strategy based on a polar parameterization, in which one edge of the parametric square is collapsed into the corner. By grading the standard mesh toward the collapsing edge, the desired locality near the singularity is obtained while maintaining the tensor-product structure. A mathematical analysis and numerical tests show that the resulting isogeometric approximation achieves optimal convergence rates with suitable grading parameters. Polar parameterizations, however, suffer from a lack of regularity at the polar point, making existing standard isogeometric approximation theory inapplicable. To address this, a new framework is developed for deriving error estimates on polar domains with corners. This involves the construction of polar function spaces on the parametric domain and a modified projection operator onto the space of -smooth polar splines. The theoretical results are verified by numerical experiments confirming both the accuracy and efficiency of the proposed approach.
Paper Structure (28 sections, 17 theorems, 220 equations, 11 figures, 1 table)

This paper contains 28 sections, 17 theorems, 220 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Fundamentals of B-splines and NURBS
  3. Univariate B-splines and NURBS
  4. Bivariate B-splines and NURBS
  5. Main concepts and results
  6. Polar parameterizations
  7. Model problem and regularity properties
  8. Graded isogeometric mesh refinement for polar domains with corners
  9. Main results
  10. Properties of graded meshes
  11. Function spaces and projectors for polar parameterizations
  12. Polar splines and NURBS
  13. Basis functions of the modified approximation space
  14. A projector into the modified approximation space
  15. Polar Sobolev spaces on the parametric domain
  16. ...and 13 more sections

Key Result

Theorem 3.5

Let $q \in \{0,1\}$ and $s, s_0 \in {\mathbb N}$ with $2 \leq s \leq s_0 \leq p +1$. Further, let $v \in V^{s}_{\beta}(\Omega)$ for all $s-1>\beta >s-1-\nu$, recall eq: weighted regularity of singular part V spaces, and let the mesh grading parameter $\mu \in (0,1]$ satisfy the condition Then, for every $h>0$ and $s\leq s_0$, there is a NURBS function $v_h \in V_h^{\mathrel{\mathsmaller{\mathsmal

Figures (11)

  • Figure 1: Polar parameterizations of exemplary domains with corners and corresponding boundary notation. (a) Circular sector with angle $\omega = \frac{5}{3} \pi$, also known as Pacman domain. (b) L-shaped domain.
  • Figure 2: (a): Isogeometric mappings $F^{(2)}_1$ and $F^{(2)}_2$ versus scaled polar angular mappings. (b): Difference between the corresponding functions.
  • Figure 3: Graded parametric and physical Bézier meshes. (a): Circular sector. (b): L-shaped domain.
  • Figure 4: Basis functions for a coarse discretization of a circular sector sector with angle $\omega = \frac{5}{3} \pi$. Some mesh lines that do not correspond to the coarse discretization are inserted for better visualization. (a) Standard singular basis functions \ref{['eq: singular basis functions']}. (b): Modified basis function \ref{['eq: modified basis function physical domain']}.
  • Figure 5: Splitting of the parametric and physical domain and the corresponding meshes. (a): Circular sector (b): L-shaped domain.
  • ...and 6 more figures

Theorems & Definitions (42)

  • Definition 3.1: Polar parameterization
  • Example 3.2: Circular sector
  • Example 3.3: L-Shape
  • Definition 3.4: Weighted Sobolev spaces on the physical domain
  • Theorem 3.5
  • Corollary 3.6
  • proof
  • Lemma 3.7
  • proof
  • Lemma 4.1
  • ...and 32 more