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Inexpensive polynomial-degree-robust equilibrated flux a posteriori estimates for isogeometric analysis

Gregor Gantner, Martin Vohralík

TL;DR

This work derives a posteriori error estimates by equilibrated fluxes, i.e., vector-valued mapped piecewise polynomials lying in the $\boldsymbol{H}({\rm div})$ space which appropriately approximate the desired divergence constraint.

Abstract

We consider isogeometric discretizations of the Poisson model problem, focusing on high polynomial degrees and strong hierarchical refinements. We derive a posteriori error estimates by equilibrated fluxes, i.e., vector-valued mapped piecewise polynomials lying in the $\boldsymbol{H}({\rm div})$ space which appropriately approximate the desired divergence constraint. Our estimates are constant-free in the leading term, locally efficient, and robust with respect to the polynomial degree. They are also robust with respect to the number of hanging nodes arising in adaptive mesh refinement employing hierarchical B-splines. Two partitions of unity are designed, one with larger supports corresponding to the mapped splines, and one with small supports corresponding to mapped piecewise multilinear finite element hat basis functions. The equilibration is only performed on the small supports, avoiding the higher computational price of equilibration on the large supports or even the solution of a global system. Thus, the derived estimates are also as inexpensive as possible. An abstract framework for such a setting is developed, whose application to a specific situation only requests a verification of a few clearly identified assumptions. Numerical experiments illustrate the theoretical developments.

Inexpensive polynomial-degree-robust equilibrated flux a posteriori estimates for isogeometric analysis

TL;DR

This work derives a posteriori error estimates by equilibrated fluxes, i.e., vector-valued mapped piecewise polynomials lying in the space which appropriately approximate the desired divergence constraint.

Abstract

We consider isogeometric discretizations of the Poisson model problem, focusing on high polynomial degrees and strong hierarchical refinements. We derive a posteriori error estimates by equilibrated fluxes, i.e., vector-valued mapped piecewise polynomials lying in the space which appropriately approximate the desired divergence constraint. Our estimates are constant-free in the leading term, locally efficient, and robust with respect to the polynomial degree. They are also robust with respect to the number of hanging nodes arising in adaptive mesh refinement employing hierarchical B-splines. Two partitions of unity are designed, one with larger supports corresponding to the mapped splines, and one with small supports corresponding to mapped piecewise multilinear finite element hat basis functions. The equilibration is only performed on the small supports, avoiding the higher computational price of equilibration on the large supports or even the solution of a global system. Thus, the derived estimates are also as inexpensive as possible. An abstract framework for such a setting is developed, whose application to a specific situation only requests a verification of a few clearly identified assumptions. Numerical experiments illustrate the theoretical developments.
Paper Structure (27 sections, 9 theorems, 167 equations, 12 figures)

This paper contains 27 sections, 9 theorems, 167 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Available results
  3. Outline
  4. Model problem and its Galerkin discretization
  5. Motivation and the new approach in a simplified spline setting
  6. Standard equilibration in finite elements
  7. A straightforward (expensive) generalization to IGA
  8. Inexpensive equilibration in IGA
  9. Hierarchical splines
  10. Multivariate splines in the parameter domain $(0,1)^d$
  11. Hierarchical splines in the parameter domain $(0,1)^d$
  12. Hierarchical splines in the physical domain $\Omega$
  13. Partitions of unity and patchwise spaces
  14. Partitions of unity based on hierarchical splines
  15. Patchwise Sobolev spaces
  16. ...and 12 more sections

Key Result

Proposition 6.9

Assumptions ass:setting--ass:min RT to cont are satisfied in the context of Sections sec:setting--sec:partitions with the choice $\bm{RT}_h^{{\bm a},{\bm b}} = \bm{RT}^{\widetilde{\bm{p}}}(\mathcal{T}_{\bm b})$. In particular, the constants in Assumption ass:CPF can be taken as $C_1=1$, $C_4=2^d$, $

Figures (12)

  • Figure 1: B-splines of degree $p=1,3$ and maximal smoothness $p-1$ on a two-dimensional uniform tensor mesh $\mathcal{T}_h$.
  • Figure 2: A straightforward (expensive) extension of the FEM equilibration to IGA with B-splines of degree $p=5$ and maximal smoothness $p-1$ on a two-dimensional uniform tensor mesh $\mathcal{T}_h$.
  • Figure 3: Inexpensive equilibration in IGA in the simplified setting of uniform tensor-product mesh, $p=5$ and maximal smoothness $p-1$.
  • Figure 4: The B-splines of degree $p=2$ corresponding to the knot vector $\mathcal{K}=(0,0,0,1/6,2/6,3/6,4/6,4/6,5/6,1,1,1)$ are depicted. They are at least $C^1$ at the knots $1/6,2/6,3/6,$ and $5/6$, and at least $C^0$ at $4/6$.
  • Figure 5: A two-dimensional hierarchical mesh $\widehat{\mathcal{T}}_h$ with level of all elements less than $4$. Levels $0$ to $3$ are respectively highlighted in white, light grey, grey, and dark grey, also denoting the corresponding domains $\widehat{\Omega}^0_h\setminus\widehat{\Omega}^1_h$, $\widehat{\Omega}^1_h\setminus\widehat{\Omega}^2_h$, $\widehat{\Omega}^2_h\setminus\widehat{\Omega}^3_h$, $\widehat{\Omega}^3_h\setminus\widehat{\Omega}^4_h$; $\widehat{\Omega}_h^4=\emptyset$.
  • ...and 7 more figures

Theorems & Definitions (33)

  • Remark 4.1
  • Remark 4.2
  • Remark 5.1
  • Remark 6.4
  • Remark 6.7
  • Proposition 6.9
  • proof
  • Remark 6.10
  • Remark 6.11
  • Definition 7.1
  • ...and 23 more