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Adaptive isogeometric methods with hierarchical splines: error estimator and convergence

Annalisa Buffa, Carlotta Giannelli

arXiv:1502.00565v2math.NAcs.NA

TL;DR

Adaptive isogeometric models require extensions of the B-spline model that relax tensor-product rigidity by allowing hanging nodes. The AIGM uses truncated hierarchical splines on admissible meshes, a residual estimator with Dörfler marking, and a refine module that preserves admissibility. The quasi-error contracts, and both the error and estimator converge geometrically to zero.

  • Problem

    Adaptive isogeometric models require extensions of the B-spline model that relax tensor-product rigidity by allowing hanging nodes.

    Source: §1 · p. 2

  • Method

    The AIGM uses truncated hierarchical splines on admissible meshes, a residual estimator with Dörfler marking, and a refine module that preserves admissibility.

    Source: §1 · pp. 2–3

  • Results

    The quasi-error contracts, and both the error and estimator converge geometrically to zero.

    Source: §5.3 · p. 22

  • Takeaways & Limitations

    Admissibility-preserving local refinement is incorporated into an AIGM with a convergence guarantee.

    Source: §1 · p. 3, §5.3 · p. 22

    The paper does not analyze refinement complexity or connect the error to the degrees of freedom required.

    Source: §6 · p. 23

Abstract

from arXiv · show

The problem of developing an adaptive isogeometric method (AIGM) for solving elliptic second-order partial differential equations with truncated hierarchical B-splines of arbitrary degree and different order of continuity is addressed. The adaptivity analysis holds in any space dimensions. We consider a simple residual-type error estimator for which we provide a posteriori upper and lower bound in terms of local error indicators, taking also into account the critical role of oscillations as in a standard adaptive finite element setting. The error estimates are properly combined with a simple marking strategy to define a sequence of admissible locally refined meshes and corresponding approximate solutions. The design of a refine module that preserves the admissibility of the hierarchical mesh configuration between two consectutive steps of the adaptive loop is presented. The contraction property of the quasi-error, given by the sum of the energy error and the scaled error estimator, leads to the convergence proof of the AIGM.

1 Introduction

The paper develops an adaptive isogeometric method using hierarchical splines for elliptic problems, addressing the need for local refinement in isogeometric analysis. It combines residual-based error estimation, marking, and admissibility-preserving refinement to prove convergence, while leaving complexity analysis to companion work.

1 Introduction

  • Hierarchical splines extend tensor-product B-splines to local refinement, supporting an automatically driven adaptive strategy despite more restrictive refinement than other spline models.The paper situates its method in isogeometric analysis, which combines finite element techniques with CAD spline models.Source: §1 · pp. 1–2
  • The method uses admissible hierarchical spline spaces with truncated bases for the Galerkin solve, and its theory depends on mesh admissibility.The adaptive loop pairs these spaces with residual estimation, marking, and refinement.Source: §1 · pp. 2–3
  • The residual estimator is reliable and efficient, bounding the error above and below up to oscillations.Spline regularity reduces it to a mesh-size-weighted element residual.Source: §1 · p. 3
  • Dörfler marking selects elements with the largest indicators until the marked set accounts for a fixed percentage of the total indicator.The marked set drives the next refinement step.Source: §1 · p. 3
  • The recursive refine module constructs meshes from marked elements while preserving admissibility between consecutive adaptive iterations.Refinement may also include suitable neighboring elements to maintain the admissible mesh class.Source: §1 · p. 3
  • Convergence and quasi-error contraction are proved, but refine-module complexity, quasi-interpolation operators, and optimality are left to companion work.The quasi-error is the sum of the error and error indicator.Source: §1 · p. 4

2 Hierarchical spline spaces

The section builds hierarchical spline spaces from nested tensor-product B-spline levels and local active elements, then uses truncation to form THB-splines. Admissibility limits the levels and supports of basis functions acting on each element, providing properties used in the later analysis.

2.1 Preliminaries: B-spline hierarchies

  • Hierarchical spaces combine nested tensor-product spline spaces and nested subdomains, selecting active elements across levels to form a locally refined mesh.The underlying grids use q-adic refinement, with dyadic refinement as the standard case considered.Source: §2.1 · pp. 4–6

2.2 The truncated basis

  • THB-splines are formed by recursively truncating hierarchical B-splines, retaining contributions from passive finer-level basis functions only through the truncation construction.The truncation operation selects passive basis contributions in the next-level B-spline expansion.Source: §2.2 · p. 6

2.3 Properties of THB-splines

  • THB-splines retain the underlying coefficients and form a partition of unity; strong stability holds under reasonable knot assumptions.The cited properties also imply the convex hull property, relevant to geometric modeling.Source: §2.3 · pp. 7–8

2.4 Admissible meshes

  • An admissible mesh of class m has at most m successive levels of nonzero THB-splines on each element, and the number of acting basis functions is independent of hierarchy depth.The support-size comparison constants depend on m, not on the number of hierarchical levels.Source: §2.4 · pp. 8–9
  • The stated hierarchy condition yields meshes admissible of class m; the class-2 examples illustrate configurations for degrees 2, 3, and 4.For admissible meshes, the number of nonzero truncated basis functions per element is bounded in terms of m and the spline degrees.Source: §2.4 · pp. 8–9, Figure 1 · p. 10

3 The module SOLVE: the Galerkin method

The SOLVE module considers a simple elliptic problem on a physical domain mapped from an admissible hierarchical mesh. It discretizes the variational problem with a Galerkin method in the mapped spline space, assuming C1 continuity to simplify the analysis.

3 The module SOLVE: the Galerkin method

  • The model is a simple elliptic problem posed on a domain represented through a bi-Lipschitz map from the parametric domain.The bilinear form is coercive and continuous, inducing an energy norm equivalent to the H1 norm.Source: §3 · pp. 10–11
  • An admissible hierarchical mesh and its truncated basis are mapped to the physical domain to generate the spline space used for discretization.The resulting discrete problem is the Galerkin approximation in that space.Source: §3 · p. 12
  • The analysis assumes the spline space is contained in C1(Ω), simplifying the estimator analysis by avoiding edge contributions in the residual.The paper notes this regularity assumption is not required to develop an adaptive strategy.Source: §3 · p. 12

4 The module ESTIMATE: the residual based error indicator

The ESTIMATE module constructs a residual-based indicator for hierarchical splines and establishes a posteriori upper and lower bounds for the Galerkin error. The analysis uses a volume residual under the stated continuity assumption and accounts for oscillations in the lower bound.

4 The module ESTIMATE: the residual based error indicator

  • With S ⊂ C1(Ω), the residual r = f − div(A∇U) belongs to L2(Ω) and has no edge contribution.The derivation integrates by parts because the discrete functions and their derivatives are continuous.Source: §4 · p. 13
  • The indicator replaces the residual’s dual norm with an elementwise quantity, while polynomial projection defines oscillations that can be made smaller than the error by choosing degrees large enough.The projector Πn maps L2(Q) to polynomials of degree nj in each space direction.Source: §4 · pp. 13–14

4.1 A posteriori upper bound

  • The residual-based indicator provides an upper bound for the Galerkin error, with a constant independent of mesh size and hierarchy level.The upper-bound theorem states the estimate in terms of εQ(U).Source: §4.1 · p. 14

4.2 A posteriori lower bound

  • The indicator is also bounded by the Galerkin error up to oscillations, with a lower-bound constant independent of Q.The paper notes this lower bound is not used for its contraction proof but is needed for optimality analysis in the companion paper.Source: §4.2 · pp. 15–16

5 The modules MARK and REFINE

This section combines Dörfler marking with recursive refinement that preserves mesh admissibility, then proves contraction of a quasi-error and convergence of the adaptive method.

5.1 MARK: the marking strategy

  • Dörfler marking selects elements whose indicators account for a fixed percentage of the total error indicator.The marking parameter θ sets the required percentage.Source: §5.1 · p. 16

5.2 REFINE: the refinement strategy

  • REFINE recursively expands refinement to neighboring active elements so the resulting mesh remains strictly admissible.The neighborhood is chosen by level and support-extension intersection; the procedure terminates and each element is refined at most once.Source: §5.2 · pp. 17–18
  • For a strictly admissible mesh of class m, REFINE terminates and returns a refined mesh of the same admissibility class.The recursive routine likewise preserves class m, and its output elements were already active or arise from a single refinement.Source: §5.2 · pp. 17, 19, 20

5.3 Contraction of the quasi-error and convergence

  • The adaptive sequence contracts a quasi-error combining energy error and a scaled estimator, although neither quantity alone is guaranteed to contract.The contraction factors are positive and independent of iteration k.Source: §5.3 · pp. 21–22
  • Both the error and estimator converge geometrically to zero; convergence for other marking strategies remains open.The authors also leave monotonicity of the error unstudied.Source: §5.3 · pp. 21–22

6 Closure

The paper combines residual-based error bounds, Dörfler marking, and admissibility-preserving refinement to prove convergence of its adaptive isogeometric method. Complexity is not analyzed here, and extension to less restrictive mesh configurations remains a challenge.

6 Closure

  • Residual-type estimators provide upper and lower bounds for Galerkin error, using truncated hierarchical bases on admissible meshes.The upper bound uses partition of unity and bounded basis overlap; the lower bound applies classical finite-element arguments.Source: §6 · p. 22
  • The refinement procedure proves contraction of the quasi-error and, consequently, convergence of the adaptive isogeometric method.The quasi-error combines energy error with a scaled error estimator.Source: §6 · p. 23
  • Complexity is not analyzed, so the paper does not yet connect error to the degrees of freedom required for an iterate.An analogous complexity estimate and optimal convergence rates are deferred to a companion paper.Source: §6 · p. 23
  • Extending the adaptivity theory to less restrictive mesh configurations, such as hierarchical constructions using analysis-suitable T-splines, remains a challenging issue.The paper identifies broader T-spline modeling capabilities as a possible direction for a more powerful refinement module.Source: §6 · p. 23

Appendix

The appendix supplies preparatory finite-element-style arguments for the convergence proof. It combines Galerkin orthogonality and estimator reduction, with parameter choices yielding contraction.

Appendix

  • Preparatory lemmas establish Galerkin orthogonality for refined meshes and reduction of the error indicator after refinement.The indicator reduction applies to marked elements and their corresponding refined mesh.Source: Appendix · p. 24
  • The contraction proof combines Galerkin orthogonality with estimator reduction, within an appendix that adapts finite-element arguments with minor changes.The authors note that some proofs are easier because their error indicator has no jump terms.Source: Appendix · pp. 24, 26
  • Choosing γ using the estimator constant and the Dörfler marking property yields the contraction inequality after δ is selected appropriately.The stated choice of δ reduces the factor involving estimator reduction.Source: Appendix · p. 26
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