Adaptive isogeometric methods with hierarchical splines: error estimator and convergence

TL;DR

This paper develops an adaptive isogeometric method using hierarchical splines for elliptic PDEs, providing error estimation, mesh refinement strategies, and convergence proof applicable in any dimension.

Contribution

It introduces a new adaptive isogeometric method with a residual error estimator and a mesh refinement algorithm that guarantees convergence for arbitrary degrees and continuities.

Findings

Provides a posteriori error bounds for hierarchical spline methods

Designs a mesh refinement module maintaining hierarchical admissibility

Proves convergence of the adaptive method based on quasi-error contraction

Abstract

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.