Adaptive IGAFEM with optimal convergence rates: T-splines

TL;DR

This paper develops an adaptive isogeometric finite element method using T-splines for elliptic PDEs, proving optimal convergence rates and linear error reduction based on a residual-based a posteriori estimator.

Contribution

It introduces an adaptive IGAFEM framework with analysis-suitable T-splines, demonstrating optimal convergence rates for elliptic PDEs in 2D and 3D.

Findings

Proves linear convergence of the error estimator.

Achieves optimal algebraic convergence rates.

Validates adaptivity effectiveness for complex geometries.

Abstract

We consider an adaptive algorithm for finite element methods for the isogeometric analysis (IGAFEM) of elliptic (possibly non-symmetric) second-order partial differential equations. We employ analysis-suitable T-splines of arbitrary odd degree on T-meshes generated by the refinement strategy of [Morgenstern, Peterseim, Comput. Aided Geom. Design 34 (2015)] in 2D and [Morgenstern, SIAM J. Numer. Anal. 54 (2016)] in 3D. Adaptivity is driven by some weighted residual a posteriori error estimator. We prove linear convergence of the error estimator (which is equivalent to the sum of energy error plus data oscillations) with optimal algebraic rates with respect to the number of elements of the underlying mesh.