Adaptive IGAFEM with optimal convergence rates: Hierarchical B-splines
Provides theoretical guarantees for adaptive IGAFEM with hierarchical B-splines, addressing a known bottleneck in isogeometric analysis for achieving optimal convergence rates.
This paper develops an adaptive isogeometric analysis finite element method (IGAFEM) using hierarchical B-splines for elliptic PDEs, proving linear convergence with optimal algebraic rates of the error estimator. Numerical experiments confirm the theoretical results.
We consider an adaptive algorithm for finite element methods for the isogeometric analysis (IGAFEM) of elliptic (possibly non-symmetric) second-order partial differential equations in arbitrary space dimension $d\ge2$. We employ hierarchical B-splines of arbitrary degree and different order of smoothness. We propose a refinement strategy to generate a sequence of locally refined meshes and corresponding discrete solutions. Adaptivity is driven by some weighted residual a posteriori error estimator. We prove linear convergence of the error estimator (resp. the sum of energy error plus data oscillations) with optimal algebraic rates. Numerical experiments underpin the theoretical findings.