Simple a posteriori error estimators in adaptive isogeometric analysis

In this article we propose two simple a posteriori error estimators for solving second order elliptic problems using adaptive isogeometric analysis. The idea is based on a Serendipity pairing of discrete approximation spaces $S_h^{p,k}\left(\mathcal{M}\right)$ – $S_h^{p+1,k+1}\left(\mathcal{M}\right)$ , where the space $S_h^{p+1,k+1}\left(\mathcal{M}\right)$ is considered as an enrichment of the original basis of $S_h^{p,k}\left(\mathcal{M}\right)$ by means of the $k$ ‐refinement, a typical unique feature available in isogeometric analysis. The space $S_h^{p+1,k+1}\left(\mathcal{M}\right)$ is used to obtain a higher order accurate isogeometric finite element approximation and using this approximation we propose two simple a posteriori error estimators. The proposed a posteriori error based adaptive $h$ ‐refinement methodology using LR B‐splines is tested on classical elliptic benchmark problems. The numerical tests illustrate the optimal convergence rates obtained for the unknown, as well as the effectiveness of the proposed error estimators.