One-dimensional fully automatic h-adaptive isogeometric finite element method package

Abstract

This paper deals with an adaptive finite element method originally developed by Prof. Leszek Demkowicz for hierarchical basis functions. In this paper, we investigate the extension of the adaptive algorithm for isogeometric analysis performed with $B$-spline basis functions. We restrict ourselves to $h$-adaptivity, since the polynomial order of approximation must be fixed in the isogeometric case. The classical variant of the adaptive FEM algorithm, as delivered by the group of Prof. Demkowicz, is based on a two-grid paradigm, with coarse and fine grids (the latter utilized as a reference solution). The problem is solved independently over a coarse mesh and a fine mesh. The fine-mesh solution is then utilized as a reference to estimate the relative error of the coarse-mesh solution and to decide which elements to refine. Prof. Demkowicz uses hierarchical basis functions, which (though locally providing $C^{p−1}$ continuity) ensure only $C^0$ on the interfaces between elements. The CUDA C library described in this paper switches the basis to $B$-spline functions and proposes a one-dimensional isogeometric version of the $h$-adaptive FEM algorithm to achieve global $C^{p−1}$ continuity of the solution.