Adaptive numerical modeling using the hierarchical Fup basis functions and control volume isogeometric analysis

A novel adaptive algorithm that is based on new hierarchical Fup (HF) basis functions and a control volume formulation is presented. Because of its similarity to the concept of isogeometric analysis (IGA), we refer to it as control volume isogeometric analysis (CV-IGA). Among other interesting properties, the IGA introduced k-refinement as advanced version of hp-refinement, where every basis function of the nth order from one resolution level are replaced by a linear combination of more basis functions of the n+1th order at the next resolution level. However, k-refinement can be performed only on whole domain, while local adaptive k-refinement is not possible with classical B-spline basis functions. HF basis functions (infinitely differentiable splines) satisfy partition of unity, and they are linearly independent and locally refinable. Their main feature is execution of the adaptive local hp-refinement because any basis function of the nth order from one resolution level can be replaced by a linear combination of more basis functions of the n+1th order at the next resolution level providing spectral convergence order. The comparison between uniform vs hierarchical adaptive solutions is demonstrated, and it is shown that our adaptive algorithm returns the desired accuracy while strongly improving the efficiency and controlling the numerical error. In addition to the adaptive methodology, a stabilization procedure is applied for advection-dominated problems whose numerical solutions “suffer” from spurious oscillations. Stabilization is added only on lower resolution levels, while higher resolution levels ensure an accurate solution and produce a higher convergence order. Since the focus of this article is on developing HF basis functions and adaptive CV-IGA, verification is performed on the stationary one-dimensional boundary value problems.