Abstract Isogeometric analysis (IGA) combines exact geometric representations with higher-order accuracy for the numerical solution of partial differential equations. However, in geometrically complex settings – such as domains with corner singularities or non-standard parameterizations – these advantages may not be fully realized by standard IGA techniques. In particular, commonly used NURBS parameterizations can result in polar mappings, where one edge of the parametric domain is collapsed onto a single point, known as the polar point. Although widely used in computer-aided design, such configurations lack a full convergence theory. Additionally, reduced solution regularity near corners can significantly limit the performance of standard IGA, as higher-order convergence is no longer attainable. In this work, both challenges are addressed by analyzing parameterizations in which the polar point coincides with a corner of the physical domain. To tackle the resulting singularity, a simple and effective local refinement strategy is proposed based on mesh grading toward the collapsed edge. This produces a locally refined mesh in the vicinity of the polar corner that accurately captures the singular behavior of the PDE solution. To support this strategy, a numerical analysis tailored to polar domains with corners is developed. The framework includes the definition of polar function spaces on the parametric domain, a quasi-interpolant for polar splines, and the derivation of error estimates in weighted Sobolev norms. Optimal convergence is proven for smooth solutions under uniform refinement and for singular solutions using appropriately graded meshes. Numerical experiments on benchmark domains confirm the theoretical predictions and demonstrate the practical efficiency of the proposed method.     «

Abstract Isogeometric analysis (IGA) combines exact geometric representations with higher-order accuracy for the numerical solution of partial differential equations. However, in geometrically complex settings – such as domains with corner singularities or non-standard parameterizations – these advantages may not be fully realized by standard IGA techniques. In particular, commonly used NURBS parameterizations can result in polar mappings, where one edge of the parametric domain is collapsed on...     »