New preprint on error estimates for IGA with polar parameterizations
15 Mai 2025
In our recent preprint, we present a new approach for locally refined isogeometric analysis (IGA) on polar domains — geometries parametrized by mappings resembling polar coordinate transformations. Although such configurations are common in practice, corresponding error estimates have so far been lacking; this gap is addressed in our work.
IGA is an efficient method for simulating physical processes with high approximation order. In non-smooth domains, such as those with corner singularities, standard methods often fail to achieve these rates. Existing refinement strategies typically require breaking the tensor-product structure of splines, which complicates implementation. We instead propose a graded refinement method based on polar parameterizations, enabling locally refined tensor-product meshes near the singularity.
To support this approach, we develop a new analytical framework using weighted Sobolev spaces tailored to polar splines. This framework is not covered by classical isogeometric approximation theory, which assumes regular parameterizations that break down in the presence of polar singularities. Our analysis establishes optimal convergence rates under suitable mesh grading, and numerical experiments confirm both the theoretical results and the practical efficiency of the method.
