Quasi Conformal and Low-Rank parametrization of planar domains: application to isogeometric analysis with Chebyshevian B-splines

This thesis focuses on the development of a parametrization technique for planar domains and its application within the framework of isogeometric analysis (IGA). Isogeometric analysis is a computational method for solving differential equations that employs the same function space to represent both the geometry of the domain and the approximate numerical solution. In this context, a high-quality parametrization of the computational domain can significantly enhance solution accuracy and reduce computational costs. In this work, we employ piecewise Chebyshevian splines, which possess properties particularly well-suited for isogeometric analysis. The study focuses on domains with complex geometries, where obtaining an appropriate parametrization becomes especially challenging. We investigate a technique that constructs a B-spline representation of the domain starting from its boundary curves. This method fulfills key requirements for isogeometric analysis, such as generating injective and quasi-conformal mappings. Finally, we evaluate the effectiveness of the proposed approach by applying it to the solution of basic boundary value problems.