By employing non-uniform rational B-splines (NURBS) as ansatz functions, isogeometric analysis (IGA) combines exact geometry representation with high-order approximation spaces for the numerical solution of partial differential equations (PDEs). However, in geometrically challenging settings, such as in the presence of corners, cracks, or non-smooth domain parameterizations, the full potential of isogeometric methods has not yet been realized. A prominent example of the latter is given by polar parameterizations, which collapse one edge of the parametric domain to a single point. Although such mappings are widely used in computer-aided design, a comprehensive convergence theory remains unavailable. In addition, reduced solution regularity near corners fundamentally limits the convergence rates attainable with high-order splines. This thesis addresses both challenges by focusing on polar parameterizations in which the polar point coincides with a corner of the physical domain. In this context, two key contributions are presented. First, a simple and effective mesh grading strategy toward the collapsed edge is introduced, yielding locally refined meshes near the polar corner that accurately capture singularities of the underlying PDE solutions. Second, a numerical analysis tailored to polar domains with corners is developed, establishing optimal convergence rates via error estimates in weighted Sobolev norms. Numerical experiments for the Poisson equation on benchmark domains confirm the theoretical findings and demonstrate the efficiency of the method. The grading concept is further transferred to the isogeometric collocation framework, where numerical results likewise confirm its effectiveness. The excellent spectral approximation properties of IGA motivate its application to eigenvalue problems (EVPs). To this end, the Laplace EVP on circular sectors is considered as a model problem, since it admits analytical solutions with corner singularities and the geometry can be represented exactly by a polar NURBS parameterization. Optimal convergence of the eigenpairs is proven under graded mesh refinement and confirmed numerically. Furthermore, the computational results indicate that smooth spline spaces exhibit superior approximation constants compared to their $C^0$-continuous counterparts, a phenomenon previously established only for rectangular domains. Graded meshes prove particularly effective for accurately approximating a limited number of eigenvalues. To reduce redundant degrees of freedom, a hierarchical mesh structure is introduced, which can be combined with the gradual refinement scheme and leads to slightly different approximation behavior. In the final part of this thesis, the spectral accuracy of the developed methods is leveraged in the context of inverse problems, with a particular focus on membranes containing cracks. Analytical results based on the short-time asymptotic expansion of the heat trace show that the existence and length of a crack and, under additional assumptions, also its angle can be recovered from the Laplace spectrum. Building on this analysis, a data-driven framework is proposed in which a neural network is trained on simulated spectral data, generated with guaranteed accuracy using graded IGA discretizations, to predict crack geometry parameters. Evaluation of the neural network on the test set demonstrates reliable and precise predictions.     «

By employing non-uniform rational B-splines (NURBS) as ansatz functions, isogeometric analysis (IGA) combines exact geometry representation with high-order approximation spaces for the numerical solution of partial differential equations (PDEs). However, in geometrically challenging settings, such as in the presence of corners, cracks, or non-smooth domain parameterizations, the full potential of isogeometric methods has not yet been realized. A prominent example of the latter is given by polar...     »