NURBS-SEM: a hybrid spectral element method on NURBS maps for the solution of elliptic PDEs on surfaces
This work addresses the accuracy limitations of high-degree NURBS in isogeometric analysis by combining IGA and SEM, benefiting computational scientists solving PDEs on complex geometries.
The authors propose a hybrid spectral element method on NURBS maps (NURBS-SEM) that achieves spectral accuracy while maintaining exact geometry representation for solving elliptic PDEs on surfaces, outperforming standard IGA methods in accuracy for high polynomial degrees.
Non Uniform Rational B-spline (NURBS) patches are a standard way to describe complex geometries in Computer Aided Design tools, and have gained a lot of popularity in recent years also for the approximation of partial differential equations, via the Isogeometric Analysis (IGA) paradigm. However, spectral accuracy in IGA is limited to relatively small NURBS patch degrees (roughly p <= 8), since local condition numbers grow very rapidly for higher degrees. On the other hand, traditional Spectral Element Methods (SEM) guarantee spectral accuracy but often require complex and expensive meshing techniques, like transfinite mapping, that result anyway in inexact geometries. In this work we propose a hybrid NURBS-SEM approximation method that achieves spectral accuracy and maintains exact geometry representation by combining the advantages of IGA and SEM. As a prototypical problem on non trivial geometries, we consider the Laplace--Beltrami and Allen--Cahn equations on a surface. On these problems, we present a comparison of several instances of NURBS-SEM with the standard Galerkin and Collocation Isogeometric Analysis (IGA).