NURBS-SEM: a hybrid spectral element method on NURBS maps for the solution of elliptic PDEs on surfaces

TL;DR

This paper introduces a hybrid spectral element method on NURBS maps that combines the geometric accuracy of IGA with the spectral precision of SEM to solve elliptic PDEs on complex surfaces.

Contribution

A novel NURBS-SEM approach that achieves spectral accuracy while preserving exact geometry, overcoming limitations of existing IGA and SEM methods.

Findings

NURBS-SEM attains spectral accuracy on complex geometries.

The method maintains exact geometry representation.

Comparative analysis shows advantages over standard IGA techniques.

Abstract

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).