Fast Barycentric-Based Evaluation Over Spectral/hp Elements

TL;DR

This paper extends barycentric interpolation techniques to high-dimensional spectral/hp elements, providing efficient algorithms for solution evaluation at arbitrary points, crucial for postprocessing and field projection in high-order methods.

Contribution

It introduces barycentric interpolation methods for 2D and 3D spectral/hp elements, with optimized algorithms demonstrated in the Nektar++ library, enhancing solution evaluation efficiency.

Findings

Efficient algorithms for barycentric interpolation in high-dimensional spectral/hp elements.

Demonstrated improvements in solution evaluation speed and accuracy.

Applicable to various element types like triangles, tetrahedra, and hexahedra.

Abstract

As the use of spectral/$hp$ element methods, and high-order finite element methods in general, continues to spread, community efforts to create efficient, optimized algorithms associated with fundamental high-order operations have grown. Core tasks such as solution expansion evaluation at quadrature points, stiffness and mass matrix generation, and matrix assembly have received tremendousattention. With the expansion of the types of problems to which high-order methods are applied, and correspondingly the growth in types of numerical tasks accomplished through high-order methods, the number and types of these core operations broaden. This work focuses on solution expansion evaluation at arbitrary points within an element. This operation is core to many postprocessing applications such as evaluation of streamlines and pathlines, as well as to field projection techniques such as mortaring. We expand barycentric interpolation techniques developed on an interval to 2D (triangles and quadrilaterals) and 3D (tetrahedra, prisms, pyramids, and hexahedra) spectral/$hp$ element methods. We provide efficient algorithms for their implementations, and demonstrate their effectiveness using the spectral/$hp$ element library Nektar++.