Fast Barycentric-Based Evaluation Over Spectral/hp Elements

Abstract As the use of spectral/ hp element methods, and high-order finite methods in general, continues to spread, community efforts create efficient, optimized algorithms associated with fundamental operations have grown. Core tasks such as solution expansion evaluation at quadrature points, stiffness mass matrix generation, assembly received tremendous attention. With types problems which are applied, correspondingly growth numerical accomplished through number these core broaden. This work focuses on arbitrary points within an element. operation is many postprocessing applications streamlines pathlines, well field projection techniques mortaring. We expand barycentric interpolation developed interval 2D (triangles quadrilaterals) 3D (tetrahedra, prisms, pyramids, hexahedra) methods. provide efficient for their implementations, demonstrate effectiveness using library Nektar++ by running a series baseline evaluations against ‘standard’ Lagrangian method, where generated matrix-multiplication applied evaluate point given location. present results from rigorous benchmarking tests variety shapes, polynomial orders dimensions. show that when interest be repeatedly evaluated, method performs worst $$50\%$$ 50 % slower, compared cached evaluation. However, changes so must regenerated approach, yields far greater performance, minimum speedup factor $$7\times $$ 7 × . Furthermore, derivatives also required, general slightly outperforms across all elements orders, up $$30\%$$ 30 speedup. Finally we investigate real-world example scalar transport non-conformal discontinuous Galerkin simulation, observe around $$6\times 6 computational time matrix-based approach. explore complexity both requires $${\mathcal {O}}(k)$$ O ( k ) storage best case space {O}}(k^2)$$ 2 method.