Nektar++

This autumn school course introduces participants to the fundamentals of spectral/hp element methods and their practical application through the open-source Nektar++ framework. Combining theoretical foundations with hands-on experience, the programme equips participants with the skills to understand this class of high-order methods and apply Nektar++ to solve partial differential equations (PDEs), particularly conservation laws governing flow dynamics. Participants will gain a foundational understanding of finite element methods (FEM), including Galerkin formulation, polynomial expansions, and numerical integration, whilst learning to use Nektar++ solvers for simulation setup, execution, and post-processing, exemplified by the classic flow-past-a-cylinder problem. Additionally, the course covers high-order mesh generation using NekMesh, enabling participants to construct meshes for complex domains and visualise simulation results.

Spectral/hp element methods are a class of high-order methods that integrate the geometric flexibility of low-order finite element methods (FEM) with the high accuracy of p-type spectral methods. In this approach, complex domains are tessellated using coarse h-type FEM elements, referred to as macro-elements, while variables within each element are approximated by piecewise polynomial basis functions of order P—thus allowing the solution of PDEs in complex domains with the desirable numerical properties of high-order methods, such as low dissipation and dispersion, as well as exponential convergence.

The Nektar++ framework is an open-source platform for spectral/hp element methods, designed for constructing efficient, high-performance solvers for a wide range of PDEs, particularly those governing flow dynamics. It comes with fully fledged solvers, including advection-reaction-diffusion, compressible and incompressible Navier–Stokes, shallow water, and acoustic solvers. It supports versatile highorder elements for 2D, quasi-3D, and 3D simulations, and its Navier–Stokes solvers are well suited for high-fidelity implicit Large Eddy Simulations (iLES) and Direct Numerical Simulations (DNS). The code, written in modern C++, is accessible via GitLab under the MIT licence.