a b s t r a c tNektar++ is an open-source software framework designed to support the development of highperformance scalable solvers for partial differential equations using the spectral/hp element method. High-order methods are gaining prominence in several engineering and biomedical applications due to their improved accuracy over low-order techniques at reduced computational cost for a given number of degrees of freedom. However, their proliferation is often limited by their complexity, which makes these methods challenging to implement and use. Nektar++ is an initiative to overcome this limitation by encapsulating the mathematical complexities of the underlying method within an efficient C++ framework, making the techniques more accessible to the broader scientific and industrial communities. The software supports a variety of discretisation techniques and implementation strategies, supporting methods research as well as application-focused computation, and the multi-layered structure of the framework allows the user to embrace as much or as little of the complexity as they need. The libraries capture the mathematical constructs of spectral/hp element methods, while the associated collection of pre-written PDE solvers provides out-of-the-box application-level functionality and a template for users who wish to develop solutions for addressing questions in their own scientific domains. Program summaryProgram title: Nektar++ Catalogue identifier: AEVV_v1_0Program summary URL:

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The spectral/hp element method can be considered as bridging the gap between the -traditionally low order -finite element method on one side and spectral methods on the other side. Consequently, a major challenge which arises in implementing the spectral/hp element methods is to design algorithms that perform efficiently for both low-and high-order spectral/hp discretisations, as well as discretisations in the intermediate regime. In this paper, we explain how the judicious use of different implementation strategies can be employed to achieve high efficiency across a wide range of polynomial orders. Furthermore, based upon this efficient implementation, we analyse which spectral/hp discretisation (which specific combination of mesh size h and polynomial order P ) minimises the computational cost to solve an elliptic problem up to a predefined level of accuracy. We investigate this question for a set of both smooth and non-smooth problems.