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From \(h\) to \(p\) efficiently: implementing finite and spectral/hp element methods to achieve optimal performance for low- and high-order discretisations. (English) Zbl 1194.65138
Summary: 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.

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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