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From \(h\) to \(p\) efficiently: selecting the optimal spectral/\(hp\) discretisation in three dimensions. (English) Zbl 1243.65136

Summary: There is a growing interest in high-order finite and spectral/\(hp\) element methods using continuous and discontinuous Galerkin formulations. In this paper we investigate the effect of \(h\)- and \(p\)-type refinement on the relationship between runtime performance and solution accuracy. The broad spectrum of possible domain discretisations makes establishing a performance-optimal selection non-trivial. Through comparing the runtime of different implementations for evaluating operators over the space of discretisations with a desired solution tolerance, we demonstrate how the optimal discretisation and operator implementation may be selected for a specified problem. Furthermore, this demonstrates the need for codes to support both low- and high-order discretisations.

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
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
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