Volume 57, Number 1, January-February 2023
|Page(s)||191 - 225|
|Published online||19 January 2023|
Homological- and analytical-preserving serendipity framework for polytopal complexes, with application to the DDR method
IMAG, Univ Montpellier, CNRS, Montpellier, France
2 School of Mathematics, Monash University, Melbourne, Australia
* Corresponding author: email@example.com
Accepted: 7 August 2022
In this work we investigate from a broad perspective the reduction of degrees of freedom through serendipity techniques for polytopal methods compatible with Hilbert complexes. We first establish an abstract framework that, given two complexes connected by graded maps, identifies a set of properties enabling the transfer of the homological and analytical properties from one complex to the other. This abstract framework is designed having in mind discrete complexes, with one of them being a reduced version of the other, such as occurring when applying serendipity techniques to numerical methods. We then use this framework as an overarching blueprint to design a serendipity DDR complex. Thanks to the combined use of higher-order reconstructions and serendipity, this complex compares favorably in terms of degrees of freedom (DOF) count to all the other polytopal methods previously introduced and also to finite elements on certain element geometries. The gain resulting from such a reduction in the number of DOFs is numerically evaluated on two model problems: a magnetostatic model, and the Stokes equations.
Mathematics Subject Classification: 65N30 / 65N99 / 65N12 / 78A30
Key words: Discrete de Rham method / Virtual Element method / compatible discretisations / polytopal methods / serendipity
© The authors. Published by EDP Sciences, SMAI 2023
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