Homological- and analytical-preserving serendipity framework for polytopal complexes, with application to the DDR method

TL;DR

This paper develops a theoretical framework for reducing degrees of freedom in polytopal complexes while preserving homological and analytical properties, and applies it to create an efficient serendipity DDR method with favorable numerical results.

Contribution

It introduces an abstract framework for transferring properties between complexes and designs a new serendipity DDR complex with improved degrees of freedom efficiency.

Findings

The serendipity DDR complex reduces DOFs compared to existing methods.

Numerical tests show favorable performance on magnetostatic and Stokes problems.

The framework ensures homological and analytical property preservation.

Abstract

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.