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Published January 15, 2014 | public
Journal Article

The chain collocation method: A spectrally accurate calculus of forms

Abstract

Preserving in the discrete realm the underlying geometric, topological, and algebraic structures at stake in partial differential equations has proven to be a fruitful guiding principle for numerical methods in a variety of fields such as elasticity, electromagnetism, or fluid mechanics. However, structure-preserving methods have traditionally used spaces of piecewise polynomial basis functions for differential forms. Yet, in many problems where solutions are smoothly varying in space, a spectral numerical treatment is called for. In an effort to provide structure-preserving numerical tools with spectral accuracy on logically rectangular grids over periodic or bounded domains, we present a spectral extension of the discrete exterior calculus (DEC), with resulting computational tools extending well-known collocation-based spectral methods. Its efficient implementation using fast Fourier transforms is provided as well.

Additional Information

© 2013 Elsevier Inc. Received 2 July 2012. Received in revised form 30 April 2013. Accepted 4 August 2013. Available online 20 August 2013. We thank Marc Gerritsma for discussions and references. We gratefully acknowledge partial funding from NSF grant CCF-1011944.

Additional details

Created:
August 22, 2023
Modified:
October 25, 2023