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Spectral Exterior Calculus and its Implementation


Rufat, Dzhelil Sabahatin (2017) Spectral Exterior Calculus and its Implementation. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/Z9VX0DKV.


Preserving geometric, topological and algebraic structures at play in partial differential equations has proven to be a fruitful guiding principle for computational methods in a variety of scientific fields. However, structure-preserving numerical methods have traditionally used spaces of piecewise polynomial basis functions with local support to interpolate differential forms. When solutions are known to be smooth, a spectral treatment is often preferred instead as it brings exponential convergence. While recent works have established spectral variants of discrete exterior calculus, no existing approach offers the full breadth of exterior calculus operators and a clear distinction between vectors and covectors. We present such a unified approach to spectral exterior calculus (SPEX) and provide detail on its implementation. Notably, our approach leverages Poincare duality through the use of a primal grid and its dual (with a natural handling of boundaries to facilitate the treatment of boundary conditions), and uses a twin representation of differential forms as both integrated and pointwise values. Through its reliance on the fast Fourier transform, the resulting framework enables computations in arbitrary dimensions that are both efficient and have excellent convergence properties.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:spectral exterior calculus discrete differential geometry numerical methods
Degree Grantor:California Institute of Technology
Division:Engineering and Applied Science
Major Option:Applied Physics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Desbrun, Mathieu
Thesis Committee:
  • Desbrun, Mathieu (chair)
  • Colonius, Tim
  • Porter, Frank C.
  • Bellan, Paul Murray
Defense Date:5 April 2017
Record Number:CaltechTHESIS:05302017-094600781
Persistent URL:
Related URLs:
URLURL TypeDescription adapted for Ch. 3
Rufat, Dzhelil Sabahatin0000-0001-8766-2338
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:10219
Deposited By: Dzhelil Rufat
Deposited On:02 Jun 2017 20:10
Last Modified:25 Oct 2023 21:04

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