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Serendipity discrete complexes with enhanced regularity.

Daniele A Di Pietro1, Marien Hanot2, Marwa Salah1

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Summary
This summary is machine-generated.

Researchers developed a new abstract construction to create serendipity versions of approximate de Rham complexes. This method enhances regularity and generates novel rot-rot and Stokes complexes for advanced mathematical applications.

Keywords:
Compatible discretizationsDiscrete de Rham methodRot–rot complexSerendipityStokes complex

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Area of Science:

  • Mathematics
  • Differential Geometry
  • Complex Analysis

Background:

  • Approximate de Rham complexes are crucial in various mathematical fields.
  • Enhanced regularity in these complexes is a significant challenge.
  • Serendipity methods offer novel approaches to complex construction.

Purpose of the Study:

  • To introduce a general abstract construction for generating new complexes.
  • To create serendipity versions of approximate de Rham complexes with enhanced regularity.
  • To devise new rot-rot and Stokes complexes within the Discrete de Rham framework.

Main Methods:

  • A novel abstract construction is presented, linking three complexes via extension and reduction maps.
  • This construction generates a fourth complex with isomorphic cohomology.
  • The method is applied to derive specific serendipity complexes.

Main Results:

  • A new general method for constructing complexes with specific cohomology properties is established.
  • Novel serendipity versions of rot-rot and Stokes complexes are successfully derived.
  • The enhanced regularity of the generated complexes is demonstrated.

Conclusions:

  • The abstract construction provides a powerful tool for developing new mathematical complexes.
  • The devised serendipity complexes offer enhanced regularity and novel properties.
  • This work advances the understanding and application of Discrete de Rham complexes.