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Serendipity discrete complexes with enhanced regularity

Daniele Di Pietro, Marien Hanot, Marwa Salah

TL;DR

This work introduces an abstract framework for deriving serendipity versions of advanced Hilbert complexes with enhanced regularity. By starting from three interconnected complexes linked through extension and reduction maps, the authors construct a fourth complex whose cohomology matches the originals, enabling systematic serendipity while preserving topological invariants. The framework is applied to the discrete de Rham (DDR) complex and its serendipity variant, deriving serendipity versions of the 2D rot-rot and 3D Stokes complexes, with explicit spaces, operators, and extension/reduction maps. Numerical experiments illustrate the benefits: serendipity formulations achieve substantial degrees-of-freedom reductions with comparable accuracy to their standard DDR counterparts across various mesh families, confirming the practical viability of enhanced-regularity serendipity discretizations for complex Hilbert problems.

Abstract

In this work we address the problem of finding serendipity versions of approximate de Rham complexes with enhanced regularity. The starting point is a new abstract construction of general scope which, given three complexes linked by extension and reduction maps, generates a fourth complex with cohomology isomorphic to the former three. This construction is used to devise new serendipity versions of rot-rot and Stokes complexes derived in the Discrete de Rham spirit.

Serendipity discrete complexes with enhanced regularity

TL;DR

This work introduces an abstract framework for deriving serendipity versions of advanced Hilbert complexes with enhanced regularity. By starting from three interconnected complexes linked through extension and reduction maps, the authors construct a fourth complex whose cohomology matches the originals, enabling systematic serendipity while preserving topological invariants. The framework is applied to the discrete de Rham (DDR) complex and its serendipity variant, deriving serendipity versions of the 2D rot-rot and 3D Stokes complexes, with explicit spaces, operators, and extension/reduction maps. Numerical experiments illustrate the benefits: serendipity formulations achieve substantial degrees-of-freedom reductions with comparable accuracy to their standard DDR counterparts across various mesh families, confirming the practical viability of enhanced-regularity serendipity discretizations for complex Hilbert problems.

Abstract

In this work we address the problem of finding serendipity versions of approximate de Rham complexes with enhanced regularity. The starting point is a new abstract construction of general scope which, given three complexes linked by extension and reduction maps, generates a fourth complex with cohomology isomorphic to the former three. This construction is used to devise new serendipity versions of rot-rot and Stokes complexes derived in the Discrete de Rham spirit.
Paper Structure (34 sections, 8 theorems, 91 equations, 12 figures)

This paper contains 34 sections, 8 theorems, 91 equations, 12 figures.

Table of Contents

  1. Introduction
  2. An abstract framework for serendipity complexes with enhanced regularity
  3. Setting
  4. Construction of a serendipity complex with enhanced regularity
  5. The discrete de Rham complex and its serendipity version
  6. Local polynomial spaces and $L^2$-orthogonal projectors
  7. The two-dimensional discrete de Rham complex
  8. Spaces
  9. Discrete vector calculus operators
  10. DDR complex
  11. The three-dimensional discrete de Rham complex
  12. Spaces
  13. Discrete vector calculus operators
  14. DDR complex
  15. Serendipity spaces
  16. ...and 19 more sections

Key Result

Lemma 4

Assume P:Identity.ER and let Then, we have the following direct decomposition: Under assumption P:E.R.cochain, this decomposition is compatible with $d_i$, in the sense that

Figures (12)

  • Figure 1: $\|\boldsymbol{e}_h\|_{\boldsymbol{\Sigma},h}$ and $\|\underline{\widehat{\boldsymbol{e}}}_h\|_{\boldsymbol{\Sigma},h}$
  • Figure 2: $\|\underline{\boldsymbol{e}}_h\|_{\mathop{\mathrm{\bf rot}}\nolimits\mathop{\mathrm{rot}}\nolimits,h}$ and $\|\underline{\widehat{\boldsymbol{e}}}_h\|_{\mathop{\mathrm{\bf rot}}\nolimits\mathop{\mathrm{rot}}\nolimits,h}$
  • Figure 3: $\|\underline{\varepsilon}_h\|_{V,h}$ and $\|\underline{\widehat{\varepsilon}}_h\|_{V,h}$
  • Figure 4: ${\|\underline{\boldsymbol{d}}_{\mathop{\mathrm{\bf grad}}\nolimits,{h}}^k \underline{\varepsilon}_h\|_{\boldsymbol{\Sigma},h}}$ and ${\|\underline{\widehat{\boldsymbol{d}}}_{\mathop{\mathrm{\bf grad}}\nolimits,h}^k\underline{\widehat{\varepsilon}}_h\|_{\boldsymbol{\Sigma},h}}$
  • Figure 6: $\|\underline{\boldsymbol{e}}_h\|_{\boldsymbol{\Sigma},h}$ and $\|\underline{\widehat{\boldsymbol{e}}}_h\|_{\boldsymbol{\Sigma},h}$
  • ...and 7 more figures

Theorems & Definitions (16)

  • Remark 3: Isomorphic cohomologies
  • Lemma 4: Decomposition of $V_i$
  • proof
  • Definition 5: Complex $(\widehat{V}_i, \widehat{d}_i)$, extension and reduction operators
  • Lemma 6: Commutation properties
  • proof
  • Theorem 7: Homological properties for $(V_i,d_i)_i$ and $(\widehat{V}_i,\widehat{d}_i)$
  • proof
  • Corollary 8: Isomorphism in cohomology
  • proof
  • ...and 6 more