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Equivalence of mixed and nonconforming methods on general polytopal partitions. Part I: Multiscale and projection methods

Simon Lemaire

Abstract

We study equivalence, in the context of a variable diffusion problem, between (conforming) mixed methods and (primal) nonconforming methods defined on potentially general polytopal partitions. In this first paper of a series of two, we focus on multiscale and projection methods. For multiscale methods, we establish the first-level equivalence between four different (oversampling-free) approaches, thereby broadening the results of [Chaumont-Frelet, Ern, Lemaire, Valentin; M2AN, 2022]. For projection methods, in turn, we provide a simple criterion (to be checked in practice) for primal/mixed well-posedness and equivalence to hold true. In the process, we also shed a new light on some self-stabilized hybrid methods. Part II of this work will address (general) polytopal element methods.

Equivalence of mixed and nonconforming methods on general polytopal partitions. Part I: Multiscale and projection methods

Abstract

We study equivalence, in the context of a variable diffusion problem, between (conforming) mixed methods and (primal) nonconforming methods defined on potentially general polytopal partitions. In this first paper of a series of two, we focus on multiscale and projection methods. For multiscale methods, we establish the first-level equivalence between four different (oversampling-free) approaches, thereby broadening the results of [Chaumont-Frelet, Ern, Lemaire, Valentin; M2AN, 2022]. For projection methods, in turn, we provide a simple criterion (to be checked in practice) for primal/mixed well-posedness and equivalence to hold true. In the process, we also shed a new light on some self-stabilized hybrid methods. Part II of this work will address (general) polytopal element methods.
Paper Structure (12 sections, 6 theorems, 71 equations, 1 figure, 1 table)

This paper contains 12 sections, 6 theorems, 71 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. An illuminating observation
  3. A finite-dimensional counterpart
  4. Polytopal discretizations
  5. Polynomial spaces
  6. Finite-dimensional functional spaces
  7. Discrete problems
  8. Primal/mixed equivalence
  9. Projection methods
  10. A silver bullet
  11. The general case
  12. Conclusion

Key Result

Proposition 1

Problems pb:c.primal and pb:c.hprimal are equivalent:

Figures (1)

  • Figure 1: Equivalence diagram (continuous case): formulations on the first row are potential-based, whereas those on the second row are flux-based.

Theorems & Definitions (20)

  • Proposition 1: Equivalence of primal forms
  • proof
  • Proposition 2: Equivalence of mixed forms
  • proof
  • Lemma 1: From primal to mixed
  • proof
  • Lemma 2: From mixed to primal
  • proof
  • Remark 1: Boundedness of the inverse DoF maps
  • Remark 2: Well-posedness
  • ...and 10 more