A Hybrid High-Order method for Leray-Lions elliptic equations on general meshes

Daniele A Di Pietro, Jerome Droniou

Research output: Contribution to journalArticleResearchpeer-review

28 Citations (Scopus)

Abstract

In this work, we develop and analyze a Hybrid High-Order (HHO) method for steady nonlinear Leray-Lions problems. The proposed method has several assets, including the support for arbitrary approximation orders and general polytopal meshes. This is achieved by combining two key ingredients devised at the local level: a gradient reconstruction and a high-order stabilization term that generalizes the one originally introduced in the linear case. The convergence analysis is carried out using a compactness technique. Extending this technique to HHO methods has prompted us to develop a set of discrete functional analysis tools whose interest goes beyond the specific problem and method addressed in this work: (direct and) reverse Lebesgue and Sobolev embeddings for local polynomial spaces, Lp-stability and Ws,p-approximation properties for L2-projectors on such spaces, and Sobolev embeddings for hybrid polynomial spaces. Numerical tests are presented to validate the theoretical results for the original method and variants thereof.

Original languageEnglish
Pages (from-to)2159-2191
Number of pages33
JournalMathematics of Computation
Volume86
Issue number307
DOIs
Publication statusPublished - 2017

Keywords

  • Convergence analysis
  • Discrete functional analysis
  • Hybrid High-Order methods
  • Nonlinear elliptic equations
  • P-Laplacian
  • W-approximation properties of L-projection on polynomials

Cite this

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title = "A Hybrid High-Order method for Leray-Lions elliptic equations on general meshes",
abstract = "In this work, we develop and analyze a Hybrid High-Order (HHO) method for steady nonlinear Leray-Lions problems. The proposed method has several assets, including the support for arbitrary approximation orders and general polytopal meshes. This is achieved by combining two key ingredients devised at the local level: a gradient reconstruction and a high-order stabilization term that generalizes the one originally introduced in the linear case. The convergence analysis is carried out using a compactness technique. Extending this technique to HHO methods has prompted us to develop a set of discrete functional analysis tools whose interest goes beyond the specific problem and method addressed in this work: (direct and) reverse Lebesgue and Sobolev embeddings for local polynomial spaces, Lp-stability and Ws,p-approximation properties for L2-projectors on such spaces, and Sobolev embeddings for hybrid polynomial spaces. Numerical tests are presented to validate the theoretical results for the original method and variants thereof.",
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A Hybrid High-Order method for Leray-Lions elliptic equations on general meshes. / Di Pietro, Daniele A; Droniou, Jerome.

In: Mathematics of Computation, Vol. 86, No. 307, 2017, p. 2159-2191.

Research output: Contribution to journalArticleResearchpeer-review

TY - JOUR

T1 - A Hybrid High-Order method for Leray-Lions elliptic equations on general meshes

AU - Di Pietro, Daniele A

AU - Droniou, Jerome

PY - 2017

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N2 - In this work, we develop and analyze a Hybrid High-Order (HHO) method for steady nonlinear Leray-Lions problems. The proposed method has several assets, including the support for arbitrary approximation orders and general polytopal meshes. This is achieved by combining two key ingredients devised at the local level: a gradient reconstruction and a high-order stabilization term that generalizes the one originally introduced in the linear case. The convergence analysis is carried out using a compactness technique. Extending this technique to HHO methods has prompted us to develop a set of discrete functional analysis tools whose interest goes beyond the specific problem and method addressed in this work: (direct and) reverse Lebesgue and Sobolev embeddings for local polynomial spaces, Lp-stability and Ws,p-approximation properties for L2-projectors on such spaces, and Sobolev embeddings for hybrid polynomial spaces. Numerical tests are presented to validate the theoretical results for the original method and variants thereof.

AB - In this work, we develop and analyze a Hybrid High-Order (HHO) method for steady nonlinear Leray-Lions problems. The proposed method has several assets, including the support for arbitrary approximation orders and general polytopal meshes. This is achieved by combining two key ingredients devised at the local level: a gradient reconstruction and a high-order stabilization term that generalizes the one originally introduced in the linear case. The convergence analysis is carried out using a compactness technique. Extending this technique to HHO methods has prompted us to develop a set of discrete functional analysis tools whose interest goes beyond the specific problem and method addressed in this work: (direct and) reverse Lebesgue and Sobolev embeddings for local polynomial spaces, Lp-stability and Ws,p-approximation properties for L2-projectors on such spaces, and Sobolev embeddings for hybrid polynomial spaces. Numerical tests are presented to validate the theoretical results for the original method and variants thereof.

KW - Convergence analysis

KW - Discrete functional analysis

KW - Hybrid High-Order methods

KW - Nonlinear elliptic equations

KW - P-Laplacian

KW - W-approximation properties of L-projection on polynomials

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