A Hybrid High-Order method for Leray–Lions elliptic equations on general meshes
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In this paper, a hybrid high-order (HHO) method for steady non-linear Leray-Lions problems is proposed, which is achieved by combining two key ingredients devised at the local level: a gradient reconstruction and a high order stabilization term.Abstract:
In this work, we develop and analyze a Hybrid High-Order (HHO) method for steady non-linear 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, $L^p$-stability and $W^{s,p}$-approximation properties for $L^2$-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.read more
Citations
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Book
The Gradient Discretisation Method
TL;DR: The core properties that are required to prove the convergence of a GDM are stressed, and the analysis of the method is performed on a series of elliptic and parabolic problems.
BookDOI
The Hybrid High-Order Method for Polytopal Meshes: Design, Analysis, and Applications
TL;DR: This monograph provides an introduction to the design and analysis of HHO methods for diffusive problems on general meshes, along with a panel of applications to advanced models in computational mechanics.
Journal ArticleDOI
Discontinuous Skeletal Gradient Discretisation Methods on polytopal meshes
TL;DR: It is proved that the novel DSGDs satisfy coercivity, consistency, limit-conformity, and compactness requirements that ensure convergence for a variety of elliptic and parabolic problems.
Journal ArticleDOI
A Hybrid High-Order method for Darcy flows in fractured porous media
TL;DR: A novel Hybrid High-Order method for the simulation of Darcy flows in fractured porous media that is fully robust with respect to the heterogeneity of the permeability coefficients, and it exhibits only a mild dependence on the square root of the local anisotropy of the bulk permeability.
Journal ArticleDOI
A discontinuous skeletal method for the viscosity-dependent Stokes problem
TL;DR: In this paper, an arbitrary-order non-conforming method for the discretization of the viscosity-dependent Stokes equations on simplicial meshes is proposed, which is inspired from the recent Hybrid High-Order (HHO) methods for linear elasticity.
References
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TL;DR: A survey of the main ideas, concepts, and methods that constitute nonlinear functional analysis can be found in this article, with extensive commentary, many examples, and interesting, challenging exercises.
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Numerical Methods for Nonlinear Variational Problems
Roland Glowinski,J. T. Oden +1 more
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An Interior Penalty Finite Element Method with Discontinuous Elements
TL;DR: In this paper, a semidiscrete finite element method for the solution of second order nonlinear parabolic boundary value problems is formulated and analyzed, where the test and trial spaces consist of discontinuous piecewise polynomial functions over quite general meshes with interelement continuity enforced approximately by means of penalties.