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2020 | Buch

Setting Kapitel lesen Erstes Kapitel lesen

The Hybrid High-Order Method for Polytopal Meshes

Design, Analysis, and Applications

verfasst von: Prof. Daniele Antonio Di Pietro, Prof. Jérôme Droniou

Verlag: Springer International Publishing

Buchreihe : MS&A — Modeling, Simulation and Applications

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This monograph provides an introduction to the design and analysis of Hybrid High-Order methods for diffusive problems, along with a panel of applications to advanced models in computational mechanics. Hybrid High-Order methods are new-generation numerical methods for partial differential equations with features that set them apart from traditional ones. These include: the support of polytopal meshes, including non-star-shaped elements and hanging nodes; the possibility of having arbitrary approximation orders in any space dimension; an enhanced compliance with the physics; and a reduced computational cost thanks to compact stencil and static condensation.

The first part of the monograph lays the foundations of the method, considering linear scalar second-order models, including scalar diffusion – possibly heterogeneous and anisotropic – and diffusion-advection-reaction. The second part addresses applications to more complex models from the engineering sciences: non-linear Leray-Lions problems, elasticity, and incompressible fluid flows. This book is primarily intended for graduate students and researchers in applied mathematics and numerical analysis, who will find here valuable analysis tools of general scope.

Inhaltsverzeichnis

Frontmatter
Correction to: The Hybrid High-Order Method for Polytopal Meshes
Daniele Antonio Di Pietro, Jérôme Droniou

Foundations

Frontmatter
Chapter 1. Setting
Abstract
In this chapter we introduce the setting for the development and analysis of Hybrid High-Order (HHO) methods. These methods are built upon general meshes possibly including polytopal elements and non-matching interfaces. In Sect. 1.1 we give a precise definition of polytopal mesh, and introduce the notion of regular sequence of h-refined polytopal meshes. In Sect. 1.2 we recall some basic notions on standard Lebesgue and Sobolev spaces, on the space H(div; Ω), and on polynomial spaces. We next introduce the first building block of HHO methods, namely local polynomial spaces, and prove some fundamental results for the analysis including, in particular, the comparison of Lebesgue and Sobolev (semi)norms defined on such spaces, as well as local trace inequalities valid on regular mesh sequences. Section 1.3 is devoted to the second key ingredient in HHO methods: projectors on local polynomial spaces.
Daniele Antonio Di Pietro, Jérôme Droniou
Chapter 2. Basic Principles of Hybrid High-Order Methods: The Poisson Problem
Abstract
In this chapter we introduce the main ideas underlying HHO methods, using to this purpose the Poisson problem: Find \(u:\Omega \to \mathbb {R}\) such that
$$\displaystyle \begin{aligned} \begin{array}{rcl} -{\Delta} u &= f \qquad \text{in}\ \Omega, \\ u &= 0 \qquad \text{on}\ \partial\Omega, \end{array} \end{aligned}$$
where Ω is an open bounded polytopal subset of \(\mathbb {R}^n\), n ≥ 2, with boundary ∂ Ω and \(f:\Omega \to \mathbb {R}\) is a given volumetric source term, assumed to be in L 2( Ω).
Daniele Antonio Di Pietro, Jérôme Droniou
Chapter 3. Variable Diffusion and Diffusion–Advection–Reaction
Abstract
In this chapter we extend the HHO method to the scalar diffusion–advection–reaction problem: Find \(u:\Omega \to \mathbb {R}\) such that
$$\displaystyle \begin{aligned} \begin{array}{rcl}{} &\displaystyle &\displaystyle \boldsymbol{\nabla}{\cdot}(-{\mathsf{K}}\boldsymbol{\nabla} u + \boldsymbol{\beta} u) + \mu u = f \qquad \text{in }\Omega{,} \\ &\displaystyle &\displaystyle \qquad \qquad \qquad u = 0 \qquad \text{on }\partial\Omega\text{,} \end{array} \end{aligned} $$
(3.1)
where \({\mathsf {K}}:\Omega \to \mathbb {R}_{\mathrm {sym}}^{d\times d}\) (with \(\mathbb {R}_{\mathrm {sym}}^{d\times d}\) denoting the space of symmetric d × d matrices) is the spatially varying and possibly anisotropic diffusion coefficient, \(\boldsymbol {\beta }:\Omega \to \mathbb {R}^d\) is the velocity, and \(\mu :\Omega \to \mathbb {R}\) is the reaction coefficient.
Daniele Antonio Di Pietro, Jérôme Droniou
Chapter 4. Complements on Pure Diffusion
Abstract
This chapter covers two unrelated topics on HHO methods for linear diffusion problems: an a posteriori error analysis for the Poisson problem and the extension of the HHO method to the case of a diffusion tensor that varies inside each element. These topics build up on Chaps. 1 and 2, and can be used in a short introductory course to present more advanced notions on HHO.
Daniele Antonio Di Pietro, Jérôme Droniou
Chapter 5. Variations and Comparison with Other Methods

In this chapter we explore variations of the Hybrid High-Order method and establish links with other polytopal methods. Specifically, in Sect. 5.1 we consider the possibility of enriching or depleting element unknowns. Section 5.2 establishes a link with the nonconforming \(\mathbb {P}^{1}\) Finite Element method on matching simplicial meshes, which can be regarded as a variation of the lowest-order depleted HHO method with a modified discretisation of the right-hand side. We next show, in Sect. 5.3, that the lowest-order version of the standard HHO method on generic polytopal meshes is intimately linked to Hybrid Mimetic Mixed methods. In Sect. 5.4 we discuss the Mixed High-Order method, which is developed using as a starting point the mixed version of the Poisson problem, and show that the HHO method corresponds to its hybridised version. Section 5.5 establishes a link between the HHO and the Nonconforming Virtual Element method. For the sake of completeness, we also discuss the Conforming Virtual Element method and prove key results for its analysis using HHO-inspired norms, which extend to the non-Hilbertian setting. Finally, we develop in Sect. 5.6 a Gradient Discretisation Method inspired by HHO. We focus on the Poisson problem (2.​1), except in Sect. 5.6 where we consider the locally variable diffusion problem (4.​33).

Daniele Antonio Di Pietro, Jérôme Droniou

Applications to Advanced Models

Frontmatter
Chapter 6. p-Laplacian and Leray–Lions

We consider in this chapter an extension of the HHO method to fully nonlinear elliptic equations of Leray–Lions kind. This class of problems contains as a special case the p-Laplace equation, which appears in the modelling of glacier motion, of incompressible turbulent flows in porous media, in airfoil design, and can be regarded as a simplified version of the viscous terms in power-law fluids. The pure diffusion linear problems treated in Chap. 2, Sects. 3.​1, and 4.​2 can also be recovered as special cases of the framework developed here.

Daniele Antonio Di Pietro, Jérôme Droniou
Chapter 7. Linear Elasticity

In this chapter, we discuss HHO discretisations of linear elasticity. This problem, central in solid mechanics, is encountered when modelling the (small) deformations of a body under a volumetric load. From the mathematical point of view, there are relevant differences with respect to the Poisson problem discussed in Chap. 2, both at the continuous and at the discrete level. The first, obvious, difference is that, in this case, the unknown is vector-valued. The second, far-reaching, difference is that the key differential operator is the symmetric part of the gradient which, applied to the displacement field, yields the infinitesimal strain tensor. As a consequence, well-posedness for the continuous problem hinges on the Korn inequality, which states that, for homogeneous Dirichlet boundary conditions, the L 2-norm of the gradient is controlled by the L 2-norm of its symmetric part.

Daniele Antonio Di Pietro, Jérôme Droniou
Chapter 8. Stokes
Abstract
In this chapter, we apply the HHO method to the discretisation of the steady Stokes problem, which models fluid flows where convective inertial forces are small compared to viscous forces. From a physical point of view, the Stokes problem is obtained writing momentum and mass balance equations. In the case of a uniform density fluid, the mass balance translates into a zero-divergence constraint on the velocity, enabling an interpretation as a constrained minimisation (saddle-point) problem with the pressure acting as the Lagrange multiplier; see Remark 8.7. As a consequence, the well-posedness of the Stokes problem hinges on an inf–sup rather than a coercivity condition. This property has to be reproduced at the discrete level, which requires to select the discrete spaces for the velocity and pressure so that the discrete divergence operator from the former to the latter is surjective.
Daniele Antonio Di Pietro, Jérôme Droniou
Chapter 9. Navier–Stokes
Abstract
In this chapter, we discuss HHO discretisations of the steady incompressible Navier–Stokes equations. These equations, which model the motion of fluids, were originally derived by Navier and Poisson using a molecular approach, while a more specific derivation is due to Saint–Venant and Stokes based on a linear relation between the stress tensor and the strain rate tensor. The main difference with respect to the Stokes equations treated in Chap. 8 is the presence of a nonlinear contribution in the momentum balance to model convective inertial forces. Our focus is therefore on the design and analysis of HHO trilinear forms to discretise this term. From a mathematical point of view, a relevant property of the convective term is that it does not contribute to the kinetic energy balance, obtained taking the velocity as a test function in the momentum equation. This property, referred to as “non-dissipativity” in what follows, is reproduced at the discrete level, as it plays an important role in the analysis.
Daniele Antonio Di Pietro, Jérôme Droniou
Backmatter
Metadaten
Titel
The Hybrid High-Order Method for Polytopal Meshes
verfasst von
Prof. Daniele Antonio Di Pietro
Prof. Jérôme Droniou
Copyright-Jahr
2020
Electronic ISBN
978-3-030-37203-3
Print ISBN
978-3-030-37202-6
DOI
https://doi.org/10.1007/978-3-030-37203-3

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