32 research outputs found
-approximation properties of elliptic projectors on polynomial spaces, with application to the error analysis of a Hybrid High-Order discretisation of Leray-Lions problems
In this work we prove optimal -approximation estimates (with
) for elliptic projectors on local polynomial spaces. The
proof hinges on the classical Dupont--Scott approximation theory together with
two novel abstract lemmas: An approximation result for bounded projectors, and
an -boundedness result for -orthogonal projectors on polynomial
subspaces. The -approximation results have general applicability to
(standard or polytopal) numerical methods based on local polynomial spaces. As
an illustration, we use these -estimates to derive novel error
estimates for a Hybrid High-Order discretization of Leray--Lions elliptic
problems whose weak formulation is classically set in for
some . This kind of problems appears, e.g., in the modelling
of glacier motion, of incompressible turbulent flows, and in airfoil design.
Denoting by the meshsize, we prove that the approximation error measured in
a -like discrete norm scales as when
and as when .Comment: keywords: -approximation properties of elliptic projector on
polynomials, Hybrid High-Order methods, nonlinear elliptic equations,
-Laplacian, error estimate
An HMM--ELLAM scheme on generic polygonal meshes for miscible incompressible flows in porous media
We design a numerical approximation of a system of partial differential
equations modelling the miscible displacement of a fluid by another in a porous
medium. The advective part of the system is discretised using a characteristic
method, and the diffusive parts by a finite volume method. The scheme is
applicable on generic (possibly non-conforming) meshes as encountered in
applications. The main features of our work are the reconstruction of a Darcy
velocity, from the discrete pressure fluxes, that enjoys a local consistency
property, an analysis of implementation issues faced when tracking, via the
characteristic method, distorted cells, and a new treatment of cells near the
injection well that accounts better for the conservativity of the injected
fluid
Structure preservation in high-order hybrid discretisations of advection-diffusion equations: linear and nonlinear approaches
We are interested in the high-order approximation of anisotropic
advection-diffusion problems on general polytopal partitions. We study two
hybrid schemes, both built upon the Hybrid High-Order technology. The first one
hinges on exponential fitting and is linear, whereas the second is nonlinear.
The existence of solutions is established for both schemes. Both schemes are
also shown to enjoy a discrete entropy structure, ensuring that the discrete
long-time behaviour of solutions mimics the PDE one. The nonlinear scheme is
designed so as to enforce the positivity of discrete solutions. On the
contrary, we display numerical evidence indicating that the linear scheme
violates positivity, whatever the order. Finally, we verify numerically that
the nonlinear scheme has optimal order of convergence, expected long-time
behaviour, and that raising the polynomial degree results, also in the
nonlinear case, in an efficiency gain
An advection-robust Hybrid High-Order method for the Oseen problem
In this work, we study advection-robust Hybrid High-Order discretizations of
the Oseen equations. For a given integer , the discrete velocity
unknowns are vector-valued polynomials of total degree on mesh elements
and faces, while the pressure unknowns are discontinuous polynomials of total
degree on the mesh. From the discrete unknowns, three relevant
quantities are reconstructed inside each element: a velocity of total degree
, a discrete advective derivative, and a discrete divergence. These
reconstructions are used to formulate the discretizations of the viscous,
advective, and velocity-pressure coupling terms, respectively. Well-posedness
is ensured through appropriate high-order stabilization terms. We prove energy
error estimates that are advection-robust for the velocity, and show that each
mesh element of diameter contributes to the discretization error with
an -term in the diffusion-dominated regime, an
-term in the advection-dominated regime, and
scales with intermediate powers of in between. Numerical results complete
the exposition
The Hybrid High-Order Method for Polytopal Meshes: Design, Analysis, and Applications
International audienceHybrid High-Order (HHO) methods are new generation numerical methods for models based on 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 to have arbitrary approximation orders in any space dimension; an enhanced compliance with the physics; a reduced computational cost thanks to compact stencil and static condensation. 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. The first part of the monograph lays the foundation 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
Bridging the Hybrid High-Order and Virtual Element methods
International audienceWe present a unifying viewpoint at Hybrid High-Order and Virtual Element methods on general polytopal meshes in dimension or , both in terms of formulation and analysis. We focus on a model Poisson problem. To build our bridge, (i) we transcribe the (conforming) Virtual Element method into the Hybrid High-Order framework, and (ii) we prove approximation properties for the local polynomial projector in terms of which the local Virtual Element discrete bilinear form is defined. This allows us to perform a unified analysis of Virtual Element/Hybrid High-Order methods, that differs from standard Virtual Element analyses by the fact that the approximation properties of the underlying virtual space are not explicitly used. As a complement to our unified analysis, we also study interpolation in local virtual spaces, shedding light on the differences between the conforming and nonconforming cases
A Hybrid High-Order Method for a Class of Strongly Nonlinear Elliptic Boundary Value Problems
In this article, we design and analyze a Hybrid High-Order (HHO) finite
element approximation for a class of strongly nonlinear boundary value
problems. We consider an HHO discretization for a suitable linearized problem
and show its well-posedness using the Gardings type inequality. The essential
ingredients for the HHO approximation involve local reconstruction and
high-order stabilization. We establish the existence of a unique solution for
the HHO approximation using the Brouwer fixed point theorem and contraction
principle. We derive an optimal order a priori error estimate in the discrete
energy norm. Numerical experiments are performed to illustrate the convergence
histories.Comment: arXiv admin note: substantial text overlap with arXiv:2110.1557
A Hybrid High-Order method for nonlinear elasticity
In this work we propose and analyze a novel Hybrid High-Order discretization
of a class of (linear and) nonlinear elasticity models in the small deformation
regime which are of common use in solid mechanics. The proposed method is valid
in two and three space dimensions, it supports general meshes including
polyhedral elements and nonmatching interfaces, enables arbitrary approximation
order, and the resolution cost can be reduced by statically condensing a large
subset of the unknowns for linearized versions of the problem. Additionally,
the method satisfies a local principle of virtual work inside each mesh
element, with interface tractions that obey the law of action and reaction. A
complete analysis covering very general stress-strain laws is carried out, and
optimal error estimates are proved. Extensive numerical validation on model
test problems is also provided on two types of nonlinear models.Comment: 29 pages, 7 figures, 4 table
Improved error estimates for Hybrid High-Order discretizations of Leray-Lions problems
We derive novel error estimates for Hybrid High-Order (HHO) discretizations
of Leray-Lions problems set in W^(1,p) with p in (1,2]. Specifically, we prove
that, depending on the degeneracy of the problem, the convergence rate may vary
between (k+1)(p-1) and (k+1), with k denoting the degree of the HHO
approximation. These regime-dependent error estimates are illustrated by a
complete panel of numerical experiments.Comment: 20 pages, 4 figures, 4 table