3,604 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
Equilibrated tractions for the Hybrid High-Order method
We show how to recover equilibrated face tractions for the hybrid high-order
method for linear elasticity recently introduced in [D. A. Di Pietro and A.
Ern, A hybrid high-order locking-free method for linear elasticity on general
meshes, Comput. Meth. Appl. Mech. Engrg., 2015, 283:1-21], and prove that these
tractions are optimally convergent
A Hybrid High-Order method for Leray-Lions elliptic equations on general meshes
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, -stability and -approximation properties for
-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
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
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
An arbitrary-order discrete rot-rot complex on polygonal meshes with application to a quad-rot problem
In this work, following the discrete de Rham (DDR) approach, we develop a
discrete counterpart of a two-dimensional de Rham complex with enhanced
regularity. The proposed construction supports general polygonal meshes and
arbitrary approximation orders. We establish exactness on a contractible domain
for both the versions of the complex with and without boundary conditions and,
for the former, prove a complete set of Poincar\'e-type inequalities. The
discrete complex is then used to derive a novel discretisation method for a
quad-rot problem which, unlike other schemes in the literature, does not
require the forcing term to be prepared. We carry out complete stability and
convergence analyses for the proposed scheme and provide numerical validation
of the results
Arbitrary-order mixed methods for heterogeneous anisotropic diffusion on general meshes
International audienceWe devise mixed methods for heterogeneous anisotropic diffusion problems supporting general polyhedral meshes. For a polynomial degree , we use as potential degrees of freedom the polynomials of degree at most inside each mesh cell, whereas for the flux we use both polynomials of degree at most for the normal component on each face and fluxes of polynomials of degree at most inside each cell. The method relies on three ideas: a flux reconstruction obtained by solving independent local problems inside each mesh cell, a discrete divergence operator with a suitable commuting property, and a stabilization enjoying the same approximation properties as the flux reconstruction. Two static condensation strategies are proposed to reduce the size of the global problem, and links to existing methods are discussed. We carry out a full convergence analysis yielding flux-error estimates of order and -potential estimates of order if elliptic regularity holds. Numerical examples confirm the theoretical results
Analysis of a discontinuous Galerkin method for heterogeneous diffusion problems with low-regularity solutions
International audienceWe study the convergence of the Symmetric Weighted Interior Penalty discontinuous Galerkin method for heterogeneous diffusion problems with low-regularity solutions only belonging to with . In 2d we infer an optimal algebraic convergence rate. In 3d we achieve the same result for p>\nicefrac65 , and for p\in(1,\nicefrac65] we prove convergence without algebraic rate
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