2,273 research outputs found
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
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
Numerical approximation of poroelasticity with random coefficients using Polynomial Chaos and Hybrid High-Order methods
In this work, we consider the Biot problem with uncertain poroelastic
coefficients. The uncertainty is modelled using a finite set of parameters with
prescribed probability distribution. We present the variational formulation of
the stochastic partial differential system and establish its well-posedness. We
then discuss the approximation of the parameter-dependent problem by
non-intrusive techniques based on Polynomial Chaos decompositions. We
specifically focus on sparse spectral projection methods, which essentially
amount to performing an ensemble of deterministic model simulations to estimate
the expansion coefficients. The deterministic solver is based on a Hybrid
High-Order discretization supporting general polyhedral meshes and arbitrary
approximation orders. We numerically investigate the convergence of the
probability error of the Polynomial Chaos approximation with respect to the
level of the sparse grid. Finally, we assess the propagation of the input
uncertainty onto the solution considering an injection-extraction problem.Comment: 30 pages, 15 Figure
On the conservativity of cell-centered Galerkin methods
International audienceIn this work we investigate the conservativity of the cell centered Galerkin method of (Di Pietro, Cell centered Galerkin methods for diffusive problems, M2AN Math. Model. Numer. Anal., 46(1):111-144, 2012) and provide an analytical expression for the conservative flux. The relation with the SUSHI method of (Eymard, Gallouët, and Herbin, Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes, SUSHI: a scheme using stabilization and hybrid interfaces, IMA J. Num. Anal., 30(4):1009-1043, 2010) and with discontinuous Galerkin methods is also explored. The theoretical results are assessed on a numerical example using standard as well as general polygonal grids
A compact cell-centered Galerkin method with subgrid stabilization
International audienceIn this work we propose a compact cell-centered Galerkin method with subgrid stabilization for anisotropic heterogeneous diffusion problems on general meshes. Both essential theoretical results and numerical validation are provided
Cell centered Galerkin methods
International audienceIn this work we propose a new approach to obtain and analyze lowest order methods for diffusive problems yielding at the same time convergence rates and convergence to minimal regularity solutions. The approach merges ideas from Finite Volume and discontinuous Galerkin methods
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