Discrete functional analysis tools for Discontinuous Galerkin methods with application to the incompressible Navier--Stokes equations

28 pagesInternational audienceTwo discrete functional analysis tools are established for spaces of piecewise polynomial functions on general meshes: (i) a discrete counterpart of the continuous Sobolev embeddings, in both Hilbertian and non-Hilbertian settings; (ii) a compactness result for bounded sequences in a suitable Discontinuous Galerkin norm, together with a weak convergence property for some discrete gradients. The proofs rely on techniques inspired by the Finite Volume literature, which differ from those commonly used in Finite Element analysis. The discrete functional analysis tools are used to prove the convergence of Discontinuous Galerkin approximations of the steady incompressible Navier--Stokes equations. Two discrete convective trilinear forms are proposed, a non-conservative one relying on Temam's device to control the kinetic energy balance and a conservative one based on a nonstandard modification of the pressure

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 $W^{2,p}$ with $p\in(1,2]$. 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

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 $k\ge 0$, we use as potential degrees of freedom the polynomials of degree at most $k$ inside each mesh cell, whereas for the flux we use both polynomials of degree at most $k$ for the normal component on each face and fluxes of polynomials of degree at most $k$ 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 $(k+1)$ and $L^2$-potential estimates of order $(k+2)$ if elliptic regularity holds. Numerical examples confirm the theoretical results

A hybrid high-order locking-free method for linear elasticity on general meshes

International audienceWe develop an arbitrary-order locking-free method for linear elasticity on general (polyhedral, possibly nonconforming) meshes without nodal unknowns. The key idea is to reconstruct the relevant differential operators in terms of the (generalized) degrees of freedom by solving an inexpensive local problem inside each element. The symmetric gradient and the divergence operators are reconstructed separately. The divergence operator satisfies a commuting diagram property, yielding robustness in the quasi-incompressible limit. Locking-free error estimates are derived for the energy norm and for the L2-norm of the displacement, with optimal convergence rates for smooth solutions. The theoretical results are confirmed numerically, and the CPU cost is evaluated on both standard and general polygonal meshes

Hybrid High-Order methods for variable diffusion problems on general meshes

We extend the Hybrid High-Order method introduced by the authors for the Poisson problem to problems with heterogeneous/anisotropic diffusion. The cornerstone is a local discrete gradient reconstruction from element- and face-based polynomial degrees of freedom. Optimal error estimates are proved

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

An arbitrary-order and compact-stencil discretization of diffusion on general meshes based on local reconstruction operators

International audienceWe develop an arbitrary-order primal method for diffusion problems on general polyhedral meshes. The degrees of freedom are scalar-valued polynomials of the same order at mesh elements and faces. The cornerstone of the method is a local (element-wise) discrete gradient reconstruction operator. The design of the method additionally hinges on a least-squares penalty term on faces weakly enforcing the matching between local element- and face-based degrees of freedom. The scheme is proved to optimally converge in the energy norm and in the L2-norm of the potential for smooth solutions. In the lowest-order case, equivalence with the Hybrid Finite Volume method is shown. The theoretical results are confirmed by numerical experiments up to order 4 on several polygonal meshes

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

A Hybrid High-Order method for multiple-network poroelasticity

We develop Hybrid High-Order methods for multiple-network poroelasticity, modelling seepage through deformable fissured porous media. The proposed methods are designed to support general polygonal and polyhedral elements. This is a crucial feature in geological modelling, where the need for general elements arises, e.g., due to the presence of fracture and faults, to the onset of degenerate elements to account for compaction or erosion, or when nonconforming mesh adaptation is performed. We use as a starting point a mixed weak formulation where an additional total pressure variable is added, that ensures the fulfilment of a discrete inf-sup condition. A complete theoretical analysis is performed, and the theoretical results are demonstrated on a complete panel of numerical tests

Fully discrete polynomial de Rham sequences of arbitrary degree on polygons and polyhedra

In this work, merging ideas from compatible discretisations and polyhedral methods, we construct novel fully discrete polynomial de Rham sequences of arbitrary degree on polygons and polyhedra. The spaces and operators that appear in these sequences are directly amenable to computer implementation. Besides proving exactness, we show that the usual sequence of Finite Element spaces forms, through appropriate interpolation operators, a commutative diagram with our sequence, which ensures suitable approximation properties. A discussion on reconstructions of potentials and discrete $L^2$-products completes the exposition