Daniele Antonio Di Pietro

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

In this work we propose a novel Hybrid High-Order method for the incompressible Navier– Stokes equations based on a formulation of the convective term including Temam’s device for stability. The proposed method has several advantageous features: it supports arbitrary approximation orders on general meshes including polyhedral elements and non-matching interfaces; it is inf-sup stable; it is locally conservative; it supports both the weak and strong enforcement of velocity boundary conditions; it is amenable to efficient computer implementations where a large subset of the unknowns is eliminated by solving local problems inside each element. Particular care is devoted to the design of the convective trilinear form, which mimicks at the discrete level the non-dissipation property of the continuous one. The possibility to add a convective stabilisation term is also contemplated, and a formulation covering various classical options is discussed. The proposed method is theoretically analysed, and an energy error estimate in h (with h denoting the meshsize) is proved under the usual data smallness assumption. A thorough numerical validation on two and three-dimensional test cases is provided both to confirm the theoretical convergence rates and to assess the method in more physical configurations (including, in particular, the well-known twoand three-dimensional lid-driven cavity problems).Abstract We devise an arbitrary-order locking-free method for linear elasticity. The method relies on a pure-displacement (primal) formulation and leads to a symmetric, positive definite system matrix with compact stencil. The degrees of freedom are vector-valued polynomials of arbitrary order k ⩾ 1 on the mesh faces, so that in three space dimensions, the lowest-order scheme only requires 9 degrees of freedom per mesh face. The method can be deployed on general polyhedral meshes. The key idea is to reconstruct the symmetric gradient and divergence inside each mesh cell in terms of the degrees of freedom by solving inexpensive local problems. The discrete problem is assembled cell-wise using these operators and a high-order stabilization bilinear form. Locking-free error estimates are derived for the energy norm and for the L 2 -norm of the displacement, with optimal convergence rates of order ( k + 1 ) and ( k + 2 ) , respectively, for smooth solutions on general meshes. The theoretical results are confirmed numerically, and the CPU cost is evaluated on both standard and polygonal meshes.

An artificial compressibility flux for the discontinuous Galerkin solution of the incompressible Navier-Stokes equations

Discontinuous Galerkin (DG) methods have proved to be well suited for the construction of robust high-order numerical schemes on unstructured and possibly nonconforming grids for a variety of problems. Their application to the incompressible Navier-Stokes (INS) equations has also been recently considered, although the subject is far from being fully explored. In this work, we propose a new approach for the DG numerical solution of the INS equations written in conservation form. The inviscid numerical fluxes both in the continuity and in the momentum equation are computed using the values of velocity and pressure provided by the (exact) solution of the Riemann problem associated with a local artificial compressibility perturbation of the equations. Unlike in most of the existing methods, artificial compressibility is here introduced only at the interface flux level, therefore resulting in a consistent discretization of the INS equations irrespectively of the amount of artificial compressibility introduced. The discretization of the viscous term follows the well-established DG scheme named BR2. The performance and the accuracy of the method are demonstrated by computing the Kovasznay flow and the two-dimensional lid-driven cavity flow for a wide range of Reynolds numbers and for various degrees of polynomial approximation.

On the flexibility of agglomeration based physical space discontinuous Galerkin discretizations

In this work we show that the flexibility of the discontinuous Galerkin (dG) discretization can be fruitfully exploited to implement numerical solution strategies based on the use of elements with very general shapes. Thanks to the freedom in defining the mesh topology, we propose a new h-adaptive technique based on agglomeration coarsening of a fine mesh. The possibility to enhance the error distribution over the computational domain is investigated on a Poisson problem with the goal of obtaining a mesh independent discretization. The main building block of our dG method consists of defining discrete polynomial spaces directly on physical frame elements. For this purpose we orthonormalize with respect to the L^2-product a set of monomials relocated in a specific element frame and we introduce an easy way to reduce the cost related to numerical integration on agglomerated meshes. To complete the dG formulation for second order problems, two extensions of the BR2 scheme to arbitrary polyhedral grids, including an estimate of the stabilization parameter ensuring the coercivity property, are here proposed.

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

Two 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 Temams device to control the kinetic energy balance and a conservative one based on a nonstandard modification of the pressure.

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

Abstract We 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 (elementwise) 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.

Discontinuous Galerkin Methods for Anisotropic Semidefinite Diffusion with Advection

We construct and analyze a discontinuous Galerkin method to solve advection-diffusion-reaction PDEs with anisotropic and semidefinite diffusion. The method is designed to automatically detect the so-called elliptic/hyperbolic interface on fitted meshes. The key idea is to use consistent weighted average and jump operators. Optimal estimates in the broken graph norm are proven. These are consistent with well-known results when the problem is either hyperbolic or uniformly elliptic. The theoretical results are supported by numerical evidence.

A pressure-correction scheme for convection-dominated incompressible flows with discontinuous velocity and continuous pressure

In this work we present a pressure-correction scheme for the incompressible Navier-Stokes equations combining a discontinuous Galerkin approximation for the velocity and a standard continuous Galerkin approximation for the pressure. The main interest of pressure-correction algorithms is the reduced computational cost compared to monolithic strategies. In this work we show how a proper discretization of the decoupled momentum equation can render this method suitable to simulate high Reynolds regimes. The proposed spatial velocity-pressure approximation is LBB stable for equal polynomial orders and it allows adaptive p-refinement for velocity and global p-refinement for pressure. The method is validated against a large set of classical two- and three-dimensional test cases covering a wide range of Reynolds numbers, in which it proves effective both in terms of accuracy and computational cost.

An extension of the Crouzeix–Raviart space to general meshes with application to quasi-incompressible linear elasticity and Stokes flow

In this work we introduce a discrete functional space on general polygonal or polyhedral meshes which mimics two important properties of the standard Crouzeix-Raviart space, namely the continuity of mean values at interfaces and the existence of an interpolator which preserves the mean value of the gradient inside each element. The construction borrows ideas from both Cell Centered Galerkin and Hybrid Finite Volume methods. The discrete function space is defined from cell and face unknowns by introducing a suitable piecewise affine reconstruction on a (fictitious) pyramidal subdivision of the original mesh. Two applications are considered in which the discrete space plays an important role, namely (i) the design of a locking-free primal (as opposed to mixed) method for quasi-incompressible planar elasticity on general polygonal meshes; (ii) the design of an inf-sup stable method for the Stokes equations on general polygonal or polyhedral meshes. In this context, we also propose a general modification, applicable to any suitable discretization, which guarantees that the velocity approximation is unaffected by the presence of large irrotational body forces provided a Helmholtz decomposition of the right-hand side is available. The relation between the proposed methods and classical finite volume and finite element schemes on standard meshes is investigated. Finally, similar ideas are exploited to mimic key properties of the lowest-order Raviart-Thomas space on general polygonal or polyhedral meshes.

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,

A Discontinuous-Skeletal Method for Advection-Diffusion-Reaction on General Meshes

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