An hp-version interior penalty discontinuous Galerkin method (DGFEM) for the numerical solution of second-order elliptic partial differential equations on general computational meshes consisting of polygonal/polyhedral elements is presented and analysed. Utilizing a bounding box concept, the method employs elemental polynomial bases of total degree p (P_p-basis) defined on the physical space, without the need to map from a given reference or canonical frame. This, together with a new specific choice of the interior penalty parameter which allows for face-degeneration, ensures that optimal a priori bounds may be established, for general meshes including polygonal elements with degenerating edges in two dimensions and polyhedral elements with degenerating faces and/or edges in three dimensions. Numerical experiments highlighting the performance of the proposed method are presented. Moreover, the competitiveness of the p-version DGFEM employing a P_p-basis in comparison to the conforming p-version finite element method on tensor-product elements is studied numerically for a simple test problem.

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hp-Version Discontinuous Galerkin Methods on Polygonal and Polyhedral 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, $L^p$-stability and $W^{s,p}$-approximation properties for $L^2$-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.

https://hal.archives-ouvertes.fr/hal-01183484/document

A Hybrid High-Order method for Leray–Lions elliptic equations on general meshes

Hybrid 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.

https://hal.archives-ouvertes.fr/hal-02151813/document

The Hybrid High-Order Method for Polytopal Meshes: Design, Analysis, and Applications

In this paper we introduce the $hp$-version discontinuous Galerkin composite finite element method for the discretization of second-order elliptic partial differential equations. This class of methods allows for the approximation of problems posed on computational domains which may contain a huge number of local geometrical features, or microstructures. While standard numerical methods can be devised for such problems, the computational effort may be extremely high, as the minimal number of elements needed to represent the underlying domain can be very large. In contrast, the minimal dimension of the underlying composite finite element space is independent of the number of geometric features. The key idea in the construction of this latter class of methods is that the computational domain $\Omega$ is no longer resolved by the mesh; instead, the finite element basis (or shape) functions are adapted to the geometric details present in $\Omega$. In this paper, we extend these ideas to the discontinuous Galer...

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$hp$-Version Composite Discontinuous Galerkin Methods for Elliptic Problems on Complicated Domains

In this paper we investigate the behavior of the condition number of the stiffness matrix resulting from the approximation of a 2D Poisson problem by means of the Virtual Element Method. It turns out that ill-conditioning appears when considering high-order methods or in presence of "bad-shaped" (for instance nonuniformly star-shaped, with small edges...) sequences of polygons. We show that in order to improve such condition number one can modify the definition of the internal moments by choosing proper polynomial functions that are not the standard monomials. We also give numerical evidence that, at least for a 2D problem, standard choices for the stabilization give similar results in terms of condition number.

https://onlinelibrary.wiley.com/doi/pdf/10.1002/num.22257

Ill-conditioning in the virtual element method: Stabilizations and bases

On the flexibility of agglomeration based physical space discontinuous Galerkin discretizations

Discontinuous Galerkin (DG) methods have proven to be perfectly suited for the construction of very high-order accurate numerical schemes on arbitrary unstructured and possibly nonconforming grids for a wide variety of applications, but are rather demanding in terms of computational resources. In order to improve the computational ef?ciency of this class of methods a p-multigrid solution strategy has been developed, which is based on a semi-implicit Runge–Kutta smoother for high-order polynomial approximations and the implicit Backward Euler smoother for piecewise constant approximations. The effectiveness of the proposed approach is demonstrated by comparison with p-multigrid schemes employing purely explicit smoothing operators for several 2D inviscid test cases.

Pietro Francesco Tesini

Papers

On the flexibility of agglomeration based physical space discontinuous Galerkin discretizations

High‐order accurate p‐multigrid discontinuous Galerkin solution of the Euler equations