Robust Hybrid High-Order method on polytopal meshes with small faces

Abstract

We design a Hybrid High-Order (HHO) scheme for the Poisson problem that is fully robust on polytopal meshes in the presence of small edges/faces. We state general assumptions on the stabilisation terms involved in the scheme, under which optimal error estimates (in discrete and continuous energy norms, as well as L 2 -norm) are established with multiplicative constants that do not depend on the maximum number of faces in each element, or the relative size between an element and its faces. We illustrate the error estimates through numerical simulations in 2D and 3D on meshes designed by agglomeration techniques (such meshes naturally have elements with a very large numbers of faces, and very small faces).

Design and analysis of the Extended Hybrid High-Order method for the Poisson problem

“…It is worth noting that Assumption 1 is independent of the size of and number of faces in each mesh element. Thus, as in [26], all error estimates in this work remain robust with respect to small and numerous faces.…”

“…For each T ∈ T h we denote by n ∂T the unit normal directed out of T , and its restriction to a face F ∈ F T is given by n T F = n ∂T | F . We further make the following assumption on the mesh inline with that stated in [26].…”

“…Hybrid High-Order methods have been used to model diffusion and diffusion-advectionreaction equations [23,17], elasticity problems [22,9], Leray Lions and p-Laplace equations [16,20], and the Stokes and Navier-Stokes equations [19,8,20]. The method has also been shown to be robust on highly irregular grids consisting of elements with arbitrarily many small faces [26] and on skewed meshes such that each element possesses a linear map to an an isotropic element [24]. All of these schemes, however, rely on error estimates which assume certain regularity of the exact solution.…”

Machine Learning based refinement strategies for polyhedral grids with applications to Virtual Element and polyhedral Discontinuous Galerkin methods

“…However, it has a considerable computational cost and it may produce elements with many small edges and faces, which implies higher costs in terms of memory storage and algorithm complexity. Moreover, although small edges and faces do not necessarily deteriorate the accuracy of numerical methods, they are in general not beneficial [49,50,51,52]. As we will see in the following, these algorithms for grids generation can be adapted to perform mesh refinement.…”

Conditioning of a Hybrid High-Order scheme on meshes with small faces

“…An analysis on skewed meshes has been carried out for a diffusion problem in [19] and identifies how the error estimate is impacted by the element distortion and local diffusion tensor. The recent work of [20] shows the HHO method to be accurate on meshes possessing elements with arbitrarily many small faces.…”