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

TL;DR

This paper introduces a robust Hybrid High-Order scheme for the Poisson problem on polytopal meshes, maintaining optimal error estimates even with small faces or edges, demonstrated through numerical simulations in 2D and 3D.

Contribution

The paper develops a new HHO method that remains stable and accurate on complex meshes with small faces, with error bounds independent of face size or number.

Findings

Optimal error estimates achieved on complex meshes.

Method remains stable with small faces and edges.

Numerical validation in 2D and 3D confirms robustness.

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).