A new approach to handle curved meshes in the hybrid high-order method

TL;DR

This paper extends the hybrid high-order method to curved meshes, enabling exact boundary condition enforcement and accurate geometry representation without complex mappings, with proven stability and optimal error estimates.

Contribution

It introduces a non-polynomial face function approach for curved meshes, enhancing the hybrid high-order method's applicability to complex geometries.

Findings

Method is stable and consistent on curved meshes.

Achieves optimal error estimates in $L^2$ and energy norms.

Successfully applied to domains with curved boundaries and discontinuous diffusion tensors.

Abstract

The hybrid high-order method is a modern numerical framework for the approximation of elliptic PDEs. We present here an extension of the hybrid high-order method to meshes possessing curved edges/faces. Such an extension allows us to enforce boundary conditions exactly on curved domains, and capture curved geometries that appear internally in the domain e.g. discontinuities in a diffusion coefficient. The method makes use of non-polynomial functions on the curved faces and does not require any mappings between reference elements/faces. Such an approach does not require the faces to be polynomial, and has a strict upper bound on the number of degrees of freedom on a curved face for a given polynomial degree. Moreover, this approach of enriching the space of unknowns on the curved faces with non-polynomial functions should extend naturally to other polytopal methods. We show the method to be stable and consistent on curved meshes and derive optimal error estimates in $L^2$ and energy norms. We present numerical examples of the method on a domain with curved boundary, and for a diffusion problem such that the diffusion tensor is discontinuous along a curved arc.