Abstract
We propose and validate a novel extension of Hybrid High-Order (HHO) methods to meshes featuring curved elements. HHO methods are based on discrete unknowns that are broken polynomials on the mesh and its skeleton. We propose here the use of physical frame polynomials over mesh elements and reference frame polynomials over mesh faces. With this choice, the degree of face unknowns must be suitably selected in order to recover on curved meshes analogous convergence rates as on straight meshes. We provide an estimate of the optimal face polynomial degree depending on the element polynomial degree and on the so-called effective mapping order. The estimate is numerically validated through specifically crafted numerical tests. We also give effective practical indications on how to choose the face polynomial degree on more standard problems closer to real life configurations, all corroborated by numerical evidence. In the worst case scenario of dealing with randomly distorted mesh sequences it is typically sufficient to select polynomial spaces over faces only one degree higher than on the elements to obtain optimal convergence rates. All test cases are conducted considering two-and three-dimensional pure diffusion problems, and include comparisons with DG discretizations. The extension to agglomerated meshes with curved boundaries is also considered.