Assessment of Hybrid High-Order methods on curved meshes and comparison with discontinuous Galerkin methods

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.MSC 2010: 65N08, 65N30, 65N12

Curvilinear Virtual Elements for 2D solid mechanics applications

“…While it is evident that the VEM cv is exact up to machine precision, the error levels associated with the VEM co are much higher. This shows that the virtual element space in the new variant (VEM cv) contains the solution of the problem at hand, see Equation (19), while the standard space (VEM co) does not contains such solution.…”

A Hybrid High-Order method for the incompressible Navier–Stokes equations based on Temam's device

“…These local reconstructions are used to formulate consistent Galerkin terms, while stability is achieved by stabilisation terms devised at the element level. The HHO approach has several advantages: it is dimension-independent; it supports arbitrary approximation orders on general meshes including polyhedral elements, non-matching junctions and, possibly, curved faces [3]; it is locally conservative; it is amenable to efficient (parallel or serial) computer implementations. The HHO method proposed in this work has additional advantageous features specific to the incompressible Navier-Stokes problem: it satisfies a uniform inf-sup condition, leading to a stable pressure-velocity coupling; it behaves robustly for large Reynolds numbers; it supports both weakly and strongly enforced boundary conditions; at each nonlinear iteration, it requires to solve a linear system where the only globally coupled unknowns are the face velocities and the mean value of the pressure inside each element.…”

HDGlab: An Open-Source Implementation of the Hybridisable Discontinuous Galerkin Method in MATLAB

“…In the context of HDG, high-order isoparametric approaches in presence of curved meshes are utilised in many references, see e.g. [148,193], whereas this technique is addressed for HHO in [29]. An alternative approach relying on meshes with planar faces and the extension to a fictitious subdomain is discussed in [101,102,250] for several linear problems and was recently extended to the semilinear Grad-Shafranov equation [233,234].…”