An Unfitted Hybrid High-Order Method for Elliptic Interface Problems

Abstract

We design and analyze a Hybrid High-Order (HHO) method on unfitted meshes to approximate elliptic interface problems. The curved interface can cut through the mesh cells in a very general fashion. As in classical HHO methods, the present unfitted method introduces cell and face unknowns in uncut cells, but doubles the unknowns in the cut cells and on the cut faces. The main difference with classical HHO methods is that a Nitsche-type formulation is used to devise the local reconstruction operator. As in classical HHO methods, cell unknowns can be eliminated locally leading to a global problem coupling only the face unknowns by means of a compact stencil. We prove stability estimates and optimal error estimates in the H 1 -norm. Robustness with respect to cuts is achieved by a local cell-agglomeration procedure taking full advantage of the fact that HHO methods support polyhedral meshes. Robustness with respect to the contrast in the material properties from both sides of the interface is achieved by using material-dependent weights in Nitsche's formulation.

A hybrid high‐order formulation for a Neumann problem on polytopal meshes

“…We have proved the convergence of the method, with the optimal rates of convergence when the exact solution is smooth enough: order k + 1 for the flux error, and k + 2 for potential error, when piecewise polynomial of degree at most k are considered for the corresponding approximations. This technique can deal with hanging nodes, as in the Refined mesh (Figure 1c), and also with triangular, quadrilateral and hexagonal meshes (Figure 1a, b, This library has been used to solve many problems as those described in [21][22][23][24][25][26][27][28][29]. On the other hand, HArDCore (Hybrid Arbitrary Degree::Core, https://github.com/jdroniou/HArDCore) is a C++ code focused on HHO methods, but it can be useful for a wide range of hybrid methods.…”

Hybrid high-order discretizations combined with Nitsche’s method for Dirichlet and Signorini boundary conditions

“…The crucial advantage of Nitsche-HHO with respect to Nitsche-FEM is the possibility to handle polyhedral meshes. This possibility has been illustrated in recent works that deal with Nitsche's technique combined with polyhedral discretization methods to treat Dirichlet boundary conditions on curved boundaries: see [6] for the Virtual Element method and [13] for HHO methods. In addition, some polyhedral discretizations of contact problems have been proposed very recently using for instance the Virtual Element method [67,60,61], weak Galerkin schemes [39], hybridizable Discontinous Galerkin methods [69] and the Scaled Boundary Finite Element method [68].…”

“…In what follows, we refer to these variants as the face version and the cell version of Nitsche-HHO, respectively. The devising of the cell version elaborates on the idea of modifying the local reconstruction operator, as proposed in [13] in the different context of geometrically unfitted methods. Our main results are on the one hand Theorem 4.7 and Theorem 5.5 which establish the optimal convergence of Nitsche-HHO for Dirichlet conditions, using face and cell versions, respectively, and on the other hand Theorem 6.4 which establishes the optimal convergence of the cell version of Nitsche-HHO for Signorini conditions.…”

Hybrid High-Order Methods for the Elliptic Obstacle Problem

“…Hybrid Higher Order (HHO) methods have been introduced for linear elasticity in [19] and linear diffusion in [21]. HHO methods have been extended to other linear PDEs, such as advection-diffusion [22], Stokes [23], and elliptic interface problems [12], and to nonlinear PDEs, such as Leray-Lions operators [17], steady incompressible Navier-Stokes equations [20], nonlinear elasticity with infinitesimal deformations [6], hyperelasticity with finite deformations [1], and plasticity with small deformations [2]. Lowest-order HHO methods are closely related to the Hybrid Finite Volume method [25] and the Mimetic Finite Difference methods [34,9,10], see also the unifying viewpoints in [24,5].…”