Section: New Results

Unfitted hybrid-high-order methods

Participants : Alexandre Ern, Guillaume Delay.

Our team contributes actively to the development of hybrid high-order (HHO) methods. Such methods support polyhedral meshes with hanging nodes, but one requirement is that the mesh cells have planar faces. This is difficult when it comes to solving with high accuracy a problem posed on a domain with curved boundaries or a problem involving a curved interface separating two materials with different properties. One key idea to treat these problems is to use an unfitted mesh, so that the curved boundary or the curved interface freely cuts through the mesh cells. This greatly simplifies the meshing process, but at the same time poses the question on how the HHO method can address the approximation of functions that are not smooth within some mesh cells. The major idea in our approach, which is inspired from similar approaches developed in the context of the more classical finite element method, is to double the discrete unknowns attached to the cut mesh faces and to introduce a consistent Nitsche-type formulation to enforce either the boundary condition or the jump conditions across the interface in a weak manner. In this context, we started a collaboration with Erik Burman (University College London) and we elaborated in [20] the numerical analysis of HHO methods in an unfitted context; further analysis for Stokes and Helmholtz equations has started recently within the postdoc of Guillaume Delay and a collaboration on the subject with CEA is on the way.