Daniela capatina

Daniela Capatina, Aimene Gouasmi

This paper deals with the local recovery of conservative fluxes for an elliptic interface problem with discontinuous coefficients. The transmission conditions on the interface are imposed weakly and the discretisation is achieved by using conforming finite elements on unfitted meshes, with the aid of the CutFEM method. In a first attempt at flux reconstruction, we define a flux belonging to the Raviart-Thomas space of each sub-domain following the method developed for a boundary problem. However, the transmission condition is not satisfied by the recovered flux. In order to overcome this shortcoming, we propose a second approach, where the flux belongs to the recently introduced immersed Raviart-Thomas space. This ensures both the continuity of the normal flux across the interface and a natural conservation property on the cut cells. Subsequently, we apply the recovered flux to a posteriori error analysis and prove the sharp reliability of the error estimator.

Daniela capatina, Aimene Gouasmi

This paper addresses the local recovery of conservative fluxes and the a posteriori error analysis for an elliptic interface problem with discontinuous coefficients. The transmission conditions on the interface are imposed by means of Nitsche's method and the discretization is carried out using conforming finite elements on unfitted meshes via the CutFEM method. A flux is subsequently defined in the global Raviart-Thomas space, ensuring that it satisfies the natural conservation property on the cut cells, and is then employed in the a posteriori error analysis. We prove here the sharp reliability of the error estimator and show a numerical experiment which illustrates the approach.

Roland Becker, Daniela Capatina, Robert Luce et al.

We study the finite element formulation of general boundary conditions for incompressible flow problems. Distinguishing between the contributions from the inviscid and viscid parts of the equations, we use Nitsche's method to develop a discrete weighted weak formulation valid for all values of the viscosity parameter, including the limit case of the Euler equations. In order to control the discrete kinetic energy, additional consistent terms are introduced. We treat the limit case as a (degenerate) system of hyperbolic equations, using a balanced spectral decomposition of the flux Jacobian matrix, in analogy with compressible flows. Then, following the theory of Friedrich's systems, the natural characteristic boundary condition is generalized to the considered physical boundary conditions. Several numerical experiments, including standard benchmarks for viscous flows as well as inviscid flows are presented.