Divergensfria skurna finita element diskretiseringar

We propose the development and analysis of Cut Finite Element Methods (CutFEM) that provide exactly divergence-free approximations of the velocity field in simulations of incompressible flows.CutFEM is a new generation of computational methods that allow external or internal boundaries such as interfaces separating immiscible fluids to cut through a background mesh in an arbitrary fashion while giving optimal convergence order.

To cure the well-known problem of ill-conditioned linear system matrices from unfitted finite element methods stabilization terms have been developed that are added in the variational formulations of CutFEM. These terms guarantee bounded condition numbers regardless of the position of the interface relative to the mesh.

However, they have not been developed with the preservation of the divergence-free condition in mind and they may contribute to spurious errors.Our aim is to improve on existing CutFEM to have discretizations with 1. Optimal rates of convergence for the velocity and the pressure; 2. Well-posed resulting linear systems; 3.

Optimal approximation of the divergence with pointwise divergence-free approximations of solenoidal velocity fields.

The new discretizations will be based on a solid mathematical foundation.We believe the new methods will provide a competitive alternative to existing computational techniques for discretizing PDEs in and on deformable geometries and stretch the state of the art in multiphase flow simulations.