Cut Finite Element Methods

In this talk I will present a new type of finite element methods that we refer to as Cut Finite Element Methods (CutFEM). CutFEM provides an efficient strategy for solving partial differential equations in evolving geometries. In CutFEM the dynamic geometry is allowed to cut through the background grid in an arbitrary fashion and remeshing processes are avoided. We will consider convection-diffusion equations on evolving surfaces (or interfaces) modeling the evolution of insoluble surfactants and a space-time cut finite element method based on quadrature in time. A stabilization term is added to the weak formulation which guarantees that the linear systems resulting from the method have bounded condition numbers independently of how the geometry cuts through the background mesh for linear as well as higher order elements.