Numerical Analysis: Cut finite element methods

We develop finite element methods for PDEs where the domain is allowed to cut through a fixed background mesh in an arbitrary fashion without loosing accuracy and without problems with ill-conditioned linear systems.

We develop finite element methods for PDEs on time dependent domains that we refer to as cut finite element methods (CutFEM). In these methods the time dependent domain is allowed to cut through a fixed background mesh in an arbitrary fashion without loosing accuracy and without problems with ill-conditioned linear systems. The cut finite element spaces are constructed by first embedding the domain in a background grid equipped with a standard finite element space and then taking the restriction of these functions to the domain. As the domain evolves with time the cut finite element space changes but no remeshing is required. Also, the same finite element space can be used for solving both problems in the bulk domain and on the surface. The CutFEM are very convenient for multiphase flow simulations where time-dependent coupled bulk-surface PDEs modeling the evolution of soluble surfactant concentrations need to be solved on evolving domains.

CutFEM can also be used to accurately capture discontinuities across moving surfaces (or interfaces). In multiphase flow problems the gradient of the velocity field and the pressure may be discontinuous across interfaces separating immiscible fluids. In CutFEM the strategy to handle discontinuities is to impose the physical jump conditions across the interface weakly by adding terms into the variational formulation. Consistent stabilization terms added to the variational formulation guarantee that the resulting equation systems have bounded condition number independently of the position of the interface.

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