Many advanced computational problems in engineering and biology require the numerical solution of multidomain, multidimension, multiphysics and multimaterial problems with interfaces. When the interface geometry is highly complex or evolving in time, the generation of conforming meshes may become prohibitively expensive, thereby severely limiting the scope of conventional discretization methods.
In this talk we focus on the so-called cut finite element methods (CutFEM) as one possible remedy. The main idea is to design a discretization method which allows for the embedding of purely surface-based geometry representations into structured and easy-to-generate background meshes. In the first part of the talk, using the Cahn-Hilliard and biharmonic equation as starting points, we explain how the CutFEM framework leads to accurate and optimal convergent discretization schemes for a variety of PDEs posed on complex geometries. Afterwards we show that the CutFEM framework can also be used to discretize surface-bound PDEs as well as mixed-dimensional problems where PDEs are posed on domains of different topological dimensionality. In the second part of the talk, we discuss how the CutFEM approach can be employed when discretizing PDEs on evolving domains, and showcase the methodology by considering fluid flow problem involving moving interfaces including Navier-Stokes on moving domains and two-phase flow problem. We conclude with a short outlook of current activities focusing on complex interface problems.