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 recent, 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. Furthermore, we demonstrate their effectiveness when discretizing PDEs on evolving domains, including Navier-Stokes equations and fluid-structure interaction problems with large deformations.

In the second part of the talk, 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.

As a particular example, we consider the so-called Extracellular-Membran-Intracellular (EMI) model which couples an elliptic partial differential equation on the extra/intracellular domains with a system of nonlinear ordinary differential equations (ODEs) over the cell membranes to model of electrical activity of explicitly resolved brain cells.

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