Sebastian Myrbäck: Discrete Continuity Equations in Cut Finite Element Methods

In modern numerical methods, solving partial differential equations often requires fulfilling associated continuity equations, such as mass conservation. This challenge is particularly pronounced in solving incompressible Navier-Stokes equations, where the divergence-free condition must be fulfilled. In this work, we focus on a convection-diffusion equation in an evolving domain, simulating the transport of a substance in immiscible fluids. By leveraging Reynold's transport theorem, we introduce an unfitted finite element discretization that not only achieves optimal higher-order convergence rates but also maintains discrete surfactant mass conservation. This approach preserves the classical properties of fitted finite element methods, which are more computationally expensive, making our method an attractive alternative.