High Order Unfitted Finite Element Methods for Interface Problems and PDEs on Surfaces

In this contribution we treat a special class of recently developed higher order unfitted finite element methods for the discretization of mass and surfactant transport equations. To achieve higher order accuracy for such PDEs on stationary geometries we combine standard techniques for numerical integration and a discretization with a special mesh transformation. This results in a new class of isoparametric unfitted finite element methods. For the treatment of such PDEs on evolving geometries we apply space-time variational formulations. These unfitted finite element techniques result in robust and accurate discretization methods for mass and surfactant transport problems in realistic two-phase flow simulations based on a sharp interface formulation. We present these finite element discretization methods, give theoretical error bounds for classes of model problems and present results of numerical simulations both for such model problems and for challenging two-phase flow applications.

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