A Space-Time Cut Finite Element Method with Quadrature in Time

Abstract

We consider convection-diffusion problems in time-dependent domains and present a space-time finite element method based on quadrature in time which is simple to implement and avoids remeshing procedures as the domain is moving. The evolving domain is embedded in a domain with fixed mesh and a cut finite element method with continuous elements in space and discontinuous elements in time is proposed. The method allows the evolving geometry to cut through the fixed background mesh arbitrarily and thus avoids remeshing procedures. However, the arbitrary cuts may lead to ill-conditioned algebraic systems. A stabilization term is added to the weak form which guarantees well-conditioned linear systems independently of the position of the geometry relative to the fixed mesh and in addition makes it possible to use quadrature rules in time to approximate the space-time integrals. We review here the space-time cut finite element method presented in Hansbo et al. (Comput. Methods Appl. Mech. Eng. 307: 96–116, 2016) where linear elements are used in both space and time and extend the method to higher order elements for problems on evolving surfaces (or interfaces). We present a new stabilization term which also when higher order elements are used controls the condition number of the linear systems from cut finite element methods on evolving surfaces. The new stabilization combines the consistent ghost penalty stabilization (Burman, C. R. Acad. Sci. Paris, Ser. I 348(21–22):1217–1220, 2010) with a term controlling normal derivatives at the interface.