Abstract
We develop a family of cut finite element methods of different orders based on the discontinuous Galerkin framework, for hyperbolic conservation laws with stationary interfaces in both one and two space dimensions, and for moving interfaces in one space dimension. Interface conditions are imposed weakly and so that both conservation and stability are ensured. A CutFEM with discontinuous elements in space is developed and coupled to standard explicit time stepping schemes for linear advection problems and the acoustic wave problem with stationary interfaces. In the case of moving interfaces, we propose a space-time CutFEM based on discontinuous elements both in space and time for linear advection problems. We show that the proposed CutFEM are conservative and energy stable. For the stationary interface case an a priori error estimate is proven. Numerical computations in both one and two space dimensions support the analysis, and in addition demonstrate that the proposed methods have the expected accuracy.
Highlights
A finite element method (FEM) that uses discontinuous piecewise polynomial spaces as trial and test spaces is commonly called a discontinuous Galerkin (DG) method
The implementation of the space-time unfitted finite element method we propose is straightforward and simple starting from an implementation of Cut Finite Element Methods (CutFEM) for a stationary interface
We have presented two high order CutFEM based on the DG framework, applicable to conservation laws with discontinuous coefficients in the flux across stationary and moving interfaces, respectively
Summary
A finite element method (FEM) that uses discontinuous piecewise polynomial spaces as trial and test spaces is commonly called a discontinuous Galerkin (DG) method. A common technique in connection with Cut Finite Element Methods (CutFEM) is to add ghost penalty stabilization terms in the weak form [3,4]. CutFEM based on discontinuous piecewise polynomial spaces and ghost penalty stabilization has been developed, e.g. see [12] where a time independent linear advection-reaction problem is considered and see [10] for time dependent nonlinear conservation laws. 3, we consider a stationary interface, propose a discontinuous cut finite element discretization in space and perform a stability analysis, and an a priori error estimate is given for the scalar problems. The stability of the semi-discrete scheme is analyzed and we present some examples to show that the method can simulate the moving interface problem with expected accuracy and with conservation.
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