Parameter-Friendly Partially Penalized Immersed Finite Element Method For Parabolic Interface Problems

Parabolic interface problems have many applications in practical scientific computations and effective numerial methods are very important for these applications.Traditional finite element methods require the used meshes to be fitted to the interface.However,for complex or moving interfaces,it is difficult to generate such fitted meshes.In contrast,immersed finite element methods are based on unfitted meshes to solve interface problems,which are independent of the interface.However,the functions in the immersed finite element space are discontinuous at edges cut by the interface.So,a penalty needs to be added to the bilinear form to ensure the stability.Theoretically,the penalty parameter should be sufficiently large while in pratice the parameter needs to be tuned manually.In order to overcome this issue,a parameter-friendly partially penalized immersed finite element method is proposed.The stability is guaranteed by constructing a lifting operator and adding it to penalty terms.The advantage of the method is that the stability can be ensured without any manually chosen parameters.The research contents are as follows:(1)A parameter-friendly partially penalized immersed finite element method is proposed for parabolic interface problems.Firstly,the immersed finite element space based on unfitted mesh is used for the spatial discretization.In order to deal with the discontinuity of the immersed finite element basis functions on interface edges,consistent terms,symmetry terms and penalty terms based on the lifting operator are added to the bilinear forms.Secondly,in the time discretization,equal partitions are adopted and the backward Euler,Crank-Nicolson and second order backward difference scheme are proposed.Finally,some numerical experiments are given to verify the effectiveness of the proposed methods.(2)Error estimates of the semi-discrete and the fully discrete schemes(backward Euler scheme,Crank-Nicolson scheme and second order backward difference scheme)for parabolic interface problems are derived rigorously.The analysis is mainly based on the traditional techniques for time discretizations and the theoretical results of immersed finite elements for the second order elliptic interface problems.In the analysis,we focus on the error caused by approximating the discontinuity coefficient β(x)by β_h(x),that is,replacing the curved interface with straight lines.(3)An error estimate for the spatial semi-discrete scheme for one-dimensional moving interface problems is derived.Since the interface is time-dependent,the immersed finite element space is also time-dependent.So,the immersed finite element interpolation operator and the partial derivative of time do not commute,which is the main difficulty in the theoretical analysis.Here we use the explicit expression of the one-dimensional immersed finite element basis function and Taylor’s expansions to derive the error estimates between the time partial derivatives of the immersed finite element interpolation and the exact solution.And then,the estimate is applied to the moving interface problem to derive the error estimate of the spatial semi-discrete scheme for the moving interface problem.