An Adaptive Immersed Finite Element Method with Arbitrary Lagrangian-eulerian Scheme for Parabolic Equations in Time Variable Domains

We first propose an adaptive immersed finite element method based on the a posteriori error estimate for solving elliptic equations with non-homogeneous boundary conditions in general Lipschitz domains. The underlying finite element mesh need not fit the boundary of the domain. Optimal a priori error estimate of the proposed immersed finite element method is proved. The immersed finite element method is then used to solve parabolic problems in time variable domains together with an arbitrary Lagrangian-Eulerian (ALE) time discretization scheme. An a posteriori error estimate for the fully discrete immersed finite element method is derived which can be used to adaptively update the time step sizes and finite element meshes at each time step. Numerical experiments are reported to support the theoretical results.