The Analysis Of The Unfitted Finite Element Methods For Some Classes Of Interface Problems

Interface problems are often encountered in material sciences, solid mechanics and fluid dynamics, such as, the heat conduction problems with different conduction coeffi-cients, elasticity problems describing various material behaviors and the two-phase flows involving with different viscosities. At present, the research of numerical methods for in-terface problems has become one of the popular research direction in the fields of scientific computing and engineering.In this thesis, we propose some unfitted finite element methods to discretize interface problems that allows for discontinuities along the interface not aligned with the mesh.Firstly, we propose a discontinuous Galerkin method, based on Nitsche’s method and LDG method, for elliptic interface problems. The key idea of this method is using weight-ed average across the interface instead of the arithmetic average in the bilinear form of the discrete problem as in the conventional discontinuous Galerkin method. We obtain opti-mal order a priori error estimates independent of the location of the interface. Numerical examples confirm our theoretical results.Secondly, to overcome the problem of ill-conditioned stiffness matrix, we present a new discontinuous Galerkin method for elliptic interface problems. Due to the mesh in-dependent of the position of the interface, very small elements typically occur near the in-terface, leading to ill-conditioned stiffness matrix. The key idea of our method is avoiding using very small elements by associating them to a sufficiently large neighboring element. This construction allows us to prove the crucial inverse inequality. Therefore, we show optimal order error estimates. And we also prove a stiffness matrix condition number of O(h-2) independent of the location of the interface.Next, we extend this method to solve elasticity interface problems and Stokes inter-face problems. For elasticity interface problems, we propose an unfitted nonsymmetric discontinuous Galerkin method and show a new extension theorem. Using the proper-ty of the classical BDM interpolant, we prove optimal error estimate independent of the location of the interface and Lame parameter λ (i.e. locking-free). For Stokes interface problems, we present a discontinuous Galerkin method based on penalizing not only the jump condition on the velocity but the jump condition on the stress tensor. We prove the inf-sup stability condition using some special techniques and optimal order a priori error estimates are obtained. Furthermore, for the two cases, the condition numbers of stiffness matrices are both independent of the location of the interface.Finally, we provide a stabilized Nitsche’s finite element method for Stokes interface problems by using the lowest equal order velocity-pressure pairs of finite element. And we combine local projection method with a ghost penalty method to prove that the method is inf-sup stable and obtain optimal order a priori error estimates. We also show that a well conditioned stiffness matrix is ensured without restrictions on the location of the interface. Numerical examples verify our theoretical results.