Interface-unfitted Finite Element Methods For Interface Problems

Interface problems widely exist in actual life and applications,such as many re-search of composite materials,blood flow and cell deformation,biological science and fluid mechanics and so on.It usually involves solving coupled partial differen-tial equations.In this dissertation,we use the finite element method to study these interface problems.According to the topological relationship between the grid and the interface,the finite element methods?FEMs?of the interface problem can be divided into two cat-egories,the interface-fitted FEMs and interface-unfitted FEMs.The advantage of interface-fitted FEMs is that the traditional finite element methods can be directly applied and the optimal error estimates can be also obtained.However,when the in-terface becomes complex or evolves with time,the work of mesh generation will be huge.And when the interface structure changes,such as cracking or merging,it is very difficult to generate meshes that fit the interface.Therefore,interface-unfitted FEMs have become an increasingly important research direction.There are mainly two kinds of interface-unfitted methods,extended finite ele-ment methods?XFEMs?and immersed finite element methods?IFEMs?.The XFEMs have many branches,but only Nitshce-XFEM has a strict theoretical analysis.In this method,because the continuity of solution is broken,additional penalty terms always need to be added in the discrete weak form.For immersed finite element method,its basic functions construction depends on the interface jump condition,which makes the error estimates very difficult.So far,only the second order elliptic interface prob-lem has complete theoretical analysis.Both methods are based on the modification of finite element space,but they have their own shortcomings.In this paper,we first propose the P₁/CR IFEM to study the elasticity interface problem.We use points values as degrees of freedom on elements,give corresponding finite element space and the theoretical estimate of interpolation error.After that,for the method which uses integral average values on the edges of elements as degrees of freedom,we also present the interpolation capability analysis.Finally,the theo-retical analysis of the finite element errors of these two kinds of methods are given by combining the partial penalty method.Then,combining the extend finite element method,we develop a new interface-unfitted method,enriched finite element method.We apply it on the mixed element form of elasticity interface problem with a non-conforming enriched finite element method and solve the phenomenon of locking.At last,for stokes interface problem,a CR immersed finite element method is presented.We give the construction of the finite element space,the interpolation error estimate and finite element error estimate.