Extended Finite Element Methods For Interface Problems

Interface problems occur widely in the real life and production,such as heat con-duction problems,oil production problems,electromagnetic wave propagations,heart blood flow problems,groundwater contamination problems and so on.These practi-cal interface applications usually can be abstracted as some coupled partial differential equation models.These equations not only have some discontinuous physical param-eters,but also should satisfy certain interface conditions at the interface.For more than four decades,researches about numerical methods for solving interface prob-lems obtain more and more attentions,and a vast of literature is available.In this dissertation,we consider Nitsche extended finite element methods to solve interface problems under interface-unfitted meshes,design accurate and efficient numerical dis-crete schemes,and obtain the stability analyses and optimal error estimates.In Chapter 1,we introduce the background of interface problems briefly,give an overview of existing finite element methods for solving interface problems and present our new Nitsche extended finite element method.At the end of this chapter,some necessary symbols,concepts and theorems in Sobolev space and discrete spaces are given.In Chapter 2,we propose nonconforming-P₁Nitsche extended finite elemen-t method for elliptic interface problems,and obtain optimal error estimates under L²and weighted H¹-norms.To deal with the problem that extended finite element meth-ods violate the continuity of solutions at the interface,an interface stabilization term is added to the discrete bilinear form,which also ensures the coercivity of the discrete scheme.The stabilization terms defined on parts of interface element edges make sure that optimal convergence results are independent of the interface position with respect to the mesh.Special weighted functions in the bilinear form make the results uniform independently of the diffusion parameters.Numerical experiments are carried out to validate theoretical results.In Chapter 3,nonconforming-P₁/P₀mixed Nitsche extended finite elemen-t method is applied successfully to solve Stokes interface problems.According to the new defined discrete bilinear formulation,the inf-sup stability result and optimal a priori error estimates are obtained,inspect of the low regularity of interface problems.It is shown that all results are independent of the discontinuous viscosity parameters.We also derive that this method produces a well-conditioned stiffness matrix indepen-dently of the interface location with respect to the mesh.The ill-condition problem caused by the extended finite element method is solved by the stabilization terms de-fined on parts of interface element edges.Numerical experiments are carried out to validate theoretical results.In Chapter 4,for H?curl?-elliptic interface problems in three dimensions,we use H?curl?-conforming Nitsche extended finite element method to discrete.The construction of the extended finite element space is based on the second family of the lowest order N?ed?elec edge element space.Theoretical proofs show that the new nu-merical scheme is stable and optimal a priori error estimates are obtained.All results are independent of the physical parameters and the interface location with respect to the mesh.Numerical experiments are carried out to validate theoretical results.In Chapter 5,the major conclusion of this dissertation and problems will be con-sidered in the future are listed.