Enriched Finite Element Methods For Interface Problems

Interface problems occur widely in practical applications,such as fluid mechan-ics,electromagnetic wave propagations,materials sciences,and biological sciences.It typically involves solving coupled systems of partial differential equations.This thesis is devoted to study finite element methods for interface problems.There are two major classes of finite element methods(FEMs)for interface prob-lems,namely,interface-fitted FEMs and interface-unfitted FEMs,categorized accord-ing to the topological relation between discrete elements and the interface.Advan-tages of interface-fitted FEMs are the ease of analysis and ability to obtain optimal convergence rates.However,letting the mesh fit the interface requires remeshing as the interface evolves with time,and leads to significant complications when topolog-ical changes occur such as breakup or coalescence.Then interface-unfitted methods have become highly attractive.There are mainly two types of interface-unfitted methods,extended finite element methods(XFEMs)and immersed finite element methods(IFEMs).The two methods modify finite element spaces in order to get optimal interpolation error estimate.Both of them have some drawbacks.Many different XFEMs exist,but only Nitsche-XFEM has a theoretical analysis.For Nitshce-XFEM,it violates the continuity of the solu-tion,thus additional penalty terms are always needed in the discrete weak form.For IFEMs,the construction of basis functions highly depend on the jump conditions of interface problems and the error analysis are not easy.To avoid these drawbacks,we propose a new interface-unfitted method for interface problems,i.e.,the conforming enriched finite element method.We start with the simplest problem,the second order elliptic interface problem.The construction of the conforming enriched finite element space is presented.And this construction does not depend on the jump conditions as IFEMs do.Optimal convergence results are also consequently derived in spite of the low regularity of interface problems.Moreover,additional penalty terms are not needed since functions in enriched finite element spaces are continuous across interfaces.Another advantage of our method is that we can easily deal with variable coefficients.Then,we apply the conforming enriched finite element method to Stokes inter-face problems.The conforming enriched finite element pair is constructed on the basis of the MINI element pair.A ghost penalty term is used in the standard discretization form as a stabilization term.An inf-sup stability result is derived,which is uniform with respect to the mesh size.Finite element errors are proved to be optimal.Finally,we extend the enriched finite element method to Stokes-elliptic interface problems.The Stokes-elliptic interface problem can be viewed as a static linearized fluid-structure interaction problem.Our method breaks the limit of the immersed finite element method which can only deal with the case of identical governing equations on either side of the interface.The well-posedness and the optimal convergence are also consequently derived.