Fitted And Unfitted Finite Element Methods For Interface Problems

A large number of real world phenomena exhibit strong or weak discontinuities.The application we have in mind is multi-physics problems,including fluid-structure interaction problems and multiphase flow problems,which involve two or more distinct materials or fluids with different densities,conductivities or permeabilities.These par-tial differential equations are usually called interface problems.In this dissertation,the fitted interface penalty finite element method for elliptic interface problems,unfitted finite element methods for elliptic and Stokes interface problems are proposed and error estimates are analyzed.We first consider the fitted finite element method for the second order elliptic interface problems with the discontinuous coefficients.An interface penalty finite ele-ment method(IPFEM)is proposed for elliptic interface problems,which allows to use different meshes in different sub-domains separated by the interface.The transmis-sion conditions across the interface are treated by the Nitsche's method(or penalty technique)with some harmonic weighted averages.Both symmetric IPFEM and non-symmetric IPFEM are analyzed.Optimal order error estimates in energy norm and L2 norm and the flux error estimate in L2 norm are proved for the symmetric IPFEM.In particular,the relative error estimate in energy norm and the flux error estimate in L2 norm are independent of the ratio of mesh sizes and the contrast of the discontinuous coefficients across the interface.Error estimates for the non-symmetric IPFEM are also obtained.Furthermore,optimal mesh size with respect to the error estimate in energy norm and the flux error estimate in L2 norm are discussed,respectively.Numerical examples are also provided to confirm the theoretical results and show that the IPFEM on a mesh with optimal mesh sizes may give much better approximations than the one on the quasi-uniform mesh with the same number of nodal points.Then we consider the unfitted finite element method for the second order elliptic interface problems with the discontinuous coefficients.We introduce a nonconforming Nitsche's extended finite element method(NXFEM)for elliptic interface problems on unfitted triangulation elements.The solution on each side of the interface is separately expanded in the standard nonconforming piecewise linear polynomials with the edge averages as degrees of freedom.The jump conditions on the interface and the dis-continuities on the cut edges(the segment of edges cut by the interface)are weakly enforced by the Nitsche's approach.In this method,the harmonic weighted fluxes are used and the extra stabilization terms on the interface edges and cut edges are added to guarantee the stability and the well conditioning.We prove that the convergence rate of the errors are optimal.Moreover,the errors are independent of the position of the interface relative to the mesh and the ratio of the discontinuous coefficients.Fur-thermore,the condition number of the system matrix is independent of the interface position.Numerical examples are given to confirm the theoretical results.Finally,we extend the nonconforming Nitsche's extended finite element scheme to the Stokes interface problems.A stabilized nonconforming extended finite element method is proposed for Stokes interface problems on unfitted triangulation elements which do not require the interface to be aligned with the meshes.The velocity so-lution and pressure solution on each side of the interface are separately expanded in the standard nonconforming piecewise linear polynomials and the piecewise constants,respectively.Harmonic weighted averages are used to ensure that errors are indepen-dent of the contrast of the discontinuous coefficients.Moreover,extra stabilization terms with respect to both velocity and pressure are added to ensure the stable inf-sup condition.It is proved that the convergence order of energy error and L2 error for velocity and L2 error for pressure are optimal and independent of the jump of viscosi-ties.Results of numerical experiments are presented which illustrate the theoretical analysis.