An Immersed Finite Element Method For Orthotropic Interface Problem

The second order elliptic interface problems are often used to model problems in fluid dynamics and material sciences when two and more fluids or materials with different densities or diffusions or conductivies are involved. These phenomenons characterized by the discontinuity of coefficient which leads to the jump of solution on the interface are generically callde interface problems. If the interface is smooth enough, then the solution of the interface is smooth, but due to the jump of the coefficient across the interface, the global regularity is usually low and has order of H1+α(Ω),0≤α≤1. Because of the low global regularity and the irregular geometry of the interface, achieving accuracy is difficult with standard finite element methods, unless the elements fit with the interface of general shape.The immersed finite element method [4,9,14,17] can solve isotropic interface prob-lem. Based on a triangular mesh,[17] get optimal-order error estimates for IFE solution in the broken H1norm and L2norm. While for rectangular mesh, although numerical experiments demonstrate the optimal-order error estimates for IFE solution, only inter-polation error and suboptimal-order energy-norm estimates are proved in [14,24].In this thesis, we propose an immersed finite element method based on rectangular meshs for the following orthotropic elliptic inerface problemThe main contribution of the thesis is to extend the provious results in the following aspects:1, extend the diffusive coefficients from a positive scalar function (representing the isotropy aquifer structure) to a diagonal positive definite matrix (representing the anisotropy aquifer structure).2, prove that the immersed finite element space is uniquely determined by the values of vertexes on the elements and the interface jump condition, then construct the IFE space of the problem when β is a diagonal positive definite matrix.3, define the IFE formulation of the problem and proved that the IFE solution is uniquely sobvable.4, prove that the IFE solution of the problem has the optimal order Hl and L2convergence rate. by using numerical analysis techniques in the nonconforming finite element estimates such as the second Stang lemma, Bilinear lemma, Scaling argument, Trace theorem and the equivalent norm on fractional Sobolev space.