| The second order elliptic interface problems are often used to model problems in material sciences and fluid dynamics when two or more distinct materials or fluids with different conductivities or densities or diffusions are involved. These phenomenons char-acterized by the discontinuity of coefficient which leads to the jump of solution on the interface are generically called interface problems.In this thesis, for the purpose of achieving better approximation properties, we propose immersed finite element methods for the following elliptic interface problemNoting the definition of Crouzeix - Raviart nonconforming finite element is simple as well as its " locking free" nature in dealing with the plane elasticity problem, we propose an immersed finite element method based on Crouzeix - Raviart nonconforming finite element. Namely, on interface triangular elements, we apply linear polynomials depended on the interface, while applying Crouzeix - Raviart elements on non-interface triangular elements, and then we construct a nonconforming interface immersed finite element space. Furthermore, the finite element formulation is proposed, and we prove the existence and uniqueness of the formulation at the same time. Ultimately, by using the scaling argument, trace theorem and equivalent norm of the fractional Sobolev spaces, we prove that the method has the optimal approximation accuracy in H1-norm and L2-norm error estimates, and the order of the convergence is O(h) and O(h2) respectively. We also present a more direct and simple proof for optimal-convergence of the im-mersed interface finite element method proposed in [26], [29], [35], [36]. And by using the conventional finite element analysis methods, such as well-known bilinear lemma, Bramble - Hilbert lemma, the scaling argument and the trace theorem, we derive the optimal approximation accuracy in H1-norm and L2-norm error estimates, and the order of the convergence is O(h) and O(h2), too.
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