We propose arbitrary order extended finite element (XFE) methods for solving both two and three dimensional elliptic interface problems. The meshes in our methods do not need to fit the interface and discontinuous Galerkin (DG) schemes are used in the vicinity of the interface to incorporate the transmission conditions between subdomains. Optimal error estimates in the piecewise H1-norm and in the L2-norm are rigorously proved for the scheme. Two implementation aspects are addressed in this paper: (1) An optimal multigrid solver is introduced for the generated linear system, which converges uniformly with respect to the mesh size. (2) A high order algorithm is presented to compute the integral over elements intersecting the interface. Numerical experiments are reported to support the theoretical results.

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