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In this paper, we study arbitrary order extended finite element (XFE) methodsbased on two discontinuous Galerkin (DG) schemes in order to solve ellipticinterface problems in two and three dimensions. Optimal error estimates in thepiecewise $H^1$-norm and in the $L^2$-norm are rigorously proved for bothschemes. In particular, we have devised a new parameter-friendly DG-XFEMmethod, which means that no "sufficiently large" parameters are needed toensure the optimal convergence of the scheme. To prove the stability ofbilinear forms, we derive non-standard trace and inverse inequalities forhigh-order polynomials on curved sub-elements divided by the interface. All theestimates are independent of the location of the interface relative to themeshes. Numerical examples are given to support the theoretical results.

High-Order Extended Finite Element Methods for Solving Interface Problems